Seminar Business Statistics (Sampling and …...Department of Social Statistics, University of Sri Jayewardenepura 17/13/2017 1 Seminar – Business Statistics (Sampling and Inferential

Department of Social Statistics, University of Sri Jayewardenepura

17/13/2017

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Seminar – Business Statistics

(Sampling and Inferential Statistics)

1) Describe 4 advantages of sampling over of a census?

2) What is a “Sampling Frame”? What are the main characteristics of a good “Sampling Frame”?

3) Explain the following terms using suitable examples.

i. Parameter ii. Statistic iii. Estimator iv. Estimate

4) What is meant by accuracy, unbiased and precision of an estimator?

5) What is “Simple Random Sampling”? Describe how you would draw a sample with size n from a

population with size N.

6) In a population with N= 3 the values of Yi are 4, 5 and 6. Simple random samples of size 2 were

drawn. Verify that sample mean (�̅�) is an unbiased estimate to population mean (�̅�) by using,

i. With replacement ii. Without replacement sampling methods.

7) Explain how to estimate population parameters and its estimators in simple random sampling

using equations.

i. Population mean ii. Population total iii. Population ratio

8) In a population with N= 5 the values of Yi are 2,3,7,8 and 10. Calculate the sample mean (�̅�) for

all possible simple random samples of size 2, without replacement. Verify that �̅� is an unbiased

estimate of population mean, �̅�. Find the variance of �̅� and compare it with the answer

obtained by Var (�̅�).

9) Define stratified random sampling and explain the reasons for stratification.

10) What is Systematic sampling? Explain how you would draw a sample with size n =5 by using linear

systematic sampling and a sample with size n =6 using cyclic systematic sampling from a population

with size N =20.

11) There is a requirement of selecting a sample of following population structures,

i. Population in random order ii. Population with periodic variations.

iii. Population with a linear trend.

Explain how each of these population structures would impact on precision of sample mean under

“Systematic Sampling”.

12) Explain the relationship between “Systematic Sampling” to “Cluster Sampling” and “Stratified

Random Sampling”.

13) What is “Cluster sampling? Explain conditions which “Cluster Sampling” is more efficient.

14) What is “Quota Sampling”? Explain how “Quota Sampling” differs from “Stratified Random

Sampling”?

15) Describe the characteristics of a good point estimator for the population parameter

16) Let 321 ,, XXX be a random sample taken from the distribution with mean and variance 2 .

Prove 4

2ˆ 321 XXX is unbiased estimator for . Find the efficiency of ̂ with respect to the

efficiency of 3

321 XXXX

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17) State the central limit theorem. Let X be the mean of a random sample of size 49 taken from a

poison distribution with mean 8. Find approximately )95.7( XP

18) Let X be the mean of a random sample with size 30 taken from a 150,106N distribution and Y

be the mean of a random sample of size 50 taken from a 200,103N distribution.

i. Write down the sampling distributions of X and Y .

ii. If X and Y are independent, write down the sampling distributions of YX .

iii. Find the probability of X exceeding Y by at least 1.2

19) 40 items were selected in random from a stock of items and the proportion of the defective items in

that stock is 0.12. Find the probability of,

i. At least 7 items being defective

ii. Number of defective items lie between 3 and 7.

20) The yield of certain kind of crop is distributed as the yields for a random sample of 5 plots

of land were as follows. 244,252,246,248,240

i. Find a point estimate for and 2 ii. Find 95% confidence interval for

21) The life time of the bulb of brand X is distributed as 725,1N and the life time of brand Y is

distributed 6302N . A random sample size 50 selected of brand X yielded 987.6 and a random

sample of size 60 from Y yielded 938.4. If X and Y are independent find 90% confidence interval

for the mean difference of life times of two kind of bulb.

22) From a survey done in 2015 it was revealed that out of 1780 registered voters, 1157 like to give

death penalty for the murderers. Accordingly construct a confidence interval at 90% for the

proportion of voters who like assigning death penalty to the murderers.

23) Explain the difference between terms of each of the following pairs.

i. Simple hypothesis and Composite hypothesis

ii. Critical region and Acceptance region

24) What do you mean by “Type I error˜ and “Type II error˜ in a hypothesis testing?

A sample of size 36 was taken from 81,N distribution to test the 96;0 H against 99;1 H

The testing procedure states that 96;0 H get reject if 46.98X .

i. Calculate type one error

ii. Calculate type two error

iii. What is the power of the test?

25) Assume that the IQ scores of a certain population has a normal distribution with mean and

variance 1002 . the mean of a random sample of size 16 taken from this population was 115.5.

test the hypothesis at 5% significance level, 112;0 H against 112;1 H

26) A random sample of 36 unemployed persons in a village showed that the mean duration of

unemployment was 87 days with a standard deviation of 16 days. A census conducted ten years

previously showed that the mean duration then was 74 days. Can we conclude that the duration of

unemployment has increased at 5% level of significance.?

27) A company manufacturing chocolates bar is particularly concerned that mean weight of a chocolate

bar is not greater than 60.03grms. a sample of 20 chocolate bars is selected and found that the mean

and SD are 60.038 and 0.02 respectively. Using 01.0 level of significance is there evidence that

the population mean weight of chocolate bars is greater than 60.3grms?

2,N


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28) A wholesaler received a shipment of goods which is reported to be containing 5% defective items.

He will accept the shipment if the claim is found true and reject if the percentage of items is more.

To verify this claim he decides to draw a sample of 45 items and find that 3 are defectives. Using

0.05 level of significance test the claim whether the shipment contain 5% defective items or more.

29) A random sample of size 10 taken from 2

11 ,N distribution has 65.3,6.492

11 sx while a

random sample of size 16 taken from 2

22 ,N distribution has 15.3,23.512

22 sx ' Test the

hypothesis 210 : H against 211 : H at 5% significance level.

30) An examination was given to two classes consisting of 40 and 50 students.in the first class mean

grade was 74 with SD of 8. In the second class the mean grade was 78 with SD of 7. Is there a

significance difference between the performance of the two classes at the (a) 0.05 level and (b) 0.01

level?

31) To investigate the claim that women are better at a certain job than men, the personnel department

of a textile firm had an appropriate aptitude test by taking a random sample of 60 male employees

and a random sample of 50 female employees. Aptitude test result showed that that the males had

an average score of 82.4 with a standard deviation of 11.8 while the females had an average score

of 89.6 with a standard deviation of 13.2. Test the claim at 0.01 level of significance and make your

conclusion

32) A firm surveyed its workers to find out whether they prefer a large increase in retirement benefits or

a small increase in salary. Out of sample of 700 male workers 462 were in favour of increased

retirement benefits. Out of a sample of 600 female workers 357 were in favour of increased

retirement benefits. Test at 0.05 level of significance whether there is a significant difference

between the proportion of males and females favouring the increase in retirement benefits.

33) The Accounts department of a firm wants to know whether the firm’s accounting process can be

simulated using the Poisson distribution to discrete the incidence of error. The manager of the

department has taken a random sample of 172 accounts from the firm’s accounting records and has

summarized the number of errors found in each account. The results are shown below.

Number of errors 0 1 2 3 4 5 6

Numbers of accounts 26 48 42 28 16 8 4

Test whether the incidence of error in the firm’s Accounting record follow a Poisson distribution at

5% significance level.

34) A company is intertesred in finding out whether there is association between the traveling time to

work of its employees and the level of stress-related problems observed on the job. The following

table gives the results obtained in a study of 120 workers. Test at 5% level of significance whether

the stress level is independent of traveling time.

Traveling to work (mts) Stress level

High Medium Low

below 15 8 6 16

15-45 16 10 18

over 45 24 14 8


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35) To test whether the prices of the certain item in 4 cities A, B, C, D a sample of 5 shops were selected

from each city and the prices of the item were recorded. The data is given in the following table.

i. Write down the model with assumptions to analyze this data.

ii. Construct ANOVA table and test whether the average prices are different in the cities at 5%

significance level.

City A City B City C City D

148 156 155 155

149 152 153 153

147 150 154 152

145 153 148 154

146 154 150 156

Department of Social Statistics

University of Sri Jayewardenepura

Supportive Seminar on Business Statistics for Advanced Level 2018

Sampling and Statistical Inference

Question 𝟎𝟏 Which of the following statement is true? (1) Simple random sampling is a method of selecting a sample giving each unit in the population a

known probability of being included in the sample. (2) Without a complete sampling frame cluster sampling cannot be used. (3) The standard error cannot be calculated in quota sampling because the selection of units is not based

on a sampling frame. (4) Circular systematic sampling is used when we want to select more than one systematic sample from

a population. (5) Cluster sampling is more effective if variation within the cluster is large.

Question 𝟎𝟐 Which of the following statement is true? (1) The probability that a specific unit in a population of size N is included in a random sample of size n

without replacement is 1

𝑁

(2) In systematic sampling the term 𝑛

𝑁 is called sampling interval.

