SLCC MATH 1040 FINAL EXAM FALL 2016 NAME: _________________________

Form A INSTRUCTOR: ________________________

TEST INSTRUCTIONS:

This exam consists of both "multiple choice" and "free response" questions. No partial credit is

available on "multiple choice" problems. For "free response" questions, you must support your

answer with relevant work. Work for these problems must be shown on the exam. Work

carefully and neatly, appropriately justifying your solutions. Partial credit may be awarded for

relevant work on "free response" problems.

You may not use your own notes, books, headphones, phones, nor any device that connects to

the internet. Any calculator may be used and formulas are provided. You have 120 minutes to

complete this exam.

A facilities manager at SLCC believes that his supply of light bulbs has too many defects. The

manager’s null hypothesis is that the supply of light bulbs has a defect rate of 0.07p (the

defect rate stated by the light bulb manufacturer). The manager takes a random sample and

finds that 27 out of the 300 in his sample are defective. Suppose he does a hypothesis test with a

significance level of 0.05 . Symbolically, his null and alternative hypotheses are:

0 : 0.07H p and 1 : 0.07H p .

1. (4 points) Choose the statement that best describes the significance level in the context of

this hypothesis test.

a. The significance level of 0.05 is the defect rate we believe is the true defect rate.

b. The significance level of 0.05 is the test statistic that we will use to compare the observed

outcome to the null hypothesis.

c. The significance level of 0.05 is the probability of concluding that the defect rate is equal

to 0.07 when in fact the defect rate is greater than 0.07.

d. The significance level of 0.05 is the probability of concluding that the defect rate is

higher than 0.07 when in fact the defect rate is equal to 0.07.

2. (4 points) For the hypothesis test, the facilities manager calculates a p-value of 0.087. Choose

the correct interpretation for the p-value.

a. The p-value tells us that the probability of concluding that the defect rate is equal to 0.07,

when in fact the rate is greater than 0.07, is approximately 0.087.

b. The p-value tells us that if the defect rate is 0.07, then the probability that the manager

will have 27 or more defective light bulbs out of 300 is approximately 0.087. At the

significance level of 0.05, this would not be an unusual outcome.

c. The p-value tells us that the true population rate of defective light bulbs is

approximately 0.087.

d. None of these.

MATH 1040 FINAL EXAM FALL 2016 A page 2

3. (4 points) For what kinds of variables would it be appropriate to create a histogram?

a. Only for a single quantitative variable

b. Only for a single categorical variable

c. Either for two quantitative or for two categorical variables together

d. It varies according to the situation

4. (4 points) The monthly salaries of the three people working in a small firm are $3,500,

$4,000, and $4,500. Suppose the firm makes a profit and everyone gets a $100 raise. How, if

at all, would the mean and standard deviation of the three salaries change?

a. Both would stay the same

b. The mean would increase and the standard deviation would stay the same

c. The mean would stay the same and the standard deviation would increase

d. The mean would stay the same and the standard deviation would decrease

e. The mean would increase and the standard deviation would decrease

f. Both would increase

g. Cannot be answered without doing calculations

5. (4 points) Given is a relative frequency histogram for the price of homes from a simple

random sample of homes sold during 2010. Based on this histogram, is a large sample

30 necessary to conduct a hypothesis test about the mean sale price? Why?

a. No; the data appear to be normally distributed

b. Yes; the data do not appear to be normally distributed, but rather skewed left

c. Yes; the data do not appear to be normally distributed, but rather skewed right

d. Yes; the data do not appear to be normally distributed but bimodal

MATH 1040 FINAL EXAM FALL 2016 A page 3

6. (4 points) Suppose a random sample of 50 college students is asked if they regularly eat

breakfast. A 95% confidence interval for the proportion of all students that regularly eat

breakfast is found to be 0.70 to 0.90. If a 99% confidence interval was calculated instead,

how would it differ from the 95% confidence interval?

a. The 99% confidence interval would have the same width as the 95% confidence interval.

b. The 99% confidence interval would be wider.

c. The 99% confidence interval would be narrower.

d. There is no way to tell if the 99% confidence interval would be wider or narrower than

the 95% confidence interval.

7. (4 points) A recent Gallup poll showed the president’s approval rating at 60%. Some friends

use this information (along with the sample size from the poll) and find confidence intervals

for the proportion of all adult Americans that approve of the president’s performance. For

the following confidence intervals, which one could have been done correctly? (The others

were definitely done incorrectly.)

a. (0.57, 0.63)

b. (0.60, 0.66)

c. (0.62, 0.69)

d. (0.47, 0.53)

8. (4 points) The following linear regression model was created to show the association

between the number of massages received per month and self-predicted stress level:

ˆ 10 0.02y x where x is the number of massages per month and y is the predicted stress

level. The coefficient of determination for the model is 2 0.066R or 6.6%. Choose the true

statement regarding this model.

a. The model shows that getting even one massage a month will decrease your stress level

by 0.02 on average.

b. The model shows that there is a negative association between stress level and number of

monthly massages, but the relatively small coefficient of determination 2R suggests

that very little of the variation is explained by the model, so the model is probably not

very good at predicting stress level.

c. The model shows that on average a person getting no massages will have a stress level

of 10.

d. All of these are true statements.

