TEST OF HYPOTHESIS In a certain perspective, we can view hypothesis testing just like a jury in a court trial. In a jury

trial, the null hypothesis is similar to the jury making a decision of not- guilty, and the alternative is the guilty verdict. Here we assume that in a jury trial that the defendant isn't guilty unless the prosecution can show beyond a reasonable doubt that defendant is guilty. If it has been established that there is evidence beyond a reasonable doubt and the jury believes that there is enough evidence to refute the null hypothesis, the jury gives a verdict in favor of the alternative hypothesis, which is a guilty verdict.

In general, when performing hypothesis testing, we set up the null (Ho) and alternative (Ha) hypothesis in such a way that we believe that Ho is true unless there is sufficient evidence (information from a sample; statistics) to show otherwise. Statistical Hypothesis A statistical hypothesis is an assertion or conjecture concerning one or more populations. Types of statistical hypothesis:

1. Null hypothesis – the hypothesis that we wish to focus our attention on. Generally, this

is a statement that a population parameter has a specified value.

The hypothesis that is tested and the one which the researcher wishes to reject or not to reject.

Specifies an exact value of the population parameter.

Denoted by Ho.

2. Alternative hypothesis -

The hypothesis that is accepted if the null hypothesis is rejected.

Allows for the possibility of several values.

Denoted by Ha or H1.

May be directional (quantifier is < or >) or non-directional (quantifier is ).

Example

State the null and alternative hypothesis in the following statements: 1. The percentage of junior high school students who pass math subjects during summer is 65%.

65 :H 65 :H ao

2. At least 5 typhoons on the average hit the country every year.

5 :H 5 :H ao

3. Forty-eight percent of high school graduates are computer illiterate.

48% :H 48% :H ao

4. At most 6 out of 10 married women in the rural areas are house wives.

60% :H 60% :H ao

Exercise State the null and alternative hypothesis.

1. At most 65% of public school children are malnourished. 2. On the average at least 2/3 of high school students who pass their math subjects pass their

physics subjects. 3. Less than half of the newly nursing board passers immediately get their visa in a year. 4. A man should have at least 8 hours of sleep everyday. 5. 55% of elected public officials came from the same university. A test of hypothesis is the method to determine whether the statistical hypothesis is true or

not. In performing statistical test of hypothesis we consider the following situations: The probability of committing a TYPE I error is also called the level of significance and is

denoted by a small Greek symbol “alpha” . Some of the common values used for the level of significance are 0.1, 0.05, and 0.01. For example, if = 0.1 for a certain test, and the null hypothesis is rejected, then it means that we are 90% certain that this is the correct decision. Important things to know before conducting a test of hypothesis:

1. Level of significance, .

The level of significance, , is the probability of committing an error of rejecting the null hypothesis when, in fact, it is true.

2. One-tailed tests vs. Two-tailed tests

One-tailed test of hypothesis

A one tailed test is performed when the alternative hypothesis is concerned with values specifically below or above an exact value of the null hypothesis.

The alternative hypothesis is directional.

Two-tailed test of hypothesis

A two-tailed test is performed when the alternative hypothesis is concerned with values that are not equal to an exact value of the null hypothesis.

The alternative hypothesis is non-directional. 3. Test Statistic

The value generated from sample data.

Test value to be compared with the critical values.

4. Critical Region (Region of rejection/region of acceptance)

Depends on the type of test to be performed.

Null hypothesis

TRUE FALSE

Reject TYPE I Error Correct Decision

Do not Reject Correct Decision TYPE II Error

If test is one tailed, then the critical region is concentrated on either the left tail (for <) or the right tail of the distribution (for >).

If test is two tailed, then the critical region is distributed on each tail of the distribution.

Critical values are obtained depending on the type of test to be performed

If the test is one tailed, the significance level will be the area either on the left tail or on the right tail of the distribution.

If the test is two tailed, the area in each tail of the distribution will be /2.

Steps in hypothesis testing

1. Set up the null and alternative hypothesis.

2. Specify the level of significance. 3. Determine the critical region and the corresponding critical values. 4. Compute the value of the test statistic. 5. Make a decision.

Reject H0 in favor of H1 if test statistic falls in the critical region.

Do not reject H0 if it falls in the acceptance region. 6. Draw appropriate conclusions.