(3) The finite population correction factor may be ignored when it is close to zero. (4) Non-response errors may be reduced by increasing the sample size. (5) Standard errors of estimators cannot be calculated in a non-probability sampling.

Question 𝟎𝟑 Which of the following statement is true? (1) Cluster sampling is more effective if the variation between clusters is large. (2) Quota sampling is an example of semi-probability sampling. (3) A systematic sample can be regarded as a simple random sample of one cluster unit from a

population of k cluster units. (4) Stratified random sampling is more efficient if the variation within a strata is large. (5) Simple random sampling with replacement is more efficient than simple random without

replacement.

Question 𝟎𝟒 Which of the following statement is true? (1) Systematic sampling can be regarded as a cluster sampling of taking one cluster form k clusters of

size n. (2) Cluster sampling cannot be used when there is no proper sampling frame. (3) When there are cyclical trends in a population, systematic sampling is always very efficient. (4) In simple random sampling, a sample is selected giving each unit of the population a known

probability to be selected. (5) If the variation within clusters is small, the cluster sampling is more effective.

Question 𝟎𝟓 Which of the following statements is true? (1) If the variation between strata is large, the precision of stratified random sampling is also large (2) If intra-class correlation coefficient is close to one, cluster sampling is more efficient than simple

random sampling.

Department of Social Statistics – University of Sri Jayewardenepura

Supportive Seminar on Business Statistics for Advanced Level 2018 2

(3) A quota sampling is usually selected using a sampling frame. (4) The main purpose of circular systematic sampling is to select a systematic sample when the sampling

interval is an integer number. (5) The field cost of cluster sampling is usually larger than that of simple random sampling.

Question 𝟎𝟔 The following three statements are about non-sampling errors. 𝐴 - Failure to measure some of the units in the selected sample in an example for non-sampling

error. 𝐵 - The non-sampling error can be reduced by increasing the sample size. 𝐶 - Non-sampling errors cannot occur in complete enumeration of the population. Which of three statements is/are true? (1) 𝐴 only (2) 𝐵 only (3) 𝐶 only (4) 𝐴 and 𝐵 only (5) 𝐵 and 𝐶 only

Question 𝟎𝟕 Which of the following statement is not true? (1) Central limit theorem implies that the sampling distribution of the mean is approximately normal if

the sample size is large. (2) An estimator is said to be unbiased if its expected value is equal the parameter being estimated. (3) An estimator is said to be sufficient if it contains all the information in the data about parameter it

estimates. (4) An estimator is said to be a consistent estimator of a population parameter if it has the smallest

variance among all the possible estimators of the parameter. (5) The standard error of mean of a sample taken from a given population decreases as the sample size

increases.

Question 𝟎𝟖 Which of the following statements are/is true? 𝐴 - Since the population size is always larger than the sample size, the sample mean can never be

larger than the population mean. 𝐵 - Sample mean can never be equal to the population mean. 𝐶 - Population standard deviation is always larger than the standard error of the sample mean if

sample size is greater than one. (1) 𝐴 only (2) 𝐵 only (3) 𝐶 only (4) 𝐴 and 𝐶 only. (5) 𝐵 and 𝐶 only

Question 𝟎𝟗 A sample statistic is said to be an unbiased estimator of population parameter if (1) it has the smallest variance of all possible sample statistics. (2) it equals the population parameter. (3) the mean of the all possible values of the sample statistic equals the population parameter. (4) its expected value is close to the population parameter being estimated. (5) it contains all the information in the data about the population parameter.

Question 𝟏𝟎 Which of the following statements is true? (1) Since the sample mean �̅� is an unbiased estimator for population mean 𝜇 , �̅�2 is an unbiased

estimator for 𝜇2. (2) If 𝐸(𝜃) = 𝜃 and 𝑉𝑎𝑟(𝜃) → 0 as the sample size 𝑛 → ∞ then 𝜃 is a consistent estimator for 𝜃.

(3) The value calculated using a sample to estimate a population parameter is called an estimator. (4) The square root of the variance of the sampling distribution of an estimator is called standard

deviation of the estimator. (5) Any function of a random sample is called a statistic.



Question 𝟏𝟏 Which of the following statements is true? 𝐴 - The distribution of sample mean of anon-normal population will tend to be increasingly normal

the sample size increases. 𝐵 - The standard deviation of the distribution of sample mean increases as the sample size

increases. 𝐶 - The mean of the distribution of sample mean is equal to the mean of the population. (1) 𝐴 only. (2) 𝐵 only. (3) 𝐴 and 𝐶 only. (4) 𝐵 and 𝐶 only. (5) All 𝐴, 𝐵 and 𝐶

Question 𝟏𝟐 Which of the following statement is/are true? 𝐴 - Central limit theorem implies that the sampling distribution of the ample mean is approximately

normal if the sample size is large. 𝐵 - An estimator is said to be unbiased, if its expected value is close to the parameter being

estimated. 𝐶 - An estimator is said to be sufficient, if it contains all the information in the sample about the

parameter it estimates. (1) 𝐴 only. (2) 𝐶 only. (3) 𝐴 and 𝐵 only.

(4) 𝐴 and 𝐶 only. (5) 𝐵 and 𝐶 only.

Question 𝟏𝟑 Which of the following statement are/is true about the 𝑡 − distribution? 𝐴 - The 𝑡 − distribution is symmetric about zero. 𝐵 - The 𝑡 − distribution has a larger variance than the standard normal distribution. 𝐶 - The 𝑡 − distribution with 𝑘 degrees of freedom has a smaller variance than the 𝑡 − distribution

with 𝑘 + 1 degrees of freedom. (1) 𝐴 only. (2) 𝐵 only. (3) 𝐴 and 𝐵 only. (4) 𝐴 and 𝐶 only. (5) All 𝐴, 𝐵 and 𝐶

Question 𝟏𝟒 Which of the following statements is/are true about the sampling distribution? 𝐴 - The 𝑡 − distribution approaches the standard normal distribution as the number of degrees of

freedom increases. 𝐵 - he shape of the 𝐹 − distribution depends on the numerator degrees of freedom and the

denominator degrees of freedom. 𝐶 - The Central Limit Theorem says that the sampling distribution of the sample mean is

approximately normal for any sample size (1) 𝐴 only. (2) 𝐴 and 𝐵 only. (3) 𝐴 and 𝐶 only. (4) 𝐵 and 𝐶 only. (5) All 𝐴, 𝐵 and 𝐶

Question 𝟏𝟓 An estimate is required for the population mean of the widths of metal bars with a margin of error of 0.1 at 95% confidence level. It is known that the variance of the width of bars is 0.25cm. What is the sample size necessary to achieve these requirements?

(1) 10 (2) 22 (3) 25 (4) 97 (5) 102

Question 𝟏𝟔 Suppose that 20% of all people in a certain population are left handed. The sampling distribution of the sample proportion of left-handers of a random sample of 100 people selected from this population is (1) approximately normal with mean 20 and standard deviation 4. (2) binomial with 𝑛 = 100 and 𝑝 = 0.20 (3) approximately normal with mean 0.20 and standard deviation 0.0016 (4) approximately normal with mean 0.20 and standard deviation 0.16 (5) approximately normal with mean 0.20 and standard deviation 0.04



Question 𝟏𝟕 A random sample of size 81 is drawn from a normal population size 501 mean 100 and standard deviation 36. The sampling distribution of the sample mean is (1) approximately normal with mean 100 and variance 16 (2) normal with mean 100 and variance 16 fõ' (3) approximately normal with mean 100 and variance 13.44 (4) normal with mean 100 and variance 13.44 (5) approximately normal with mean 100 and variance 3.36

Question 𝟏𝟖 A manufacturing company measures the weight of boxes produced before transporting them to the customers. If the weight of boxes have the population mean of 20 kg and a population standard deviation ofn2.8 kg, find the probability that the average weight of a sample of 49 boxes will be less than 19.2 kg? (1) 0.0793 (2) 0.1586 (3) 0.4207 (4) 0.5793 (5) 0.9207

Question 𝟏𝟗 If �̅� and �̅� are sample means of sample size 25 each from 𝑁(2,16) and 𝑁(1,9) respectively, 𝑃(�̅� > �̅�) is (1) 0.1587 (2) 0.3174 (3) 0.3413 (4) 0.6826 (5) 0.8413

Question 𝟐𝟎 Under which of the following circumstances is it impossible to construct a confidence interval for the population mean using a sampling distribution? (1) A non-normal population with a large sample and an unknown population variance. (2) A normal population with a large sample and a known population variance. (3) A non-normal population with a small sample and an unknown population variance. (4) A normal population with a small sample and an unknown population variance. (5) A non-normal population with a large sample and a known population variance.

