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# Hypothesis Testing of Single Mean and Single Proportion: Homework

Summary: This module provides a homework of Hypothesis Testing of Single Mean and Single Proportion as a part of Collaborative Statistics collection (col10522) by Barbara Illowsky and Susan Dean.

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## Exercise 1

Some of the statements below refer to the null hypothesis, some to the alternate hypothesis.

State the null hypothesis, HoHo size 12{H rSub { size 8{o} } } {}, and the alternative hypothesis, HaHa size 12{H rSub { size 8{a} } } {}, in terms of the appropriate parameter (μμ size 12{μ} {} or pp size 12{p} {}).

• a. Americans work an average of 34 years before retiring.
• b. At most 60% of Americans vote in presidential elections.
• c. The average starting salary for San Jose State University graduates is at least $100,000 per year. • d. 29% of high school seniors get drunk each month. • e. Fewer than 5% of adults ride the bus to work in Los Angeles. • f. The average number of cars a person owns in her lifetime is not more than 10. • g. About half of Americans prefer to live away from cities, given the choice. • h. Europeans have an average paid vacation each year of six weeks. • i. The chance of developing breast cancer is under 11% for women. • j. Private universities cost, on average, more than$20,000 per year for tuition.

### Solution

• a. H o : μ = 34 H o : μ = 34 size 12{H rSub { size 8{o} } :μ="34"} {} ; H a : μ 34 H a : μ 34 size 12{H rSub { size 8{a} } :μ <> "34"} {}
• c. H o : μ 100 , 000 H o : μ 100 , 000 size 12{H rSub { size 8{o} } :μ >= "100","000"} {} ; H a : μ < 100 , 000 H a : μ < 100 , 000 size 12{H rSub { size 8{a} } :μ<"100","000"} {}
• d. H o : p = 0 . 29 H o : p = 0 . 29 size 12{H rSub { size 8{o} } :p=0 "." "29"} {} ; H a : p 0 . 29 H a : p 0 . 29 size 12{H rSub { size 8{a} } :p <> 0 "." "29"} {}
• g. H o : p = 0 . 50 H o : p = 0 . 50 size 12{H rSub { size 8{o} } :p=0 "." "50"} {} ; H a : p 0 . 50 H a : p 0 . 50 size 12{H rSub { size 8{a} } :p <> 0 "." "50"} {}
• i. H o : p 0 . 11 H o : p 0 . 11 size 12{H rSub { size 8{o} } :p >= 0 "." "11"} {} ; H a : p < 0 . 11 H a : p < 0 . 11 size 12{H rSub { size 8{a} } :p<0 "." "11"} {}

## Exercise 2

For (a) - (j) above, state the Type I and Type II errors in complete sentences.

### Solution

• a. Type I error: We believe the average is not 34 years, when it really is 34 years. Type II error: We believe the average is 34 years, when it is not really 34 years.
• c. Type I error: We believe the average is less than $100,000, when it really is at least$100,000. Type II error: We believe the average is at least $100,000, when it is really less than$100,000.
• d. Type I error: We believe that the proportion of h.s. seniors who get drunk each month is not 29%, when it really is 29%. Type II error: We believe that 29% of h.s. seniors get drunk each month, when the proportion is really not 29%.
• i. Type I error: We believe the proportion is less than 11%, when it is really at least 11%. Type II error: WE believe the proportion is at least 11%, when it really is less than 11%.

## Exercise 3

For (a) - (j) above, in complete sentences:

• a. State a consequence of committing a Type I error.
• b. State a consequence of committing a Type II error.

## Directions:

For each of the word problems, use a solution sheet to do the hypothesis test.

## Note:

If you are using a student-t distribution for a homework problem below, you may assume that the underlying population is normally distributed. (In general, you must first prove that assumption, though.)

## Exercise 4

A particular brand of tires claims that its deluxe tire averages at least 50,000 miles before it needs to be replaced. From past studies of this tire, the standard deviation is known to be 8000. A survey of owners of that tire design is conducted. From the 28 tires surveyed, the average lifespan was 46,500 miles with a standard deviation of 9800 miles. Do the data support the claim at the 5% level?

