Math 156 Summer 04 Final Exam

1. Consider the data set below - neck size in cms31.5 31.8 32.0 36.0 37.0 34.4 31.332.0 31.3 30.2 30.1 32.0 31.1 33.633.9 32.6 31.7 39.1 35.8 30.4 27.642.7 37.2 38.4 32.2 37.9

a) Make a stem and leaf plot

b) Find the 5 number summary. Indicate at what positions the values are found in thesorted data.

c) Make a boxplot of this data

d) Make a histogram using 6 classes.

e) Use your calculator to find the mean and standard deviation

f) Would you use the mean and standard deviation, or the five number summary todescribe this data set? Which is best and why?

g) Describe the shape of the data.

2. Use x =1n

xi∑ and s = 1n −1

(xi − x )2∑ to find the mean and standard deviation of:

2.5 3.1 2.2 4.3 4.1

3. Define the terms:

Central Limit Theorem

Law of Large Numbers

Sampling distribution of a statistic

Lurking variable

Confounding

Statistically significant

4. When are the following valid computations?

a) P(A and B) = P(A)P(B) b) P(A or B) = P(A) + P(B)

5. Recent SAT results had a normal distribution with mean 500 and standard deviation100.

a) Use the 68-95-99.7 rule to give a range of scores including nearly all scores

b) What is the probability a randomly chosen student scores between 450 and 600?

c) How high a score does a student need to be in the top 15%?

d) What is the chance the average of 5 students' scores is between 450 and 600?

e) What is the probability that the average of 80 randomly chosen students' scores isbelow 490?

f) What would the mean and standard deviation of the sampling distribution be for theaverage of 80 scores?

6.  Consider the 2 seater data attached to the end of the test

a) Which of the seven variables are categorical? Quantitative?

b) Make a scatterplot using city mileage as the explanatory variable

c) Describe the relationship (form direction strength)

d) Find the equation of the regression line y = a + bx using b = rsysx

and a = y − bx .

e) Find the coordinates of two points you would use to plot the regression line.

f) What hwy mileage would be predicted for a city mileage of 30 mpg?

g) What is this type of prediction in f) called?

h) Find the residual for the Porsche Boxster S

i) What percent of the variation in hwy mileage can be attributed to a linear relationshipbetween mileages?

j) Which points probably have the most influence on the regression line?

7. Consider the probabilities listed for colors of a randomly selected M&M candy:Brown Red Green Blue Orange.15 .2 .1 .2 ??

a) What must be the orange probability?

b) What is the chance an M&M is not Red?

c) If you select one M&M and then another, what is the chance your selection consists ofan Orange followed by a Green?

d) In a sample of 10 M&Ms, what is the chance there are no Red candies?

8. a) In 6 rolls of a die, what is the probability of rolling at least 2 sixes?

b) In 60 rolls of a die, what is the approximate probability of rolling at least 20 sixes?

9. Use the tables to perform an SRS of size 12 from a list of 900 people. Use line 102.

10. Diagram the experiment and assign treatments to 30 subjects in an experimentcomparing 3 kinds of therapy. Use line 120.

2 SEATER DATA

Manufacturer carline name displ cyl drv cty hwyBMW Z4 ROADSTER 2.5 6 R 20 28MAZDA MX-5 MIATA 1.8 4 R 23 28PORSCHE BOXSTER 2.7 6 R 20 29TOYOTA MR2 1.8 4 R 26 32ACURA NSX 3.2 6 R 17 24AUDI TT ROADSTER 1.8 4 4 20 28BMW Z4 ROADSTER 3 6 R 21 29CHEVROLET CORVETTE 5.7 8 R 19 28CHRYSLER CROSSFIRE 3.2 6 R 17 25DODGE VIPER CONVERTIBLE 8.3 10 R 12 20HONDA S2000 2.2 4 R 20 25MAZDA MX-5 MIATA 1.8 4 R 23 28MERCEDES-BENZ SLK230 2.3 4 R 21 29MERCEDES-BENZ SLK320 3.2 6 R 19 26NISSAN 350Z 3.5 6 R 20 26PORSCHE BOXSTER S 3.2 6 R 18 26PORSCHE CARRERA 2 911 GT3 3.6 6 R 15 23PORSCHE TURBO 2 911 GT2 3.6 6 R 15 23