Question 𝟐𝟏 If �̅� is the mean of a random sample of size n from 𝑁(𝜇, 100), find the value of n such that 𝑃(−5 < �̅� − 𝜇 < 5) = 0.9544 (1) 4 fõ' (2) 8 fõ' (3) 15 fõ' (4) 16 fõ' (5) 18 fõ'

Question 𝟐𝟐

If 𝑝 is the sample proportion and 𝜋 is the population proportion, assuming 𝜋 =1

2. Find the value of sample

size n such that 𝑃(−0.1 < 𝑝 − 𝜋 < 0.1) = 0.9544. (1) 10 (2) 25 (3) 50 (4) 100 (5) 200

Question 𝟐𝟑 A quality control inspector needs to test whether a machine that packages potato chips is working properly. The inspector selects a random sample of packages and weight the content of potato chips in each. If an estimate for the mean content must be given with 98% confidence and a margin of error no more than 20 grams, what would be the minimum sample size of packages the inspector must select? Assume that the weights of potato chips in packages have a normal distribution with a standard deviation of 50 grams. (1) 6 (2) 17 (3) 25 (4) 27 (5) 34

Question 𝟐𝟒 Which of the following statements about a confidence interval for the population mean is/are true? 𝐴 - All other things remain constant a 99% confidence interval will be wider than a 95% confidence

interval. 𝐵 - All other things remain constant, a confidence interval based on a sample of size 100 will be

narrower than a confidence interval based on a sample size 50. 𝐶 - There is 5% chance that a 95% confidence interval will not include the population mean. (1) 𝐴 only. (2) 𝐵 only. (3) 𝐴 and 𝐵 only. (4) 𝐴 and 𝐶 only. (5) All 𝐴, 𝐵 and 𝐶



Question 𝟐𝟓 Which of the following statements is/are true about confidence intervals? 𝐴 - The width of a confidence interval increases as confidence level decreases. 𝐵 - Confidence intervals can be used to test some hypothesis. 𝐶 - For small samples, the width of the confidence intervals based on 𝑡 − distribution is larger than

the width of the confidence intervals based on 𝑍 − distribution. (1) 𝐴 only. (2) 𝐵 only. (3) 𝐴 and 𝐵 only. (4) 𝐵 and 𝐶 only. (5) All 𝐴, 𝐵 and 𝐶

Question 𝟐𝟔 Which of the following statement/s is/are true about the confidence intervals? 𝐴 - The end values of an interval estimator are random variables 𝐵 - The width of the confidence interval for the mean of a normal population is larger when 𝜎2 is

known than when the variance 𝜎2 is unknown. 𝐶 - In a (1 − 𝛼)100% confidence interval for the mean of a normal population with known

variance, the term 𝑍𝛼 2⁄𝜎

√𝑛 is called the probable error of estimator.

(1) 𝐴 only. (2) 𝐴 and 𝐵 only. (3) 𝐴 and 𝐶 only. (4) 𝐵 and 𝐶 only. (5) All 𝐴, 𝐵 and 𝐶

Question 𝟐𝟕 The analysis of a random sample of 300 households in a small town indicates that a 98% confidence interval for the mean family income is (Rs. 42, 520, Rs. 49, 860). Could this information be used to conduct a test of the null hypothesis 𝐻0: 𝜇 = 40, 000 against the alternative hypothesis 𝐻1: 𝜇 ≠ 40, 000 at a 0.02 level of significance? (1) No, because it is not known whether the data are normally distributed. (2) No, because the sample standard deviation is not known. (3) Yes, since the sample mean Rs. 46, 190 greater than Rs. 40, 000 𝐻0 would be rejected. (4) Yes, since Rs. 40, 000 is not contained in 98% confidence interval, 𝐻0 would be rejected. (5) Yes, since Rs. 40, 000 is not contained in 98% confidence interval, 𝐻0 would not be rejected.

Question 𝟐𝟖 A sample of size 25 is taken from a normal population. The sample mean and the sample variance are calculated as 15 and 16 respectively. What is the upper limit of the 95% confidence interval for the population mean 𝜇?

(1) 16.31 (2) 16.37 (3) 16.57 (4) 16.65 (5) 21.27

Question𝟐𝟗 The lifetime of a certain type of light bulb is known to have a standard deviation of 40 hours. How large a sample should be taken if it is desired to have a margin of error of 10 hours or less at 95% level of confidence? (1) 8 (2) 32 (3) 44 (4) 62 (5) 66

Question 𝟑𝟎 According to past records, the mean lifetime for a certain type of battery has been 196 hours. The manufacturer has introduced a change in the production method and wants to perform a hypothesis test to determine whether the mean lifetime has increased as a result. The null hypothesis (𝐻0) and a alternative hypothesis (𝐻1) for this test are, (1) 𝐻0: 𝜇 ≥ 196 hours; 𝐻1: 𝜇 < 196 hours; (2) 𝐻0: 𝜇 > 196 hours; 𝐻1: 𝜇 ≤ 196 hours; (3) 𝐻0: 𝜇 = 196 hours; 𝐻1: 𝜇 ≠ 196 hours; (4) 𝐻0: 𝜇 < 196 hours; 𝐻1: 𝜇 ≥ 196 hours;

(5) 𝐻0: 𝜇 = 196 hours; 𝐻1: 𝜇 > 196 hours;

Question 𝟑𝟏 The mean of a random sample of size 16 from 𝑁(𝜇, 100) distribution is observed as �̅� = 114.5. For testing 𝐻0: 𝜇 = 112 against 𝐻1: 𝜇 ≠ 112 the 𝑃 − value is" (1) 0.1587 (2) 0.1706 (3) 0.3413 (4) 0.6286 (5) 0.6826



Question 𝟑𝟐 A car manufacturer claims that at least 90% of their cars do not experience engine failures before reaching a mileage of 150 000 kilometers. A sample of 100 cars is investigated and 84 of the cars did not have an engine failure before reaching 150 000 kilometers. What is the 𝑃 − value of the test? (1) 0.1010 (2) 0.0228 (3) 0.0456 (4) 0.0505 (5) 0.2644

Question 𝟑𝟑 Which of the following statements can be regarded as a null hypothesis? (1) The coin is not fair. (2) The variables X and Y are related in the population. (3) The mean delivery time of an accepted order is at most six days. (4) The defendant is guilty of committing the crime. (5) There are differences among the mean marks of Statistics scored by the students of three classes.

Question 𝟑𝟒 Which of the following statements about hypothesis testing is not true? (1) When the researcher rejects a true null hypothesis, a type I error occurs. (2) At the beginning of the test null hypothesis is assumed to be true. (3) The maximum probability of a type I error that the researcher will tolerate is called the level of

significance. (4) If the 𝑝 − value id larger compared to the significance level then the null hypothesis should be

rejected. (5) A hypothesis test in which rejection of the null hypothesis occurs for values of the point estimators

in either tail of the sampling distribution is called a two tailed test.

Question 𝟑𝟓 Which of the following statements is/are true about hypothesis testing? 𝐴 - If 𝜎2 of a normal population is unknown the hypothesis 𝐻0: 𝜇 = 𝜇0 is a composite hypothesis.

𝐵 - If 𝜎2 of a normal population is unknown 𝑍 =�̅�−𝜇0

𝜎 √𝑛⁄ is a test statistic.

𝐶 - If the probability of occurring type II error is β the power of the test is 1-β. (1) 𝐴 only. (2) 𝐶 only. (3) 𝐴 and 𝐶 only. (4) 𝐵 and 𝐶 only. (5) All 𝐴, 𝐵 and 𝐶

Question 𝟑𝟔 Which of the following statement/s is/are true about hypothesis testing? 𝐴 - If the probability distribution of the population is completely specified when a hypothesis is

true, it is a simple hypothesis. 𝐵 - A hypothesis test with smaller type I error is always better than a hypothesis test with greater

type I error. 𝐶 - The 𝑝 − value of a hypothesis test is a measure of the credibility of the null hypothesis. (1) 𝐴 only. (2) 𝐴 and 𝐵 only. (3) 𝐴 and 𝐶 only. (4) 𝐵 and 𝐶 only. (5) All 𝐴, 𝐵 and 𝐶

Question 𝟑𝟕 Which of the following statement is true? (1) A hypothesis test with 10% significance level is better than a hypothesis test with 5% significance

level. (2) In a hypothesis test at 5% significance level, 𝐻0 is not rejected if 𝑃 − value < 0.05. (3) The decision to use one tailed or two tailed test depends on the form of the null hypothesis. (4) In a hypothesis test both type I error and type II error can be reduced only by increasing the sample

size. (5) The value obtained by substituting sample data to test statistic is called the critical value.

Question 𝟑𝟖 Which of the following statements is true? (1) In hypothesis testing type II error is considered as the most serious error. (2) A confidence interval can also be constructed using the sampling distribution of a test statistic.