## Exercise 5

From generation to generation, the average age when smokers first start to smoke varies. However, the standard deviation of that age remains constant of around 2.1 years. A survey of 40 smokers of this generation was done to see if the average starting age is at least 19. The sample average was 18.1 with a sample standard deviation of 1.3. Do the data support the claim at the 5% level?

### Solution

• e. z = 2 . 71 z = 2 . 71 size 12{z= - 2 "." "71"} {}
• f. 0.0034
• h. Decision: Reject null; Conclusion: μ < 19 μ < 19 size 12{μ<"19"} {}
• i. ( 17 . 449 , 18 . 757 ) ( 17 . 449 , 18 . 757 ) size 12{ $$"17" "." "449","18" "." "757"$$ } {}

## Exercise 6

The cost of a daily newspaper varies from city to city. However, the variation among prices remains steady with a standard deviation of 6¢. A study was done to test the claim that the average cost of a daily newspaper is 35¢. Twelve costs yield an average cost of 30¢ with a standard deviation of 4¢. Do the data support the claim at the 1% level?

## Exercise 7

An article in the San Jose Mercury News stated that students in the California state university system take an average of 4.5 years to finish their undergraduate degrees. Suppose you believe that the average time is longer. You conduct a survey of 49 students and obtain a sample mean of 5.1 with a sample standard deviation of 1.2. Do the data support your claim at the 1% level?

### Solution

• e. 3.5
• f. 0.0005
• h. Decision: Reject null; Conclusion: μ > 4 . 5 μ > 4 . 5 size 12{μ>4 "." 5} {}
• i. ( 4 . 7553 , 5 . 4447 ) ( 4 . 7553 , 5 . 4447 ) size 12{ $$4 "." "7553",5 "." "4447"$$ } {}

## Exercise 8

The average number of sick days an employee takes per year is believed to be about 10. Members of a personnel department do not believe this figure. They randomly survey 8 employees. The number of sick days they took for the past year are as follows: 12; 4; 15; 3; 11; 8; 6; 8. Let xx size 12{x} {} = the number of sick days they took for the past year. Should the personnel team believe that the average number is about 10?

## Exercise 9

In 1955, Life Magazine reported that the 25 year-old mother of three worked [on average] an 80 hour week. Recently, many groups have been studying whether or not the women's movement has, in fact, resulted in an increase in the average work week for women (combining employment and at-home work). Suppose a study was done to determine if the average work week has increased. 81 women were surveyed with the following results. The sample average was 83; the sample standard deviation was 10. Does it appear that the average work week has increased for women at the 5% level?

### Solution

• e. 2.7
• f. 0.0042
• h. Decision: Reject Null
• i. ( 80 . 789 , 85 . 211 ) ( 80 . 789 , 85 . 211 ) size 12{ $$"80" "." "789","85" "." "211"$$ } {}

## Exercise 10

Your statistics instructor claims that 60 percent of the students who take her Elementary Statistics class go through life feeling more enriched. For some reason that she can't quite figure out, most people don't believe her. You decide to check this out on your own. You randomly survey 64 of her past Elementary Statistics students and find that 34 feel more enriched as a result of her class. Now, what do you think?

## Exercise 11

A Nissan Motor Corporation advertisement read, “The average man’s I.Q. is 107. The average brown trout’s I.Q. is 4. So why can’t man catch brown trout?” Suppose you believe that the average brown trout’s I.Q. is greater than 4. You catch 12 brown trout. A fish psychologist determines the I.Q.s as follows: 5; 4; 7; 3; 6; 4; 5; 3; 6; 3; 8; 5. Conduct a hypothesis test of your belief.