Testing a single population mean,

Suppose a random sample of size n is taken from a normal population with mean

and standard deviation . To test the claim that the population mean is equal to a certain

value 0, we perform the test of hypothesis for the population mean, .

Null and alternative hypothesis

H0 : = 0

H1: < 0 , > 0, or 0.

CASE 1: 2 is known or 2 unknown but n 30.

Test statistic: The test statistic depends on the case where the problem falls under.

n

xz

0

Remark

Although most textbooks in statistics use the term “accept” and “reject”

when interpreting results of statistical test of hypothesis, it is very important to

understand that the rejection of a hypothesis is to conclude that it is false. While the

acceptance of a hypothesis merely implies that there is no significant evidence to say

that it is false.

Critical region/ value/s:

For a one tailed test: Reject H0 if

z < - z for a left-tailed test

z > z for a right-tailed test

For a two tailed test: Reject H0 if

z < -z/2 or z > z/2 for a two tailed test

Example

1. A random sample of 100 recorded deaths in Mindanao during the past year showed an average life span of 71.8 years. The sample also showed a 8.9 years of standard deviation. Does this data indicate that the average life span of people living in Mindanao is greater than 70 years? Use a 0.05 level of significance. Solution:

Step 1. 700 :H and 701 :H

Note that the alternative hypothesis is directional, we are performing a one tailed test; specifically a right tailed test.

Step 2. 050. (Level of significance) Step 3. Critical Region

6451050 .zzzz .

Step 4. Test Statistic

Critical

Region

Note

z denotes the z value with an area of “ ” to the right.

z

Since the population standard deviation is unknown but the sample size is large

enough, that is, 30n , we can substitute the sample standard deviation for “ ”, thus we have

02210098

708710 ./.

.

n

xz

Step 5. Since the value of the test statistic falls under the critical region, we reject the null hypothesis in favor of the alternative hypothesis. Step 6.

Since the null hypothesis is rejected, we say that there is sufficient evidence to say that the average life span of people living in Mindanao is greater than 70 years. 2. A salon owner believes that the average number of their regular customers gets a haircut and pedicure is 25. A random sample of 25 regular customers showed that 20 of them did a

haircut and pedicure. With =5% is the belief of the salon owner true? Assume that the standard deviation is 10.5. Solution

Step 1. 250 :H and 251 :H

Note that the alternative hypothesis is bi-directional, we are performing a two tailed test.

Step 2. 050. (Level of significance) Step 3. Critical Region

96102502 .zzzz ./ or 96102502 .zzzz ./

Step 4. Test Statistic

Since the population standard deviation is known we have

382

25510

25200 ..

n

xz

Critical

Region

1.96 -1.96

Critical

Region

Step 5. Since the value of the test statistic falls under the critical region, we reject the null hypothesis in favor of the alternative hypothesis. Step 6.

Since the null hypothesis is rejected, we say that there is sufficient evidence to refute the claim of the salon. And hence, we can say that based on the sample evidence, the average number of customers who did haircut and pedicure is not 25.

CASE 2: 2 unknown and n < 30

Test statistic

n

s

xt 0

Critical Values/Regions

For a one tailed test: Reject H0 if

t < - t with df = n -1 for a left-tailed test

t > t with df = n -1 for a right-tailed test

For a two tailed test: Reject H0 if

t < -t/2 or t > t/2 with df - n -1 for a two tailed test

Example

1. In the past a study has been made on call center agents on their sleeping habits. The result showed that the average number of hours they took for sleeping is at most 8 hours. A random sample of 25 call center agents where asked and showed that the average number of hours they took for sleeping is 6.5 with a sample variance of 2 hours. Test whether past the study still true. Use α=0.01. Solution

Step 1. 80 :H and 81 :H

Note that the alternative hypothesis is directional, we are performing a one tailed test, specifically a right tailed test.

Step 2. 010. (Level of significance) Step 3. Critical Region

96102502 .zzzz ./

Step 4. Test Statistic

Critical

Region

1.96

Since the population standard deviation is unknown but the sample size is large

enough, that is, 30n , we can substitute the sample standard deviation for “ ”, thus we have

022100

25200 .

ns

xz

Step 5. Since the value of the test statistic falls under the critical region, we reject the null hypothesis in favor of the alternative hypothesis. Step 6.