(3) The statistic is defined under the assumption that the null hypothesis is true. (4) The power of a test is related to the type I error. (5) If the 𝑝 − value for a test is 0.014, then 𝐻0 is acceptable at the 5% level and also at 1% level. Question 𝟑𝟗 Suppose we have obtained the 𝑝 − value after performing a t-test for comparison of means. Which of the following statements are/is true at a 5% level of significance? 𝐴 - If 𝑝 − value < 0.05 , we accept 𝐻0 and reject 𝐻1. 𝐵 - If 𝑝 − value < 0.05 , we reject 𝐻0 and accept 𝐻1. 𝐶 - If 𝑝 − value < 0.05 , we reject 𝐻0 but do not accept 𝐻1.

(1) 𝐴 only. (2) 𝐵 only. (3) 𝐶 only. (4) 𝐴 and 𝐵 only. (5) 𝐴 and 𝐶 only.

Question𝟒𝟎 Let �̅� be the mean of a random sample of size 𝑛1 drawn from a normal population with mean 𝜇1, variance 𝜎2 and �̅� be the mean of a random sample of size 𝑛2 drawn from a normal population with mean 𝜇2, variance 𝜎2. If the pooled estimate of 𝜎2is given by 𝑆𝑃

2, the equality of two population means is tested by the rest statistic,

(1) 𝑡 =�̅�−�̅�

𝑆𝑃 (2) 𝑡 =

�̅�−�̅�

𝑆𝑃√

𝑛1+𝑛2

𝑛1𝑛2

(3) 𝑡 =�̅�−�̅�

𝑆𝑃√

𝑛1𝑛2

𝑛1+𝑛2 (4) 𝑡 =

�̅�−�̅�

𝑆𝑃√

𝑛1𝑛2

𝑛1−𝑛2

(5) 𝑡 =�̅�−�̅�

𝑆𝑃√

𝑛1−𝑛2

𝑛1𝑛2

Question 𝟒𝟏 Let �̅� be the mean of a random sample of size 25 from 𝑁(𝜇, 100) distribution. If the critical region for testing 𝐻0: 𝜇 = 60 against 𝐻1: 𝜇 > 60 is given by �̅� > 63, the probability of type I error is (1) 0.0668 (2) 0.1336 (3) 0.2266 (4) 0.4332 (5) 0.5668

Question 𝟒𝟐 Let �̅� be the mean of a random sample of size 20 from 𝑁(𝜇, 80) distribution. If the critical region for testing 𝐻0: 𝜇 = 68 against 𝐻1: 𝜇 = 68 is given by �̅� > 67, the probability of type II error is (1) 0.0987 (2) 0.1915 (3) 0.3085 (4) 0.4013 (5) 0.8085

Question 𝟒𝟑 A company is ordering a large lot of items. Letting p denotes the probability that an item is defective, the company decides to test the null hypothesis 𝐻0: 𝑃 ≤ 0.1. If 𝐻0 is rejected, the company returns the lot of items, while it retains the lot of items if 𝐻0 is not rejected. What occurs if it commits a type II error? (1) The company returns a lot which has less than or equal to 10% defective items. (2) The company retains a lot which has more than 10% of defective items. (3) The company returns a lot which has more than 10% of defective items. (4) The company samples some more items. (5) The company retains a lot which has less than or equal to 10% defective items.

Question 𝟒𝟒 Which of the following statements about chi-square test of independency is/are true? 𝐴 - The null hypothesis states the two variables are statistically dependent. 𝐵 - The null hypothesis states the two variables are statistically independent. 𝐶 - The test statistics follows a chi-square distribution with (𝑟 − 1)(𝑐 − 1) degrees of freedom. (1) 𝐴 only. (2) 𝐵 only. (3) 𝐴 and 𝐵 only. (4) 𝐴 and 𝐶 only. (5) 𝐵 and 𝐶 only.



Question 𝟒𝟓 A Poisson distribution was fitted to a observed frequency distribution with ten classes, 0, 1, 2, … , 9. The expected frequencies of last two classes are less than 5. The critical region for testing the goodness of fit at 5% level is given by (1) 𝜒2 > 14.1 (2) 𝜒2 > 15.5 (3) 𝜒2 > 16.0 (4) 𝜒2 > 16.9 (5) 𝜒2 > 17.5

Question 𝟒𝟔 Each person in a random sample of 50 was asked to state his/her gender and preferred colour of car. The results are shown below.

Gender Colour of car

Red Blue White

Male 5 14 6

Female 15 6 4

If Chi-square test is used to test the null hypothesis that the gender and preferred colour are independent, then what is the expected number of females who preferred red colour?

(1) 10 (2) 15 (3) 20 (4) 25 (5) 50

Question 𝟒𝟕 Which of the following statements are /is true about the assumptions made in the analysis of variance? 𝐴 - Populations, from which samples are drawn, are normally distributed. 𝐵 - The mean of the populations are equal. 𝐶 - The variances of populations are equal. (1) 𝐴 only. (2) 𝐵 only. (3) 𝐴 and 𝐶 only. (4) 𝐵 and 𝐶 only. (5) All 𝐴, 𝐵 and 𝐶

Question 𝟒𝟖 Consider the following table:

Analysis of Variance (ANOVA) Source of variation

Sum of squares

Degrees of freedom

mean square 𝐹 - statistic

Between samples 722.7 4 180.68 15.8 Within samples 473.3 40 11.83 Total 1196.0

If all sample sizes are equal, then the number of samples and number of observations in a sample are respectively (1) 4 and 9 (2) 4 and 10 (3) 4 and 11 (4) 5 and 9 (5) 5 and 10

Question 𝟒𝟗 An analysis of variance table constructed to compare means of three normal populations with equal variances gives sums of squares for between population as 70 and sums of square for errors as 36 with 12 degrees of freedom. The F value in the analysis of variance table is (1) 1.94 h' (2) 2.83 h' (3) 2.91 h' (4) 7.78 h' (5) 11.6 h'

Question 𝟓𝟎 To compare the mean of 5 normal population with equal variances, random samples of sizes 10, 9, 8, 8 were taken respectively. The F-table value for testing the equality of means at 1% significant level is (1) 3.13 (2) 3.51 (3) 3.83 (4) 9.24 (5) 13.70


Department of Social Statistics

University of Sri Jayewardenepura

Supportive Seminar on Business Statistics for Advanced Level 2019

Sampling and Statistical Inference

Part One

Question 𝟎𝟏 Which of the following statement is true? (1) Simple random sampling is a method of selecting a sample giving each unit in the population a

known probability of being included in the sample. (2) Without a complete sampling frame cluster sampling cannot be used. (3) Quota sampling is an example of semi-probability sampling. (4) The finite population correction factor may be ignored when it is close to zero. (5) Cluster sampling is more effective if variation within the cluster is large. Question 𝟎𝟐 Which of the following statements is true? (1) If the variation between strata is large, the precision of stratified random sampling is small. (2) If intra-class correlation coefficient is close to one, cluster sampling is more efficient than simple

random sampling. (3) A quota sampling is usually selected using a sampling frame. (4) The main purpose of circular systematic sampling is to select a systematic sample when the

sampling interval is an integer number. (5) Systematic sampling can be regarded as a cluster sampling of taking one cluster form k clusters

of size n. Question 𝟎𝟑 Which of the following statements is true? (1) The accuracy of an estimate is measured by the standard error of that estimator. (2) The variance of the sample mean in sampling with replacement is smaller than variance of the

sample mean in sampling without replacement. (3) The standard error of an estimator can be measured only in a probability sampling. (4) The failure to interview the units in the selected sample is an example for a sampling error.

(5) The term 𝑁

𝑛 is called the sampling fraction.

Question 𝟎𝟒 Which of the following statement is not true? (1) Central limit theorem implies that the sampling distribution of the sample mean is

approximately normal if the sample size is large. (2) An estimator is said to be unbiased if its expected value is equal the parameter being estimated. (3) An estimator is said to be sufficient if it contains all the information in the data about parameter

it estimates. (4) An estimator is said to be a consistent estimator of a population parameter if it has the smallest

variance among all the possible estimators of the parameter. (5) The standard error of mean of a sample taken from a given population decreases as the sample

size increases.



Question 𝟎𝟓 Which of the following statements is true? 𝐴 - The distribution of sample mean of a non-normal population will tend to be increasingly

normal when the sample size increases. 𝐵 - The standard deviation of the distribution of sample mean increases as the sample size

increases. 𝐶 - The mean of the distribution of sample mean is equal to the mean of the population. (1) 𝐴 only. (2) 𝐵 only. (3) 𝐴 and 𝐶 only. (4) 𝐵 and 𝐶 only. (5) All 𝐴, 𝐵 and 𝐶

Question 𝟎𝟔 Which of the following statement are/is true about the 𝑡 − distribution? 𝐴 - The 𝑡 − distribution is symmetric about zero. 𝐵 - The 𝑡 − distribution has a larger variance than the standard normal distribution. 𝐶 - The 𝑡 − distribution with 𝑘 degrees of freedom has a smaller variance than the 𝑡 −

distribution with 𝑘 + 1 degrees of freedom. (1) 𝐴 only. (2) 𝐵 only. (3) 𝐴 and 𝐵 only. (4) 𝐴 and 𝐶 only. (5) All 𝐴, 𝐵 and 𝐶 Question 𝟎𝟕 An estimate is required for the population mean of the widths of metal bars with a margin of error of 0.1 at 95% confidence level. It is known that the variance of the width of bars is 0.25cm. What is the sample size necessary to achieve these requirements?