### Solution

• d. t 11 t 11 size 12{t rSub { size 8{"11"} } } {}
• e. 1.96
• f. 0.0380
• h. Decision: Reject null when a = 0 . 05 a = 0 . 05 size 12{a=0 "." "05"} {} ; do not reject null when a = 0 . 01 a = 0 . 01 size 12{a=0 "." "01"} {}
• i. ( 3 . 8865 , 5 . 9468 ) ( 3 . 8865 , 5 . 9468 ) size 12{ $$3 "." "8865",5 "." "9468"$$ } {}

## Exercise 12

Refer to the previous problem. Conduct a hypothesis test to see if your decision and conclusion would change if your belief were that the average brown trout’s I.Q. is not 4.

## Exercise 13

According to an article in Newsweek, the natural ratio of girls to boys is 100:105. In China, the birth ratio is 100: 114 (46.7% girls). Suppose you don’t believe the reported figures of the percent of girls born in China. You conduct a study. In this study, you count the number of girls and boys born in 150 randomly chosen recent births. There are 60 girls and 90 boys born of the 150. Based on your study, do you believe that the percent of girls born in China is 46.7?

### Solution

• e. -1.64
• f. 0.1000
• h. Decision: Do not reject null
• i. ( 0 . 3216 , 0 . 4784 ) ( 0 . 3216 , 0 . 4784 ) size 12{ $$0 "." "3216",0 "." "4784"$$ } {}

## Exercise 14

A poll done for Newsweek found that 13% of Americans have seen or sensed the presence of an angel. A contingent doubts that the percent is really that high. It conducts its own survey. Out of 76 Americans surveyed, only 2 had seen or sensed the presence of an angel. As a result of the contingent’s survey, would you agree with the Newsweek poll? In complete sentences, also give three reasons why the two polls might give different results.

## Exercise 15

The average work week for engineers in a start-up company is believed to be about 60 hours. A newly hired engineer hopes that it’s shorter. She asks 10 engineering friends in start-ups for the lengths of their average work weeks. Based on the results that follow, should she count on the average work week to be shorter than 60 hours?

Data (length of average work week): 70; 45; 55; 60; 65; 55; 55; 60; 50; 55.

### Solution

• d. t 9 t 9 size 12{t rSub { size 8{9} } } {}
• e. -1.33
• f. 0.1086
• h. Decision: Do not reject null
• i. ( 51 . 886 , 62 . 114 ) ( 51 . 886 , 62 . 114 ) size 12{ $$"51" "." "886","62" "." "114"$$ } {}

## Exercise 16

Use the “Lap time” data for Lap 4 (see Table of Contents) to test the claim that Terri finishes Lap 4 on average in less than 129 seconds. Use all twenty races given.