N MEAN MEDIAN TRMEAN STDEV SEMEANcty 18 19.222 20.000 19.250 3.282 0.774hwy 18 26.500 27.000 26.562 2.854 0.673

MIN MAX Q1 Q3cty 12.000 26.000 17.000 21.000hwy 20.000 32.000 24.750 28.250 Correlation of cty and hwy = 0.917

 

 

 

Formulas

Regression b = rsy

sx

, a = y − bx

Binomial µ = np σ = np(1 − p) P(X = k) = n nCk k pk (1− p)n− k

z-score

z = x − µσ

Confidence intervals

x ± z * σn

n =z *σm

⎛ ⎝ ⎜

⎞ ⎠ ⎟ 2

�

x ± t* sn

x − y ± t * sx2

nx

+sy2

ny

ˆ p ± z *ˆ p (1 − ˆ p )

n n =

z *m

⎛ ⎝ ⎜

⎞ ⎠ ⎟ 2

p* (1 − p*)

p = successes + 2n + 4

p ± z * p(1− p)n

Hypothesis tests

z = x − µσ / n

t = x − µs / n

t = xdsd / n

z =ˆ p − p

p(1− p)n

t = x − y sx2

nx

+sy2

ny

χ 2 =(observed - expected)2

expected∑ 

Math 156 Final Exam Part 2

1. a) If you were to examine 100 independently generated 95% confidence intervals, howmany would you expect to be providing correct results?

b) Your friend tells you that a 95% confidence interval gives a range of values where wewould likely find the sample mean in 95% of sampling situations. What do you tell yourfriend?

c) A 95% confidence interval for the mean contains the number 6.5. What hypothesis testand what conclusion can you state based in this evidence?

2. Produce the required confidence intervals:

a) 90% confidence, for the population mean, if x = 11.2, n = 50, σ = 4.3

b) 95% confidence, for the population mean, if x = 26.5, n = 15, s = 2.5

c) 99% confidence, for difference of the population means, if x1 = 26, n1 = 12, s1 = 1.5x2 = 24.1, n1 = 16, s1 = 1.75

d) 98% confidence, for population proportion, if 56 of 200 polled are in favor ofincreasing taxes.

3. Perform the hypothesis tests. Quote a p-value or approximate p-value in your results.Data: x = 10.1, n = 200, σ = .95

a) H0 :µ = 10 Ha :µ < 10

b) H0 :µ = 10 Ha :µ > 10

c) H0 :µ = 10 Ha :µ ≠ 10

4. Perform the hypothesis tests. Quote a p-value or approximate p-value in your results.Data: x = 58.1, n = 100, s = 12

a) H0 :µ = 60 Ha :µ < 60

b) H0 :µ = 60 Ha :µ > 60

c) H0 :µ = 60 Ha :µ ≠ 60

5. Find the sample sizes needed if:

a) the margin of error in a 90% z-interval for the mean is to be .04 , and σ = 1.22

b) the margin of error in a 99% z-interval for the population proportion is to be .04 , andyou think the population proportion is around 75%.

6. Define the terms

Confidence level

Margin of error

Standard error of the mean

p-value

Significant at level alpha

7. What conditions need to be verified for results to be valid that are based on:

a) ˆ p ± z *ˆ p (1 − ˆ p )

n

b) z = x − µσ / n

c) t = x − µs / n

d) χ 2 =(observed - expected)2

expected∑

8. Could you conclude that more than half of all CSU-Pueblo students are in favor of afee to support the Math Learning Center, based on an SRS of 200 students yielding 124in favor of the fee? Explain.

9. Perform a chi-square test of the data below. State null and alternative hypotheses, carryout the test and quote a p-value in your results.

men womenhate pets 22 15tolerate pets 26 21love pets 51 52