(1) 10 (2) 22 (3) 25 (4) 97 (5) 102

Question 𝟎𝟖 Suppose that 20% of all people in a certain population are left handed. The sampling distribution of the sample proportion of left-handers of a random sample of 100 people selected from this population is (1) approximately normal with mean 20 and standard deviation 4. (2) binomial with 𝑛 = 100 and 𝑝 = 0.20 (3) approximately normal with mean 0.20 and standard deviation 0.0016 (4) approximately normal with mean 0.20 and standard deviation 0.16 (5) approximately normal with mean 0.20 and standard deviation 0.04

Question 𝟎𝟗 A manufacturing company measures the weight of boxes produced before transporting them to the customers. If the weight of boxes have the population mean of 20 kg and a population standard deviation of 2.8 kg, find the probability that the average weight of a sample of 49 boxes will be less than 19.2 kg? (1) 0.0793 (2) 0.1586 (3) 0.4207 (4) 0.5793 (5) 0.9207

Question 𝟏𝟎 If �̅� is the mean of a random sample of size n from 𝑁(𝜇, 100), find the value of n such that 𝑃(−5 < �̅� − 𝜇 < 5) = 0.9544 (1) 4 (2) 8 (3) 15 (4) 16 (5) 18

Question 𝟏𝟏 Which of the following statements is/are true about confidence intervals? 𝐴 - The width of a confidence interval increases as confidence level decreases. 𝐵 - Confidence intervals can be used to test some hypothesis. 𝐶 - For small samples, the width of the confidence intervals based on 𝑡 − distribution is larger

than the width of the confidence intervals based on 𝑍 − distribution. (1) 𝐴 only. (2) 𝐵 only. (3) 𝐴 and 𝐵 only. (4) 𝐵 and 𝐶 only. (5) All 𝐴, 𝐵 and 𝐶



Question 𝟏𝟐 The analysis of a random sample of 300 households in a small town indicates that a 98% confidence interval for the mean family income is (Rs. 42, 520, Rs. 49, 860). Could this information be used to conduct a test of the null hypothesis 𝐻0: 𝜇 = 40, 000 against the alternative hypothesis 𝐻1: 𝜇 ≠40, 000 at a 0.02 level of significance? (1) No, because it is not known whether the data are normally distributed. (2) No, because the sample standard deviation is not known. (3) Yes, since the sample mean Rs. 46, 190 greater than Rs. 40, 000 𝐻0 would be rejected. (4) Yes, since Rs. 40, 000 is not contained in 98% confidence interval, 𝐻0 would be rejected. (5) Yes, since Rs. 40, 000 is not contained in 98% confidence interval, 𝐻0 would not be rejected. Question 13 A sample of size 25 is taken from a normal population. The sample mean and the sample variance are calculated as 15 and 16 respectively. What is the upper limit of the 95% confidence interval for the population mean 𝜇?

(1) 16.31 (2) 16.37 (3) 16.57 (4) 16.65 (5) 21.27

Question 𝟏𝟒 The lifetime of a certain type of light bulb is known to have a standard deviation of 40 hours. How large a sample should be taken if it is desired to have a margin of error of 10 hours or less at 95% level of confidence? (1) 8 (2) 32 (3) 44 (4) 62 (5) 66 Question 𝟏𝟓 According to past records, the mean lifetime for a certain type of battery has been 196 hours. The manufacturer has introduced a change in the production method and wants to perform a hypothesis test to determine whether the mean lifetime has increased as a result. The null hypothesis (𝐻0) and a alternative hypothesis (𝐻1) for this test are, (1) 𝐻0: 𝜇 ≥ 196 hours; 𝐻1: 𝜇 < 196 hours; (2) 𝐻0: 𝜇 > 196 hours; 𝐻1: 𝜇 ≤ 196 hours; (3) 𝐻0: 𝜇 = 196 hours; 𝐻1: 𝜇 ≠ 196 hours; (4) 𝐻0: 𝜇 < 196 hours; 𝐻1: 𝜇 ≥ 196 hours; (5) 𝐻0: 𝜇 = 196 hours; 𝐻1: 𝜇 > 196 hours; Question 16 If the mean of a random sample of size 80 taken from a population with mean 128 and variance 20, the approximate probability that �̅� lies between 127 and 129 is (1) 0.2280 (2) 0.3413 (3) 0.4772 (4) 0.6826 (5) 0.9544

Question 1𝟕 Which of the following statement is true? (1) A test statistic cannot have parameters. (2) The 𝑃 − value for a test is calculated under the assumption that the alternative hypothesis is

true. (3) The sampling distribution of a test statistic is decided under the assumption that the alternative

hypothesis is true. (4) The observed value for a test statistic is called a critical value. (5) The probability of accepting the correct null hypothesis is called the power of the test. Question 𝟏𝟖 Which of the following statements is true? (1) In hypothesis testing type II error is considered as the most serious error. (2) A confidence interval can also be constructed using the sampling distribution of a test statistic. (3) The statistic is defined under the assumption that the null hypothesis is true. (4) The power of a test is related to the type I error. (5) If the 𝑝 − value for a test is 0.014, then 𝐻0 is acceptable at the 5% level and also at 1% level.



Question 𝟏𝟗 Let �̅� be the mean of a random sample of size 𝑛1 drawn from a normal population with mean 𝜇1, variance 𝜎2 and �̅� be the mean of a random sample of size 𝑛2 drawn from a normal population with mean 𝜇2, variance 𝜎2. If the pooled estimate of 𝜎2is given by 𝑆𝑃

2, the equality of two population means is tested by the rest statistic,

(1) 𝑡 =�̅�−�̅�

𝑆𝑃 (2) 𝑡 =

�̅�−�̅�

𝑆𝑃√

𝑛1+𝑛2

𝑛1𝑛2 (3) 𝑡 =

�̅�−�̅�

𝑆𝑃√

𝑛1𝑛2

𝑛1+𝑛2

(4) 𝑡 =�̅�−�̅�

𝑆𝑃√

𝑛1𝑛2

𝑛1−𝑛2 (5) 𝑡 =

�̅�−�̅�

𝑆𝑃√

𝑛1−𝑛2

𝑛1𝑛2

Question 𝟐𝟎 Let �̅� be the mean of a random sample of size 25 from 𝑁(𝜇, 100) distribution. If the critical region for testing 𝐻0: 𝜇 = 60 against 𝐻1: 𝜇 > 60 is given by �̅� > 63, the probability of type I error is (1) 0.0668 (2) 0.1336 (3) 0.2266 (4) 0.4332 (5) 0.5668

Question 𝟐𝟏 A company is ordering a large lot of items. Letting p denotes the probability that an item is defective, the company decides to test the null hypothesis 𝐻0: 𝑃 ≤ 0.1. If 𝐻0 is rejected, the company returns the lot of items, while it retains the lot of items if 𝐻0 is not rejected. What occurs if it commits a type II error? (1) The company returns a lot which has less than or equal to 10% defective items. (2) The company retains a lot which has more than 10% of defective items. (3) The company returns a lot which has more than 10% of defective items. (4) The company samples some more items. (5) The company retains a lot which has less than or equal to 10% defective items.

Question 𝟐𝟐 Which of the following statements about chi-square test of independency is/are true? 𝐴 - The null hypothesis states the two variables are statistically dependent. 𝐵 - The null hypothesis states the two variables are statistically independent. 𝐶 - The test statistics follows a chi-square distribution with (𝑟 − 1)(𝑐 − 1) degrees of freedom. (1) 𝐴 only. (2) 𝐵 only. (3) 𝐴 and 𝐵 only. (4) 𝐴 and 𝐶 only. (5) 𝐵 and 𝐶 only.

Question 𝟐𝟑 A Poisson distribution was fitted to a observed frequency distribution with ten classes, 0, 1, 2, … , 9. The expected frequencies of last two classes are less than 5. The critical region for testing the goodness of fit at 5% level is given by (1) 𝜒2 > 14.1 (2) 𝜒2 > 15.5 (3) 𝜒2 > 16.0 (4) 𝜒2 > 16.9 (5) 𝜒2 > 17.5

Question 𝟐𝟒 Which of the following statements are /is true about the assumptions made in the analysis of variance? 𝐴 - Populations, from which samples are drawn, are normally distributed. 𝐵 - The mean of the populations are equal. 𝐶 - The variances of populations are equal. (1) 𝐴 only. (2) 𝐵 only. (3) 𝐴 and 𝐶 only. (4) 𝐵 and 𝐶 only. (5) All 𝐴, 𝐵 and 𝐶

Question 𝟐𝟓 In an analysis of variance table constructed to compare the mean scores of 4 teaching methods, the between teaching methods sums of squares was 42 and error sums of square was 60 with 30 degrees of freedom. The null hypothesis should be rejected at 5% significance level since the value of test statistics is (1) 7 > 2.92 (2) 7 > 4.51 (3) 5.25 > 4.02 (4) 5.25 > 2.69 (5) 7 > 3.59



Part Two

01. (a) Describe four advantages of sampling over a census/complete enumeration.