Since the null hypothesis is rejected, we say that there is sufficient evidence to say that the average life span of people living in Mindanao is greater than 70 years.

Exercise 1. Ritz grocery store declared that their average daily income is P 15, 000 with a standard deviation of P 2, 000. A random sample of 20 grocery stores of the same kind had been asked of their daily income and said to have P19, 000 on the average. If we assume that the daily income is normally distributed can we conclude that Ritz grocery store daily income declaration is right with 95% confidence? 2. Last 2000, Central Luzon farmers demanded more supply on the fertilizers the government is providing them. They said that the supply should be 20 kilos on the average for 10 hectares of land. In 2007, a random sample of 35 farmers was asked the number kilos of fertilizers a 10-hectare land would need. The result showed 24 kilos on the average with 3 kilos standard deviation. With 90% confidence is it true that the average kilos of fertilizers needed for a 10 hectare land have changed since their last demand? Assume normal distribution. 3. The daily average number of major defects detected per module by the Quality Assurance team that is considered normal is less than or equal to 6. To access the Software Development team’s quality of developed modules a random sample of 35 modules was tested and found out that the mean number of major defects per module is 8 with a standard deviation of 2 major defects. Based on the result with α=5% is the daily average number of major defects not normal? 4. The mayor of Las Piñas City wanted to hire a batch of teachers for his newly constructed elementary school. Based from previous studies by his education committee, the average age of elementary school teachers in Las Piñas is 40 years old with a standard deviation of 5. A sample of 36 newly hired elementary teachers was taken and the following information is obtained: average age is 35. Does this indicate that the average age of elementary school teachers decreased? Use a 0.05 level of significance and assume normality. 5. A random sample of 8 cigarettes of a Marlboro has an average nicotine content of 4.2 milligrams and a standard deviation of 1.4 milligrams. Is this I line with the manufacturer's

claim that the average nicotine content does not exceed 3.5 milligrams? Use a 0.01 level of significance and assume the distribution of nicotine contents to be normal. 6. A new process for producing synthetic diamonds can be operated at a profitable level only if the average weight of the diamonds is greater than 0.5 karat. To evaluate the profitability of the process, six diamonds are generated, with recorded weights: 0.46, 0.61, 0.52, 0.48, 0.57, and 0.54 karat.

a. Is there evidence to suggest that the variance of these measurements is greater than 0.02 karat?

b. Do the six measurements present sufficient evidence that the process will be profitable? Test at the 0.01 level.

Testing a value of a single population proportion, 𝝅

Suppose there are x successes in a random sample of size n drawn from a normal population. We wish to test whether the proportion of successes in a certain population is equal to some specified value.

Null and alternative hypothesis

H0 : 𝜋 = 𝜋0

H1 : 𝜋 ≠ 𝜋0; 𝜋 > 𝜋0; or 𝜋 < 𝜋0

Critical values/ critical region

For a one-tailed test: Reject H0 if 𝑧 < −𝑧𝛼 for a left tailed test. Reject H_0 if

𝑧 > 𝑧𝛼for a right tailed test.

For a two-tailed test: Reject H0 if 𝑧 < −𝑧𝛼2 or 𝑧 > 𝑧𝛼

2

Test statistic

n

pz

00

0

1

ˆ

where

n

xp ˆ (assumption 𝑛�̂� ≥ 5, 𝑛(1 − �̂�) ≥ 5)

Example

1. A chocolate manufacturer targets an 8 out of 10 public approval of their new chocolate recipe to release in the market. A random sample of 70 people where given a taste test and resulted a 75% approval of the product. Will the company release the product in the market with 0.05 level of significance?

Solution

Step 1. 800 .:H and 801 .:H

Note that the alternative hypothesis is non-directional, we are performing a two tailed test.

Step 2. 050. (Level of significance) Step 3. Critical Region

96102502

.zzzz .

96102502

.zzzz .

Step 4. Test Statistic

05.1

70

16.0

05.0

70

2.0*8.0

8.075.0

1

ˆ

00

0

n

pz

Step 5. Since the value of the test statistic falls under the critical region, we reject the null hypothesis in favor of the alternative hypothesis. Step 6. Since 1.05 is not greater than 1.96 , we do not reject Ho. There is no sufficient sample evidence to refute the claim of the Manufacturer.