## Exercise 17

Use the “Initial Public Offering” data (see Table of Contents) to test the claim that the average offer price was $18 per share. Do not use all the data. Use your random number generator to randomly survey 15 prices. ## Note: The following questions were written by past students. They are excellent problems! ## Exercise 18 18. "Asian Family Reunion" by Chau Nguyen Every two years it comes around We all get together from different towns. In my honest opinion It's not a typical family reunion Not forty, or fifty, or sixty, But how about seventy companions! The kids would play, scream, and shout One minute they're happy, another they'll pout. The teenagers would look, stare, and compare From how they look to what they wear. The men would chat about their business That they make more, but never less. Money is always their subject And there's always talk of more new projects. The women get tired from all of the chats They head to the kitchen to set out the mats. Some would sit and some would stand Eating and talking with plates in their hands. Then come the games and the songs And suddenly, everyone gets along! With all that laughter, it's sad to say That it always ends in the same old way. They hug and kiss and say "good-bye" And then they all begin to cry! I say that 60 percent shed their tears But my mom counted 35 people this year. She said that boys and men will always have their pride, So we won't ever see them cry. I myself don't think she's correct, So could you please try this problem to see if you object?  ## Exercise 19 "The Problem with Angels" by Cyndy Dowling Although this problem is wholly mine, The catalyst came from the magazine, Time. On the magazine cover I did find The realm of angels tickling my mind.  Inside, 69% I found to be In angels, Americans do believe.  Then, it was time to rise to the task, Ninety-five high school and college students I did ask. Viewing all as one group, Random sampling to get the scoop. So, I asked each to be true, "Do you believe in angels?" Tell me, do! Hypothesizing at the start, Totally believing in my heart That the proportion who said yes Would be equal on this test.  Lo and behold, seventy-three did arrive, Out of the sample of ninety-five. Now your job has just begun, Solve this problem and have some fun.  ### Solution • e. 1.65 • f. 0.0984 • h. Decision: Do not reject null • i. ( 0 . 6836 , 0 . 8533 ) ( 0 . 6836 , 0 . 8533 ) size 12{ $$0 "." "6836",0 "." "8533"$$ } {} ## Exercise 20 "Blowing Bubbles" by Sondra Prull Studying stats just made me tense, I had to find some sane defense. Some light and lifting simple play To float my math anxiety away.  Blowing bubbles lifts me high Takes my troubles to the sky. POIK! They're gone, with all my stress Bubble therapy is the best. The label said each time I blew The average number of bubbles would be at least 22. I blew and blew and this I found From 64 blows, they all are round!  But the number of bubbles in 64 blows Varied widely, this I know. 20 per blow became the mean They deviated by 6, and not 16.  From counting bubbles, I sure did relax But now I give to you your task. Was 22 a reasonable guess? Find the answer and pass this test!  ## Exercise 21 21. "Dalmatian Darnation" by Kathy Sparling A greedy dog breeder named Spreckles Bred puppies with numerous freckles The Dalmatians he sought Possessed spot upon spot The more spots, he thought, the more shekels.  His competitors did not agree That freckles would increase the fee. They said, “Spots are quite nice But they don't affect price; One should breed for improved pedigree.” The breeders decided to prove This strategy was a wrong move. Breeding only for spots Would wreak havoc, they thought. His theory they want to disprove.  They proposed a contest to Spreckles Comparing dog prices to freckles. In records they looked up One hundred one pups: Dalmatians that fetched the most shekels.  They asked Mr. Spreckles to name An average spot count he'd claim To bring in big bucks. Said Spreckles, “Well, shucks, It's for one hundred one that I aim.”  Said an amateur statistician Who wanted to help with this mission. “Twenty-one for the sample Standard deviation's ample:  They examined one hundred and one Dalmatians that fetched a good sum. They counted each spot, Mark, freckle and dot And tallied up every one.  Instead of one hundred one spots They averaged ninety six dots Can they muzzle Spreckles’ Obsession with freckles Based on all the dog data they've got?  ### Solution • e. -2.39 • f. 0.0093 • h. Decision: Reject null • i. ( 91 . 854 , 100 . 15 ) ( 91 . 854 , 100 . 15 ) size 12{ $$"91" "." "854","100" "." "15"$$ } {} ## Exercise 22 "Macaroni and Cheese, please!!" by Nedda Misherghi and Rachelle Hall As a poor starving student I don't have much money to spend for even the bare necessities. So my favorite and main staple food is macaroni and cheese. It's high in taste and low in cost and nutritional value. One day, as I sat down to determine the meaning of life, I got a serious craving for this, oh, so important, food of my life. So I went down the street to Greatway to get a box of macaroni and cheese, but it was SO expensive!$2.02 !!! Can you believe it? It made me stop and think. The world is changing fast. I had thought that the average cost of a box (the normal size, not some super-gigantic-family-value-pack) was at most $1, but now I wasn't so sure. However, I was determined to find out. I went to 53 of the closest grocery stores and surveyed the prices of macaroni and cheese. Here are the data I wrote in my notebook: ### Price per box of Mac and Cheese: • 5 stores @$2.02
• 15 stores @ $0.25 • 3 stores @$1.29
• 6 stores @ $0.35 • 4 stores @$2.27
• 7 stores @ $1.50 • 5 stores @$1.89
• 8 stores @ 0.75.