(04 marks)

(b) What is a “Sampling Frame”? Explain the main properties of a good sampling frame.

(04 marks)

(c) Explain the meaning of “Cluster Sampling” and compare Cluster Sampling over Simple Random

Sampling.

(04 marks)

(d) State the central limit theorem. Let 𝑋 be the mean of a random sample of size 49 taken from a

poison distribution with mean 8. Find approximately 𝑃(7.5 ≤ 𝑋 ≤ 9).

(04 marks)

(e) The yield of a certain crop per hectare is distributed as 𝑁(𝜇, 𝜎2). The yield per hectare for a

randomly selected sample of 5 plots of land were 240, 248, 246, 252 𝑎𝑛𝑑 244 in kilograms.

i. Find point estimates for 𝜇 and 𝜎2

ii. Find approximately 90% confidence interval for 𝜇.

(04 marks)

02.

(a) Explain the relationship between “Systematic Sampling” to “Cluster Sampling” and “Stratified

Random Sampling”.

(04 marks)

(b) Define estimator and explain the meaning of accuracy, unbiasedness and precision of an estimator.

(04 marks)

(c) Let 𝑋1, 𝑋2, 𝑋3 be a random sample of size 3 taken from the distribution with mean 𝜇 and variance

𝜎2. Prove �̂� =𝑋1+2𝑋2+𝑋3

4 is unbiased estimator for 𝜇. Find the efficiency of �̂� with respect to the

efficiency of 𝑋 =𝑋1+𝑋2+𝑋3

3.

(06 marks)

(d) 40 items were simple randomly selected from a large stock of items with 0.12 proportion of the

defective items. Find the probabilities of,

i. At least 7 items being defective

ii. Number of defective items lies between 3 and 7.

(06 marks)



03.

(a) What do you mean by “Type I error” and “Type II error” in a hypothesis testing?

(04 marks)

(b) A sample of size 36 was taken from 𝑁(𝜇, 81) distribution to test the hypothesis 𝐻0: 𝜇 = 96 against

the hypothesis 𝐻1: 𝜇 = 99. The testing procedure states that if 𝑋 > 98.46, hypothesis 𝐻0: 𝜇 = 96

will be rejected.

i. Calculate the probability of type I error (𝛼).

ii. Calculate the probability of type II error (𝛽)

iii. What is the power of the test?

(05 marks)

(c) Assume that the IQ scores of a certain population has a normal distribution with mean 𝜇 and

variance 𝜎2 = 100. The mean of a random sample of size 16 taken from the population was 115.5.

Test the hypothesis 𝐻0: 𝜇 = 112 against the hypothesis 𝐻1: 𝜇 > 112 at 5% level of significance.

(04 marks)

(d) A certain Department of Accounts in a firm wants to know whether the incidence of errors occurred

in their accounting process follows a Poisson distribution. The manager of the department has

taken a random sample of 172 accounts and summarized the number of errors found in those

accounts. The results are shown below.

Number of errors 0 1 2 3 4 5 6

Numbers of accounts 26 48 42 28 16 8 4

Test whether the incidence of errors in the firm’s Accounting records follows a Poisson

distribution at 5% significance level.

(07 marks)

04.

(a) Explain the difference between each pair of terms below.

i. Simple hypothesis and Composite hypothesis

ii. Critical region and Acceptance region

(04 marks)

(b) An examination was conducted in two classes consisting of 40 and 50 students. The mean and

standard deviation of the students marks in the first class was 74 and 8 while the mean and standard

deviation of the students marks in second class was 78 and 7. Test whether there is a significance

difference between the average marks of students in two classes at the 5% level of significance.

(04 marks)



(c) A company is intertesred in finding out whether there is an association between the level of work-

life balance of employees and their job positions. The following table provides the results obtained

from the study conducted by using 120 workers. Test at 5% level of significance whether the job

position of the employee is independent of the level of work-life balance.

Job position Level of work-life balance

High Medium Low

Manager 8 6 16

Supervisor 16 10 18

Machine Operator 24 14 8

(06 marks)

(d) To test whether there is a price difference of a certain product in four cities (A, B, C, D), 5 number

of shops were selected from each city and price of the product were recorded from each shop.

Those prices are given in the following table.

i. Write down the model with assumptions to analyze the data.

ii. Test whether the average prices are the same in the cities at 5% level of significance.

(06 marks)

∗∗∗

City A City B City C City D

248 256 255 255

249 252 253 253

247 250 254 252

245 253 248 254

246 254 250 256

METHODIST COLLEGE, COLOMBO 03

FIRST TERM EXAMINATION-2017

BUSINESS STATISTICS I GRADE : 14

TIME: 02 HOURS

1 When is secondary data more suitable than primary data?

a. When there maybe errors when copying the data

b. When the population is large and difficult to collect information c. When there are restrictions in collecting data

d. When the details are in a descriptive manner e. All of the above 2 Which of the following is not checked when editing? a. Completeness b. Consistency c. Convenience d. Accuracy e. Homogeneity 3 When a grouped frequency distribution is considered, from which of the following the frequency of

each class interval cannot be found? a. Cumulative frequency curve b. Histogram c. Frequency polygon d. Relative frequency polygon e. Ogive 4 Which of the following is/ are true about a frequency distribution with unequal class width? A -Class which has the highest frequency is the modal class B -The highest class in the histogram is the modal class C -Class which has the highest frequency density is the modal class a.A only b.B only c.A and B only d.A and C only e. A,B and C

5 Which one of the following is not 2 dimensional?

a. Histogram

b. Lorenz curve

c. Z chart

d. Multiple Bar chart

e. Scatter plot

6 If the mean and mode of a distribution are 18 and 30 respectively, the value of the median is

a. 12 b. 20 c. 14 d. 24 e. None of the above

7 The PH- level of rain water was measured in 30 different working sites located around a particular

industrial region. The mean and standard deviation of those measures were 4.60 and 1.10 respectively.

It was later revealed that the PH- meter was erroneous and pointed out that each observation should be

added 0.1 PH units and then the results to be multiplied by 1.2 to recorrect the errors. The mean and

standard deviation of the recorrected PH measures are respectively

a.5.64 and 1.44 b.5.64 and 1.32 c.5.40 and 1.44 d.5.40 and 1.32 e.5.65 and 1.20

8.

Which one of the following statements is true?

a. The value of every observation in the data set is taken into account when we calculate its median.

b. A measure of the peakedness of a distribution curve is its skewness

c. With ungrouped data, the mode is most frequently used as the measure of central tendency

d. One advantage of using the range to measure dispersion is that it ignores the nature of variations

among most of the observations

e. For a data array with 50 observations, the median will not be the value of the 25th observation in the

array.

9. If in a class of 100 students, the arithmetic mean of amount of pocket money is Rs.35 per student and

the arithmetic mean is Rs.25 for girls and Rs.50 for boys, then the number of girls in the class will be

a. 20 b. 40 c. 60 d. 80 e.Cannot be determined

10 The standard deviation of the first n natural numbers is

a. 𝑛(𝑛+1)(2𝑛+1)

6 b.

𝑛2 −1

12 c. √

𝑛2 −1

12 d.

𝑛2+ 1

6 e. None of the above

11 The quartile deviation includes

a. Middle 50% of the data observations

b. All the observations

c. First 50% of the data observations

d. Least 50% of the data observations

e. Least 25% of the data observations.

12 If the 2 observations 𝑥1 and 𝑥2 are such that 𝑥1 = −1

2 𝑥2, their harmonic mean is

a. 0 b. 1

2𝑥1 c. 4𝑥1 d. ∞ e.none of the above

13 Which one of the following statements is/are true?

A -If the regression coefficient is positive, there is a positive correlation between the variables

B -The rank correlation coefficient cannot be negative

C -The square of the coefficient of determination is the coefficient of correlation

a. A only b. B only c. C only d.A and C only e.A, B and C

14 The rank correlation coefficient of 10 pair of data was -1. What is the value of ∑ 𝑑2 ?

a. 170 b. 165 c. 0 d. 195 e. 330

15 A study found that there is a strong positive relation between height of a person and his income, it

implies that

a. The employers prefer to pay higher wages to taller people

b. Though taller people receive a higher salary there is no linear relationship between income and height

c. The strong positive relationship is due to an error in the study

d. The strong positive relationship has occurred due to randomness

e. Taller people receive higher salaries.