MATH 1040 FINAL EXAM FALL 2016 A page 4

9. (4 points) A General Social Survey (GSS) asks many questions of American adults regularly.

Identify which of the following questions elicit an answer that is coded as a quantitative

variable and which as a categorical variable. (Circle the correct response for each.)

a. Are you currently employed? QUANTITATIVE CATEGORICAL

b. What is your annual income? QUANTITATIVE CATEGORICAL

c. Do you have your driver’s license? QUANTITATIVE CATEGORICAL

d. What is your age? QUANTITATIVE CATEGORICAL

10. (6 points) The medal counts for the top 10 countries at the Rio Olympics are given below,

ranked by total medals (according to rio2016.com 10/17/2016).

Country

Total

Medals Gold Silver Bronze

1 United States 121 46 37 38

2 China 70 26 18 26

3 England 67 27 23 17

4 Russia 56 19 18 19

5 Germany 42 17 10 15

6 France 42 10 18 14

7 Japan 41 12 8 21

8 Australia 29 8 11 10

9 Italy 28 8 12 8

10 Canada 22 4 3 15

a. Report the 5-number summary for Total Medals won by the top 10 countries at the Rio

Olympics.

b. Create a boxplot for Total Medals won by the top 10 countries at the Rio Olympics. Be

sure to scale it appropriately.

MATH 1040 FINAL EXAM FALL 2016 A page 5

11. (6 points) USA Today (January 24, 2012) reported that ownership of tablet computers and e-

readers is soaring. Suppose you want to estimate the proportion of students at SLCC who

own at least one tablet or e-reader. What sample size would you use in order to estimate this

proportion at the 95% confidence level, with a margin of error of 3%?

12. (6 points) MATCHING: Choose which statistical procedure (from the second column)

would most likely be used to answer each of the three research questions in the first

column. Assume all requirements have been met for using the procedures.

I. Does the proportion of females

participating in major college sports

(e.g. basketball), differ from the

proportion participating in minor

sports (e.g. swimming)?

II. Is the number of females participating

in major college sports (e.g. basketball)

associated with the number of

scholarships available to female

athletes?

III. What is the average number of females

participating in major college sports at

SLCC?

a. Test the difference between two

proportions

b. Construct a confidence interval

estimate for the proportion.

c. Test one mean against a

hypothesized value.

d. Test the difference in means

between two paired or dependent

samples.

e. Construct a confidence interval

estimate for the mean.

f. Test if the correlation coefficient is

significant.

MATH 1040 FINAL EXAM FALL 2016 A page 6

13. (4 points) The data given in the table represent the amount of pressure (psi) exerted by a

stamping machine ( x ), and the amount of scrap brass shavings (in pounds) that are

collected from a machine each hour ( y ). A scatterplot of the data confirms that there is a

linear association and the correlation coefficient is found to be significant.

x y

2.00 2.30

7.80 15.14

14.51 28.65

2.80 4.15

4.01 6.35

6.21 10.52

11.84 24.05

5.11 8.75

11.67 22.22

8.70 17.02

a. Find the equation of the regression line. Round values to the nearest thousandth.

b. What is the best predicted amount of scrap brass shavings for a stamping machine

pressure of 6.25? (Report to the nearest hundredth of a pound.)

14. (6 points) Suppose the amount of money spent by SLCC students each semester on

textbooks is normally distributed with a mean of $195 and a standard deviation of $20. If

you were to randomly sample 100 SLCC students from this population, what is the

probability that the sample mean x amount spent will be between $193 and $197?

Support your answer.

MATH 1040 FINAL EXAM FALL 2016 A page 7

15. (6 points) A survey of 35,353 people asked questions about their happiness and health. One

would think that health plays a role in one’s happiness. You are interested in determining if

there is evidence that healthier people tend also to be happier, treating level of health as the

explanatory variable. A conditional distribution of happiness by health is given below.

Level of Health L

evel

of

Hap

pin

ess

Poor Fair Good Excellent

Not too happy 0.35 0.21 0.10 0.07

Pretty happy 0.48 0.58 0.61 0.47

Very happy 0.17 0.21 0.29 0.46

Totals 1 1 1 1

a. Draw a bar graph of the conditional distribution. Draw three bars, for each level of

happiness (the first bar for people who are not too happy, the second bar for people who

are pretty happy, and the third bar for people who are very happy). The horizontal axis

will represent health and the vertical axis will represent the relative frequency.

b. Does your bar graph indicate that there is an association between health and happiness?