## Exercise 33

La Leche League International reports that the average age of weaning a child from breastfeeding is age 4 to 5 worldwide. In America, most nursing mothers wean their children much earlier. Suppose a random survey is conducted of 21 U.S. mothers who recently weaned their children. The average weaning age was 9 months (3/4 year) with a standard deviation of 4 months. Conduct a hypothesis test to determine is the average weaning age in the U.S. is less than 4 years old. (Source: http://www.lalecheleague.org/Law/BAFeb01.html)

### Solution

• e. -44.7
• f. 0.0000
• h. Decision: Reject null
• i. ( 0 . 60 , 0 . 90 ) ( 0 . 60 , 0 . 90 ) size 12{ $$0 "." "60",0 "." "90"$$ } {} - in years

## Try these multiple choice questions.

### Exercise 34

When a new drug is created, the pharmaceutical company must subject it to testing before receiving the necessary permission from the Food and Drug Administration (FDA) to market the drug. Suppose the null hypothesis is “the drug is unsafe.” What is the Type II Error?

• A. To claim the drug is safe when in, fact, it is unsafe
• B. To claim the drug is unsafe when, in fact, it is safe.
• C. To claim the drug is safe when, in fact, it is safe.
• D. To claim the drug is unsafe when, in fact, it is unsafe

The next two questions refer to the following information: Over the past few decades, public health officials have examined the link between weight concerns and teen girls smoking. Researchers surveyed a group of 273 randomly selected teen girls living in Massachusetts (between 12 and 15 years old). After four years the girls were surveyed again. Sixty-three (63) said they smoked to stay thin. Is there good evidence that more than thirty percent of the teen girls smoke to stay thin?

B

### Exercise 35

The alternate hypothesis is

• A. p < 0 . 30 p < 0 . 30 size 12{p<0 "." "30"} {}
• B. p 0 . 30 p 0 . 30 size 12{p <= 0 "." "30"} {}
• C. p 0 . 30 p 0 . 30 size 12{p >= 0 "." "30"} {}
• D. p > 0 . 30 p > 0 . 30 size 12{p>0 "." "30"} {}

D

### Exercise 36

After conducting the test, your decision and conclusion are

• A. Reject HoHo size 12{H rSub { size 8{o} } } {}: More than 30% of teen girls smoke to stay thin.
• B. Do not reject HoHo size 12{H rSub { size 8{o} } } {}: Less than 30% of teen girls smoke to stay thin.
• C. Do not reject HoHo size 12{H rSub { size 8{o} } } {}: At most 30% of teen girls smoke to stay thin.
• D. Reject HoHo size 12{H rSub { size 8{o} } } {}: Less than 30% of teen girls smoke to stay thin.

#### Solution

C

The next three questions refer to the following information: A statistics instructor believes that fewer than 20% of Evergreen Valley College (EVC) students attended the opening night midnight showing of the latest Harry Potter movie. She surveys 84 of her students and finds that 11 of attended the midnight showing.

### Exercise 37

An appropriate alternative hypothesis is

• A. p = 0 . 20 p = 0 . 20 size 12{p=0 "." "20"} {}
• B. p > 0 . 20 p > 0 . 20 size 12{p>0 "." "20"} {}
• C. p < 0 . 20 p < 0 . 20 size 12{p<0 "." "20"} {}
• D. p 0 . 20 p 0 . 20 size 12{p <= 0 "." "20"} {}

C

### Exercise 38

At a 1% level of significance, an appropriate conclusion is:

• A. The percent of EVC students who attended the midnight showing of Harry Potter is at least 20%.
• B. The percent of EVC students who attended the midnight showing of Harry Potter is more than 20%.
• C. The percent of EVC students who attended the midnight showing of Harry Potter is less than 20%.
• D. There is not enough information to make a decision.

A

### Exercise 39

The Type I error is believing that the percent of EVC students who attended is:

• A. at least 20%, when in fact, it is less than 20%.
• B. 20%, when in fact, it is 20%.
• C. less than 20%, when in fact, it is at least 20%.
• D. less than 20%, when in fact, it is less than 20%.