16 The two events, born is January and holding a higher post in a company are

a. Mutually exclusive events

b. Independent events

c. Exhaustive events

d. Dependent events

e. Complementary events

17 In a container there are 2 blue balls and 5 green balls and in another container there are 6 blue balls and

3 green balls. If one ball is taken from the first container and placed in the second container and then a

ball is picked from the second container, what is the probability of that ball being a green ball?

a. 22

35 b.

7

10 c.

6

70 d.

20

70 e.

13

35

18 If n is the number of values in a data set and if 𝑛𝑝4 = 12 𝑛𝑝2

,find n

a. 2 b. 6 c. 1 d. -1 e. 3

19 Statements of 3 students regarding the mutually exclusive events are as follows

A -Such events are always totally exhaustive

B -Such events are independent

C -Such events are dependent

Which one of the above statements is/are true?

a. A only b. B only c. C only d.A and B only e.A and C only

20

If A and B are two independent events such that P(A)= 1

2 and P(A∪B)=

2

3 , What is P(B)?

𝑎.

1

2 𝑏.

1

3 𝑐.

1

6 d.

2

3 𝑒.

1

4

21 Which one of the following statements is/ are true?

A -If P(A/B)=0 when P(B) ≠ 0, then the events A and B are mutually exclusive

B -Once A and B are any 2 events defined in a sample space the probability of A occurring is less than

the probability of only A occurring

C -When A⊂B, P(A∩B)=P(A) and P(A∪ 𝐵)=P(B)

a. A only b. B only c. C only d.A and C only e.A and B only

22

The expected value and the variance of the random variable X are 0.4 and 0.84 respectively. If another

random variable can be defined as Y=2X+3, the expected value and the variance of Y respectively are

a. 3.8 and 6.36 b. 3.8 and 3.36 c. 10.6 and 6.36 d. 0.4 and 0.84

e. 0.8 and 1.36

23 Which one of the following distributions is negatively skewed?

a. Binomial distribution with 𝜇 = 3 𝑎𝑛𝑑 𝜎2 = 2.1

b. Binomial distribution with 𝜇 = 16 𝑎𝑛𝑑 𝜎2 = 3.2

c. Binomial distribution with 𝜇 = 20 𝑎𝑛𝑑 𝜎2 = 10

d. Poisson distribution with 𝜆 = 1.5

e. Normal distribution with 𝜇 = 6 𝑎𝑛𝑑 𝜎 = 2

24 X follows Poisson distribution such that 3P(X=1) = P(X=2). What is the coefficient of variation of the

distribution?

a. 36 b. 6 c. √6 d. 1

√6 e. 1

25 If the random variable Y is normally distributed with mean 8, variance 0.04 and

P( 7.6≤ 𝑌 ≤ 𝐾) = 0.957, what is the value of K?

a. 8.41 b. 2.05 c. 2.5 d. 8.5 e. 8.14

26 In which of the following methods each unit in the population has equal chance of being selected to the

sample?

a. Cluster sampling

b. Systematic sampling

c. Simple random sampling

d. Stratified sampling

e. Quota sampling

27

Which one of the sampling methods can be used without a sampling frame?

A- Systematic sampling B- Stratified sampling C- Simple random sampling

a. A only b. B only c.A and B only d.B and C only e.A,B and C

28 Out of several unbiased estimators, the estimator which has the least variance is called as

a. Unbiased estimator

b. Least square estimator

c. Consistent estimator

d. Sufficient estimator

e. Efficient estimator

29 Which one of the following is not used to find the width of the confidence interval?

a. Confidence level

b. Population standard error

c. Population size

d. Sample size

e. Distribution of the random variable

30

The following are 2 confidence intervals for the population weight.

(114.4,115.6) (114.1,115.9)

What is the mean of the distribution?

a. 114.5 b. 115 c. 114 d. Data is insufficient e.None of the

above

31 A sample size of 20 is selected from a normal population. It was found that ∑ 𝑥 = 20 𝑎𝑛𝑑 ∑ 𝑥2 = 25

What is the unbiased estimator for variance?

a. 0.2375 b. 4.25 c. 5.263 d. Insufficient data e. None of the above

32 Which one of the following is a name not suitable for critical region in hypothesis testing?

a. Size of the test

b. Power of the test

c. Level of significance

d. Type I error

e. Probability of rejecting H0 when H0 is true

33 When n = 100, what is the probability of 𝜇 in the following confidence interval?

(�̅� - 2.14𝜎

√𝑛 , �̅� + 2.14

𝜎

√𝑛 )

a. 0.42 b. 0.97 c. 0.13 d. 0.95 e. None of the

above

34 In the above question if 𝜎 = 5 𝑎𝑛𝑑 𝑛 = 100 ,what is the size of the confidence interval?

a. 0.97 b. 0.42 c. 0.13 d. 0.84 e. None of the

above

35 If 𝜋 is he population proportion, which one the following is the standard error?

a. 𝜋(1−𝜋)

𝑛 b.

𝜋(1−𝜋)

√𝑛 c.

√𝜋(1−𝜋)

√𝑛 d. √

𝜋(1−𝜋)

𝑛(

𝑁−𝑛

𝑁−1)

e.none of the

above

36 What is the degrees of freedom to test the independency between doing a job and gender?

a. 4 b. 1 c. 2 d. 3.84 e.none of the above

37 We cannot use a 𝜒2 test

a. To compare the means of populations

b. To check the independency of 2 variables

c. To check if the observations are distributed evenly

d. To check the goodness of fit

e. All of the above

38 Which one of the following distribution is most suitable in the one way classification of variance?

a. Normal b. 𝜒2 c. t d. F e. Binomial

39 If F8,10(0.95) , what is the critical value?

a. 0.2985 b. 3.35 c. 3.07 d. 3.85 e. data is insufficient

40 If the equation for sales for one year (Rs.’000 000) is T=20+10X, the origin is 2010 and the unit of X

is 1 year, what is the trend value for 2017 March?

a. 80 b. 815 c. 835 d. 2445 e.none of the above

41 If 614,615,652,678,681,655,717,719…are yearly data, find the 1st and 2nd 4 yearly moving averages

a.639.75,656.5 b.648.125,661.5 c.2559,2772 d.639.75,693 e.none of the above

42 Which one of the following is the monthly trend equation when the annual trend equation is

Y= 81.6 + 28.8 x

a. Y= 81.6 + 2.4x b. Y= 6.8 + 28.8x c. Y= 6.8+ 2.4 x d. Y=6.8+ 0.2 x e. Y= 81.6+ 28.8x

43 Which of the following indices satisfies the time reversal test and the factor reversal test

A - Laspeyre’s index

B - Paasche’s index

C - Fisher’s index

a. A only b. B only c. C only d.A and C only e. B and C only

44 What is the weight used in Laspeyre’s index?

a. Current year quantity

b. Base year quantity

c. Average of all the years

d. Typical period quantity

e. None of the above

45

If the cost of living index and a labourer’s wage in 2010 are 100 and Rs.3050 respectively, and if the

cost of living index is 175 what should be his salary?

a. Rs.5337.5 b. Rs.1742.85 c. Rs.1750 d. Rs.3225 e.None of the above

46 Which one of the following is obtained when one’salary is divided by the index

a. Cost of living index

b. Average wage

c. Real wage

d. Inflation index

e. Deflation index

47 Which one of the following statements s/are true?

A - A control chart in which only a single point falls outside the upper control limit can be

considered to be under control.

B - A defect occurred due to application of low quality materials is an assignable variation while a

defect occurred due to negligence of a worker is a chance variation

C - Although a point falls below the lower limit of range chart, the process is not disregarded as out

of control.

a. A only b. B only c. A and B only d. B and C only e. A, B and C

48

The predetermined standard deviation for an R chart is 𝜎′ =2.5. The UCL and LCL of the range chart

to be constructed using a set of samples of size 10 are

a. 4.4425 and 0.5575 b. 13.6725 and 1.7175 c. 0.77 and 2.57

d. 1.9925 and 1.7175 e. 6.058 and 1.854

49 n= 50 and c=2 of a particular acceptance sampling plan. If AQL= 0.01 and LTPD=0.08, the type I error

and type II error respectively are

a. 0.0902 and 0.0916 b. 0.1246 and 0.2381 c. 0.0144 and 0.2381

d. 0.9856 and 0.7619 e. 0.9098 and 0.0902

50 Which of the following charts is used to monitor an attribute?

a. 𝑥 ̅ chart

b. A chart

c. P chart

d. R chart

e. None of the above

METHODIST COLLEGE, COLOMBO 03

FIRST TERM EXAMINATION- 2017

BUSINESS STATISTICS II GRADE: 14

TIME: 03 HOURS

Answer any 5 questions selecting atleast 2 from each section

Part I

1 a. (i) Give 4 differences between a questionnaire and a schedule.