Exercise

1. A new papaya soap, claims that 80% of women who used it observed skin whitening within two weeks of use. A known competitor of the said product surveyed if the claim is true. The survey result said that 6 out of 10 women observed whitening on their skin in two weeks of use. With α= 5% is the claim of the newly produced papaya soap true?

2. An environmental non-government organization (NGO) declared that 7 out of 10 endangered

birds in the country dies by hunting. The alarming report pushed the government environmental department to conduct their survey if such report is true to be able to put an immediate action to it. The result showed that 67% of the endangered bird species dies in hunting. With 90% confidence can the government environmental department conclude that the NGO’s report is true?

Critical

Region

Critical

Region

1.96 -1.96

3. The student government is preparing for this year school’s foundation day. They are having a hard time where to conduct the talent’s night to make sure there are enough seats for attendees. In the past, 70% of the total number of students attends talent’s night. They conducted a survey on 120 students and found out that 80 will be attending. Test whether the number of attendees this year will be the like the past with 95% confidence.

Testing the value of a population variance, 2

Suppose we wish to test whether the population variance is equal to a specified value 20 .

Null and alternative hypothesis

H0 : 2 = 20

H1 : 2 < 20 ; 2 > 2

0 ; or 2 20

Critical values/region

For a one-tailed test: Reject H0 if 21

2 for a left tailed test with df = n -1

22 for a right tailed test with df = n -1.

For a two-tailed test: Reject H0 if : 2

1

2

2

or 22

2

with df = n -1.

Test statistic:

20

2)1(2

sn

Example

1. A known candy manufacturer claims that their high-speed machines minimize the defected

candies produced. They claim that their daily production only contains a variance of 100 defected candies. A random sample of 15 days candy production resulted a sample

variance of 165 defected candies. Using =5% is the claim true?

Step 1. 10020 :H and 1002 :Ha

Note that the alternative hypothesis is non-directional, we are performing a one tailed test; specifically, a right tailed test.

Step 2. 050. (Level of significance) Step 3. Critical Region

919161101 22050

222 .)()n( .

5.62873n .

22975

22

1

2 1412

or

26.11895)()n( .

220250

222 1412

Step 4. Test Statistic

123

100

165115120

22 .

s)n(

Step 5. Since the value of the test statistic does not fall under the critical region, we do not reject the null hypothesis in favor of the alternative hypothesis. Step 6. Since 23.1 is not greater than 26.11895, we do not reject Ho. There is no sufficient sample evidence to refute the claim of the manufacturer that the variance of defected candies is 100.

2. The agency that monitors earthquakes said intensity of earthquakes in the Philippines has

a variance of 2. A study was conducted on 30 earthquakes that hit the country since 1980.

A standard deviation of 1.2 resulted. With =1% is the agency correct?

Exercise

1. A bottling company wants to access the state of their machines in terms of defects or over

spilling on their daily production of bottled juices. If the over spilling does not exceed a variance of 20 bottled juices daily, the machines are said to be in good condition. A random sample of 60 juices resulted standard deviation of 25 bottles. Using α= 5% are the machines still in good condition?

2. A publishing company considers a book to be error free and ready to be out in the market if

the variance error is less than 5 per page. A random sample of 30 pages was gathered on a book and found a variance of 7 errors. Using α= 10% is the book ready to be out in the market?

3. Ten encoders from Clark Data Center Inc. have applied for a higher position in the typing pool of company and took a typing test. The following are the duration of each encoders for the test: 75, 70, 59, 60, 63, 55, 52, 70, 45, and 85 seconds. Construct a 90% confidence interval for the true variance of the times required by encoders to complete the test paragraph.

4. In an experiment with rats, a behavioral scientist used an auditory signal hat food is available through an open door in the cage. The scientist counted the number of trials needed by each rat to learn to recognize the signal. Assuming that the population of number of trials is approximately normal, calculate a 95% confidence interval for the population variance with the given data below: 18 19 15 14 18 12 14 21 14 11 Does this interval support the claim that the variability of this data set is about 2?