(circle one)

i. No, the change in the percent that are very happy cancels out the change in the

percent who are not too happy

ii. Yes; the percent that are not happy decreases when health improves.

iii. No; the percent that are very happy decreases when health improves.

iv. Yes; the percent that are pretty happy stays constant when health improves.

v. No; the percent that are not happy increases when health improves.

MATH 1040 FINAL EXAM FALL 2016 A page 8

16. (8 points) In the initial test of the Salk vaccine for polio, 400,000 children were selected and

divided into two groups of 200,000. One group was vaccinated with the Salk vaccine while

the second group was vaccinated with a placebo. Of those vaccinated with the Salk vaccine,

33 later developed polio. Of those receiving the placebo, 115 later developed polio.

A hypothesis test is performed at the 0.01 significance level to test the hypothesis that

the Salk vaccine is effective in lowering the polio rate. The test yields a standardized test

statistic of 0 6.742z and a p-value of nearly zero.

a. What are the hypotheses for this test?

b. What decision should be made about the null hypothesis? (circle one)

REJECT 0H or FAIL TO REJECT 0H

c. Does the test provide evidence to support the claim that the Salk vaccine is effective

in lowering the polio rate? Support your decision by explaining the statistical

evidence.

17. (4 points) A set of 8 observations yields a linear correlation coefficient of 0.335r . Does

this indicate a significant linear relationship between the variables? Respond with a

complete sentence, include the value of the correlation coefficient and the appropriate

critical value in your response.

MATH 1040 FINAL EXAM FALL 2016 A page 9

18. (6 points) Clarinex-D is a medication whose purpose is to reduce the symptoms associated

with a variety of allergies. In clinical trials of Clarinex-D, 5% of the patients in the study

experienced insomnia as a side effect. We will randomly select 240 users of Clarinex-D.

a. Demonstrate that this experiment meets all the criteria for a Binomial Experiment.

b. How many of the 240 users of Clarinex-D would we expect to experience insomnia as a

side effect?

c. Would it be unusual to observe exactly 20 patients experiencing insomnia as a side effect

in 240 trials of the probability experiment? Why? Include the appropriate probability

computation in your response.

Statistics Formulas and Tables Descriptive Statistics Probability

� = ∑ ��� mean

� = ∑����∑ � approximate mean from grouped data

� = �∑��� ��� � sample standard deviation

� = �∑��� ��∙��∑ � � approx. std. dev. from grouped data

Sample variance = ��

Interquartile Range: ��� = �� − ��

Lower fence = �� − 1.5����

Upper fence = �� + 1.5����

General Addition Rule

��� �� = ��� + �� − ��� !"#

multiplication rule for independent events

��� !"# = ��� ∙ ��

multiplication rule for dependent events

��� !"# = ��� ∙ �� |�

complement rule ���% = 1 − ���

�&� = �!�� &! Permutations (no elements alike)

�!�(!��!⋯�*! Permutations ("� alike, ⋯)

+&� = �!�� &!&! Combinations

Probability Distributions Normal Distribution and Sampling Distributions

mean (expected value) of a discrete random variable

,� = ∑-� ∙ ���.

standard deviation of a discrete random variable

/� = 0∑�� − ,� ∙ ���

��� = +�� ∙ 1� ∙ �1 − 1� � Binomial probability

,� = " ∙ 1 mean for a Binomial distribution

/� = 0" ∙ 1 ∙ �1 − 1 std. dev. for a Binomial distribution

2 = � 34 standard normal score

mean and std. dev. of the sampling distribution of � ,� = , /� = 4

√� Standard Error

mean and std. dev. of the sampling distribution of 1

,78 = 1 /78 = �7�� 7�

Estimating a Population Parameter Hypothesis Testing

Proportion:

�� �( )

2

1p pp z

nα

−±

( )

2

2

2

0.25zn

E

α= sample size, 1 unknown

( ) ( )

2

2

2

ˆ ˆ1z p pn

E

α −= sample size, 1 known

Mean:

2

sx t

nα±

2

2z s

nE

α =

sample size

Difference of Two Proportions (Independent Samples):

� �( )� �( ) � �( )1 1 2 2

21 2

1 2

1 1p p p pp p z

n nα

− −− ± +

proportion – one population

( )0

0

0 0

ˆ

1

p pz

p p

n

−=

−

mean – one population 0

0

xt

s n

µ−=

two population proportions (Independent Samples)

� �( ) ( )

� �( )

1 21 2

0

1 2

1 11

p p p pz

p pn n

− − −=

− +

where � 1 2

1 2

x xp

n n

+=

+

two population proportions (Dependent Samples)

12 21

0

12 21

1f fz

f f

− −=

+

Linear Correlation and Regression

Correlation Coefficient � = ∑9:�;:<=: >?@�;@<

=@ A� �

B8 = CD + C�� estimated eqn. of linear regression line

�� = �� coefficient of determination

Residual = B − B8