(ii) Write a situation is which personal interview is more suitable. Give 2 advantages and

Disadvantages of the method.

b. Represent the following information in a suitable tabular form

Out of a total number of 1807 men who were interviewed for employment in a textile factory of

Bombay, 512 were from textile areas and the rest were from non textile areas. Amongst the married

men who belonged to the textile areas , 247 were experienced and 73 inexperienced. While for non

textile areas, the corresponding figures were 49 and 521. The total number of inexperienced men was

1341 of whom 111 resided in textile areas. Of the total number of men 918 were unmarried, and of

these the number of experienced men in the textile and non textile areas were154 and 16 respectively.

c. The sales distribution of employees in two firms of a factory A and B are given below.

% of employees % of sales

Firm A Firm B

0 0 0

20 3 6

40 12 20

60 25 32

80 60 65

100 100 100

Construct Lorenz curves on the same graph to show the distribution of sales of the two firms. Give

your conclusions about the sales distribution.

2 a. Discuss the merits and demerits of the mean and mode as measures of central tendency. ( 04 marks)

b. The first of the 2 samples has 100 items with mean 15 and standard deviation 3. If the whole group has

250 items with mean 15.6 and standard deviation √13.44 , find the standard deviation of the second

group. (06 marks)

c. What do you mean by skewness?

Distribution of wages of 230 persons has a mean of Rs. 110.4, mode of Rs. 116.2 and standard

deviation of Rs. 17.3. Find the skewness value and comment on the distribution. (04 marks)

d. What do you understand by coefficient of variation?

During the first 10 weeks of a session, the marks of two students, X and Y, taking the course were:

X: 58 59 60 54 65 66 52 75 69 52

Y: 56 87 89 78 71 73 84 65 66 46

Which of the two students you would consider to be more consistent? (06 marks)

3

a.

Define index numbers. Give its limitations

b. The group indices and the corresponding weights for the working class cost of living index numbers in

an industrial city for the years 2000 and 2016 are given below

Group index

Group Weights 2000 2016

Food 71 370 380

Clothing 3 423 304

Fuel etc. 9 469 336

Housing rent 7 110 116

Miscelleneous 10 279 283

Compute the cost of living indices for two years 2000 and 2016 . If a worker was getting 300 per

month in 2000, do you think that he should be given extra allowance so that he can maintain his 2000

standard of living in 2016? If so, what should be the minimum amount of this extra allowance?

c. Explain the use of the components of time series in the prediction of future in business by giving an

example each.

d. Fit a straight line trend equation by the method of least squares and forecast the value of 2017. Also

find the trend value of 2016.

Year: 1984 1985 1986 1987 1988 1989 1990 1991

Value: 80 90 92 83 94 99 82 104

4 a. Explain the properties of regression coefficient. (02 marks)

b. The following data gives the experience of machine operators and their performance ratings as given

by the number of good parts turned out per 100 pieces.

Operator: 1 2 3 4 5 6 7 8

Experience (X): 16 12 18 4 3 10 5 12

Performance ratings (Y): 87 88 89 68 78 80 75 83

For the above data ∑ 𝑥 = 80, ∑ 𝑥2 = 218, ∑ 𝑦 = 648, ∑ 𝑦2 = 368 𝑎𝑛𝑑 ∑ 𝑥 𝑦 = 247

(i) Compute the coefficient of correlation between X and Y, assuming a linear relationship.

Comment on you result.

(ii) Estimate the least squares regression line of Y on X.

(iii) Predict the expected performance if an operator has 7 years experience. (06 marks)

c. Explain the following concepts.

Chance variations and assignable variations. (02 marks)

The following table gives the number of errors of alignment observed at final inspection of a certain

model of bus.

Bus number: 1001 1002 1003 1004 1005 1006 1007 1008 1009 1010

No. of alignment defects: 6 10 8 7 12 9 5 7 3 4

Construct a control chart for this process and state whether the process is in control (04 marks)

d. Consider a single sampling plan with a lot size of 2000, a sample size of 100 and acceptance number 2.

(i)Using poisson approximation find the probability of accepting a lot containing 0.5%, 1% and 5%

defectives.

(ii)Sketch the operating characteristic curve for the above plan.

(iii)If AQL=0.05, LTPD=0.2, producer’s risk=0.05 and consumer’s risk=0.10. comment on the

performance of this plan using the operating characteristic curve sketched in part (ii) (06 marks)

PART II

5 a. (i) Explain the classical approach and axiomatic approach to probability. (02 marks)

(ii) An integer is chosen at random from the first 200 digits. What is the probability that the integer

Chosen is divisible by 6 or 8? (04 marks)

b. The records of 400 examinees are given below

Educational qualification

Score B.A B.Sc B.com Total

Below 50 90 30 60 180

Between 50 &60 20 70 70 160

Above 60 10 30 20 60

Total 120 130 150 400

If an examinee is selected from this group of examinees, find

(i) Find the probability that he is a commerce graduate.

(ii) The probability that he is a commerce graduate and does not have a score below 50

(iii) The probability that he is a commerce graduate or an examinee who has a score below 50

(iv) The probability that he is a science graduate, given that his score is above 60.

(v) The probability that his score is below 50, given that he has a qualification of B.A

(08 marks)

c. State the law of total probability and Bayes’ theorem. (02 marks)

d. A manufacturing firm produces T.V sets in three plants with daily production volume of 250,500 and

1000 units repectively. According to past experience, it I known that the fractions of defective output

produced by the three plants respectively are 0.005,0.008 and 0.010. If a T.V set selected from a day’s

total production, find

(i) The probability of the T.V set being defective.

(ii) The probability that the defective T.V set is from the second plant. (04 marks)

6 a. (i) Define the binomial distribution, stating the conditions under which it maybe applied. (02 marks)

(ii)The probability of any ship of a company being destroyed on a certain voyage is 0.3. The

Company owns 5 ships for the voyage. What is the probability of

Losing one ship

Losing at most 2 ships (04 marks)

b. State the conditions under which the binomial distribution maybe approximated by the poisson

Distribution. (02 marks)

A manufacturer of pins knows that 5% of his product are defective. If he sells pins in boxes of 100 and

guarantees that not more than 4 pins will be defective, what is the approximate probability that a box

will fail to meet the guaranteed quality? (02 marks)

c. A car-hire firm has two cars, which it hires out day by day. The number of demands for a car on each

day is distributed as poisson distribution with mean 1.5. calculate

(i) The proportion of days on which neither car is used.

(ii) The proportion of days on which some demand is refused. (04 marks)

d. State the main properties of the normal distribution.

The results of particular examination are given below.

Results % of candidates

Passed with distinction 10

Passed 60

Failed 30

It is known that a candidate fails if he obtains less than 40 marks out of 100 while he must obtain

atleast 75 marks in order to pass with distinction. Determine the mean and standard deviation of the

distribution of marks assuming this to be normal. (06 marks)

7 a. What is stratified sampling? Give two advantages and disadvantages of this method.

�̅� is the mean of sample means of samples of size 15 taken from N(25,9) and �̅� is the mean of sample

means of samples size 8, taken from N(20,4)

(i) State the distributions for �̅� 𝑎𝑛𝑑 �̅� separately.

(ii) If X and Y are independent state the distribution of �̅� − �̅�

b. (i) what do you mean by statistical estimation?

(ii)What is an estimator? How does an estimate differ from an estimator?

(iii)Distinguish between point estimation and interval estimation.

(iv)What are the desirable properties of a good point estimator?

(v)A commuter regularly uses a train service which should arrive in Colombo Fort at 08.31. He

decided to test this stated arrival time. For each working day for a period of 4 weeks he recorded the

number of minutes x that the train was late on arrival in Colombo Fort. If the train arrived early then

the value of x was taken as negative. His results are summarized as follows

n=20.∑ 𝑥 = 150.0, ∑ 𝑥2 = 1600.00

calculate unbiased estimates of the mean and variance of the number of minutes late of this train

service.

8 a. Explain the differences of the following concepts

(i) Type I error and Type II error

(ii) Alternative hypothesis and null hypothesis

b. A random sample of 400 flower stems has an average length of 10cm. Can this be regarded as a sample

from a large population with mean of 10.2 cm and a standard deviation of 2.25 cm?

c. In order to make a survey of the buying habits, two markets A and B are chosen at two different parts

of the city.

400 women shoppers are chosen at random in market A. their average weekly expenditure on food is

found to be Rs. 250 with a standard deviation of Rs. 40. The figures are Rs. 220 and Rs. 55

respectively in the market B where also 400 women shoppers are chosen at random. Test at 1% level of

significance whether the average weekly food expenditure of the two populations of shoppers are

equal.

d. A die is thrown 150 times with the following results:

No. turned up: 1 2 3 4 5 6

Frequency: 19 23 28 17 32 31

Test the hypothesis that the die is unbiased at 5% level of significance.

Documents

Seminar Business Statistics (Sampling and …...Department of Social Statistics, University of Sri Jayewardenepura 17/13/2017 1 Seminar – Business Statistics (Sampling and Inferential