#### Solution

C

The next two questions refer to the following information:

It is believed that Lake Tahoe Community College (LTCC) Intermediate Algebra students get less than 7 hours of sleep per night, on average. A survey of 22 LTCC Intermediate Algebra students generated an average of 7.24 hours with a standard deviation of 1.93 hours. At a level of significance of 5%, do LTCC Intermediate Algebra students get less than 7 hours of sleep per night, on average?

### Exercise 40

The distribution to be used for this test is X¯X ~

• A. N ( 7 . 24 , 1 . 93 22 ) N ( 7 . 24 , 1 . 93 22 ) size 12{N $$7 "." "24", { { size 8{1 "." "93"} } over { size 8{ sqrt {"22"} } } }$$ } {}
• B. N ( 7 . 24 , 1 . 93 ) N ( 7 . 24 , 1 . 93 ) size 12{N $$7 "." "24",1 "." "93"$$ } {}
• C. t 22 t 22 size 12{t rSub { size 8{"22"} } } {}
• D. t 21 t 21 size 12{t rSub { size 8{"21"} } } {}

D

### Exercise 41

The Type II error is “I believe that the average number of hours of sleep LTCC students get per night

• A. is less than 7 hours when, in fact, it is at least 7 hours.”
• B. is less than 7 hours when, in fact, it is less than 7 hours.”
• C. is at least 7 hours when, in fact, it is at least 7 hours.”
• D. is at least 7 hours when, in fact, it is less than 7 hours.”

#### Solution

D

The next three questions refer to the following information: An organization in 1995 reported that teenagers spent an average of 4.5 hours per week on the telephone. The organization thinks that, in 2007, the average is higher. Fifteen (15) randomly chosen teenagers were asked how many hours per week they spend on the telephone. The sample mean was 4.75 hours with a sample standard deviation of 2.0.

### Exercise 42

The null and alternate hypotheses are:

• A. Ho:x¯=4.5Ho:x¯=4.5 size 12{H rSub { size 8{o} } : {overline {x}} =4 "." 5} {}, Ha:x¯>4.5Ha:x¯>4.5 size 12{H rSub { size 8{a} } : {overline {x}} >4 "." 5} {}
• B. H o : μ 4 . 5 H o : μ 4 . 5 size 12{H rSub { size 8{o} } :μ >= 4 "." 5} {} H a : μ < 4 . 5 H a : μ < 4 . 5 size 12{H rSub { size 8{a} } :μ<4 "." 5} {}
• C. H o : μ = 4 . 75 H o : μ = 4 . 75 size 12{H rSub { size 8{o} } :μ=4 "." "75"} {} H a : μ > 4 . 75 H a : μ > 4 . 75 size 12{H rSub { size 8{a:} } μ>4 "." "75"} {}
• D. H o : μ = 4 . 5 H o : μ = 4 . 5 size 12{H rSub { size 8{o} } :μ=4 "." 5} {} H a : μ > 4 . 5 H a : μ > 4 . 5 size 12{H rSub { size 8{a} } :μ>4 "." 5} {}

D

### Exercise 43

At a significance level of a=0.05a=0.05 size 12{a=0 "." "05"} {}, the correct conclusion is:

• A. The average in 2007 is higher than it was in 1995.
• B. The average in 1995 is higher than in 2007.
• C. The average is still about the same as it was in 1995.
• D. The test is inconclusive.

C

### Exercise 44

The Type I error is:

• A. To conclude the average hours per week in 2007 is higher than in 1995, when in fact, it is higher.
• B. To conclude the average hours per week in 2007 is higher than in 1995, when in fact, it is the same.
• C. To conclude the average hours per week in 2007 is the same as in 1995, when in fact, it is higher.
• D. To conclude the average hours per week in 2007 is no higher than in 1995, when in fact, it is not higher.

B

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