Summary of the Test Statistics and Critical Regions for Hypothesis Testing

0H Value of Test Statistic aH Critical Region

0

n

xz

0 , known or 30n

0

0

0

zz

zz

2

zz and 2

zz

0 n

s

xt 0 , =n-1,

unknown and 30n

0

0

0

tt

tt

2

tt and 2

tt

0

n

pz

00

0

1

ˆ

where

n

xp ˆ

0

0

0

zz

zz

2

zz and 2

zz

2 = 20

20

2)1(2

sn

2 < 20

2 > 20

2 20

21

2

22

2

1

2

2

and 22

2

Chapter Review Choose the letter of the correct answer. (For numbers 1-4)The average monthly income of a gasoline boy is said to be P6,500. A random sample was conducted to 35 gasoline boys and found out that the mean monthly income is P 6,000 with P 300 standard deviation. With α=10% can we conclude that the average monthly income of a gasoline boy is P6,500?

1. What is the correct alternative hypothesis? a. The average monthly income of a gasoline boy is P6,500. b. The average monthly income of a gasoline boy is less than P6,500. c. The average monthly income of a gasoline boy is not P6,500. d. The average monthly income of a gasoline boy is at least P6,500.

2. Base on the alternative what type of test is it? a. one-tailed b. two-tailed c. both a and b d. none

3. What is the value for Z ? a. 11.23 b. 9.86 c. 21.05 d. 12.57

4. What is the decision? a. Reject Ho. b. Reject H1. c. Both a and b d. None

(For items 5-8)A certain music club believes that 78% of musicians plays guitar. In random sample of 25 musicians 22 of them play guitar. With 95% confidence is the belief of the music club right?

5. What is the value of po ? a. 0.78 b. 0.88 c. 0.22 d. 0.12

6. What is the value of qo ? a. 0.78 b. 0.88 c. 0.22 d. 0.12

7. What is the value of Zα/2? a. 1.64 b. 1.645 c. 1.96 d. 1.7

8. What is the decision? a. Reject Ho. b. Reject H1. c. Both a and b d. None

9. If σ2 = 2.35, s2 = 3.56 and n=33 what is the value of χ2? a. 48.91 b. 48.48 c. 21.123 d. 22.23

10. If n= 20 and α = 10% what is the tabular value of χ2 if we are dealing with a two-tailed test? a. 28.412 b. 30.144 c. 31.410 d. 38.582

11. Commuter students at the University of the Philippines claim that the average distance they have to commute to campus is 26 kilometers per day. A random sample of 16 commuter students was surveyed and resulted to the following data: The average distance of 31 km and a variance of 64. The value of the test statistic for this is

a. z = -2.5. b. z = 0.0394. c. t = 2.5. d. t = 1.25.

12. For a small-sample left-tailed test for the population mean, given the sample size was 18 and

010. . The critical (table) value for this test is a. -2.878. b. -2.552. c. 2.878. d. -2.567.

13. Rejection of the null hypothesis when it is true is called a. type I error b. type II error c. no error d. statistical error 14. The type of test is determined by

a. Ho b. Ha

c. 𝛼

d. 1 − 𝛼 15. This denotes the probability of committing a TYPE I error.

a. Ho b. Ha

c. 𝛼

d. 1 − 𝛼 16. Under the Philippine judicial system, an accused person is presumed innocent until proven guilty. Suppose we wish to test the hypothesis that the accused is innocent (H0) against the alternative that he is guilty (H1). A type I error is committed, if any, if the court

a. convicts the accused when, in fact, he is innocent? b. convicts the accused when, in fact, he is guilty? c. acquits the accused when, in fact, he is innocent? d. acquits the accused when, in fact, he is guilty?

17. Which statement is/are correct?

I. A null hypothesis is a claim (or statement) about a population parameter that is assumed to be true until it declared false.

II. An alternative hypothesis is a claim about a population parameter that will be true if the null hypothesis is false.

a. I b. II c. Both I and II d. Neither I and II For numbers 18-20: According to a company’s records, the average length of all long-distance calls places through this company in a year is 12.55 minutes. The company’s management wants to check if the mean length of the current long-distance calls is different from 12.55 minutes. A random sample of 50 such calls placed through this company produced a mean length of 13.55 minutes with a standard deviation of 2.65 minutes. Use a 0.05 level of significance. 18. What is the computed test value?

a. z = 2.67 b. t = 2.67 c. z = -2.67 d. t = -2.67 19. What is the corresponding critical value?

a. 1.645 b. 2.575 c. 2.633 d. 1.96 20. What is the type of test to be used?

a. right tailed test b. left tailed test c. two tailed test d. cannot be determined