# Introductory Statistics

## Educators

LE
+ 4 more educators

### Problem 1

You are testing that the mean speed of your cable Internet connection is more than three Megabits per second. What is the random variable? Describe in words.

Bon Z.

### Problem 2

You are testing that the mean speed of your cable Internet connection is more than three Megabits per second. State the null and alternative hypotheses.

Bon Z.

### Problem 3

The American family has an average of two children. What is the random variable? Describe in words.

Bon Z.

Robin C.

### Problem 12

A sleeping bag is tested to withstand temperatures of –15 °F. You think the bag cannot stand temperatures that low. State the Type I and Type II errors in complete sentences.

Robin C.

### Problem 13

For Exercise $9.12,$ what are $\alpha$ and $\beta$ in words?

Robin C.

### Problem 14

In words, describe $1-\beta$ For Exercise 9.12

Robin C.

### Problem 15

A group of doctors is deciding whether or not to perform an operation. Suppose the null hypothesis, H0, is: the surgical procedure will go well. State the Type I and Type II errors in complete sentences.

Robin C.

### Problem 16

A group of doctors is deciding whether or not to perform an operation. Suppose the null hypothesis, $H_{0},$ is: the surgical procedure will go well. Which is the error with the greater consequence?

Robin C.

### Problem 17

The power of a test is $0.981 .$ What is the probability of a Type II error?

Robin C.

### Problem 18

A group of divers is exploring an old sunken ship. Suppose the null hypothesis, H0, is: the sunken ship does not contain buried treasure. State the Type I and Type II errors in complete sentences.

Robin C.

### Problem 19

A microbiologist is testing a water sample for E-coli. Suppose the null hypothesis, H0, is: the sample does not contain E-coli. The probability that the sample does not contain E-coli, but the microbiologist thinks it does is 0.012. The probability that the sample does contain E-coli, but the microbiologist thinks it does not is 0.002. What is the power of this test?

Robin C.

### Problem 20

A microbiologist is testing a water sample for E-coli. Suppose the null hypothesis, H0, is: the sample contains E-coli. Which is the error with the greater consequence?

Robin C.

### Problem 21

Which two distributions can you use for hypothesis testing for this chapter?

Bryan M.

### Problem 22

Which distribution do you use when you are testing a population mean and the standard deviation is known? Assume sample size is large.

Bryan M.

### Problem 23

Which distribution do you use when the standard deviation is not known and you are testing one population mean? Assume sample size is large.

Bryan M.

### Problem 24

A population mean is 13. The sample mean is 12.8, and the sample standard deviation is two. The sample size is 20. What distribution should you use to perform a hypothesis test? Assume the underlying population is normal.

Bryan M.

### Problem 25

A population has a mean is 25 and a standard deviation of five. The sample mean is 24, and the sample size is 108. What distribution should you use to perform a hypothesis test?

Bryan M.

### Problem 26

It is thought that 42% of respondents in a taste test would prefer Brand A. In a particular test of 100 people, 39% preferred Brand A. What distribution should you use to perform a hypothesis test?

Bryan M.

### Problem 27

You are performing a hypothesis test of a single population mean using a Student’s t-distribution. What must you assume about the distribution of the data?

Bryan M.

### Problem 28

You are performing a hypothesis test of a single population mean using a Student’s t-distribution. The data are not from a simple random sample. Can you accurately perform the hypothesis test?

Bryan M.

### Problem 29

You are performing a hypothesis test of a single population. What must be true about the quantities of $n p$ and $n q ?$

Bryan M.

### Problem 30

You are performing a hypothesis testof a single population proportion. You find out that $n p$ is less than five. What must you do to be able to perform a valid hypothesis test?

Bryan M.

### Problem 31

You are performing a hypothesis test of a single population. The data come from which distribution?

Bryan M.

### Problem 32

When do you reject the null hypothesis?

Joe L.

### Problem 33

The probability of winning the grand prize at a particular carnival game is 0.005. Is the outcome of winning very likely or very unlikely?

Joe L.

### Problem 34

The probability of winning the grand prize at a particular carnival game is 0.005. Michele wins the grand prize. Is this considered a rare or common event? Why?

Joe L.

### Problem 35

It is believed that the mean height of high school students who play basketball on the school team is 73 inches with a standard deviation of 1.8 inches. A random sample of 40 player. The sample mean was, and the sample standard deviation was 1.5 years. Do the data support the claim that the mean height is less than 73 inches? The $p$ -value is almost zero. State the null and alternative hypotheses and interpret the $p$ -value.

Joe L.

### Problem 36

The mean age of graduate students at a University is at most 31 y ears with a standard deviation of two years. A random sample of 15 graduate students is taken. The sample mean is 32 years and the sample standard deviation is three years. Are the data significant at the 1$\%$ level? The p-value is $0.0264 .$ State the null and alternative hypotheses and interpret the $p$ -value.

Joe L.

### Problem 37

Does the shaded region represent a low or a high p-value compared to a level of significance of 1$\% ?$

Joe L.

### Problem 38

What should you do when $\alpha > p$ -value?

Joe L.

### Problem 39

What should you do if $\alpha=p$ -value?

Joe L.

### Problem 40

If you do not reject the null hypothesis, then it must be true. Is this statement correct? State why or why not in complete sentences.

Joe L.

### Problem 41

Use the following information to answer the next seven exercises: Suppose that a recent article stated that the mean time spent in jail by a first-time convicted burglar is 2.5 years. A study was then done to see if the mean time has increased in the new century. A random sample of 26 first-time convicted burglars in a recent year was picked. The mean length of time in jail from the survey was three years with a standard deviation of 1.8 years. Suppose that it is somehow known that the population standard deviation is 1.5. Conduct a hypothesis test to determine if the mean length of jail time has increased. Assume the distribution of the jail times is approximately normal.

Is this a test of means or proportions?

Joe L.

### Problem 42

Use the following information to answer the next seven exercises: Suppose that a recent article stated that the mean time spent in jail by a first-time convicted burglar is 2.5 years. A study was then done to see if the mean time has increased in the new century. A random sample of 26 first-time convicted burglars in a recent year was picked. The mean length of time in jail from the survey was three years with a standard deviation of 1.8 years. Suppose that it is somehow known that the population standard deviation is 1.5. Conduct a hypothesis test to determine if the mean length of jail time has increased. Assume the distribution of the jail times is approximately normal.

What symbol represents the random variable for this test?

Joe L.

### Problem 43

Use the following information to answer the next seven exercises: Suppose that a recent article stated that the mean time spent in jail by a first-time convicted burglar is 2.5 years. A study was then done to see if the mean time has increased in the new century. A random sample of 26 first-time convicted burglars in a recent year was picked. The mean length of time in jail from the survey was three years with a standard deviation of 1.8 years. Suppose that it is somehow known that the population standard deviation is 1.5. Conduct a hypothesis test to determine if the mean length of jail time has increased. Assume the distribution of the jail times is approximately normal.

In words, define the random variable for this test.

Joe L.

### Problem 44

Use the following information to answer the next seven exercises: Suppose that a recent article stated that the mean time spent in jail by a first-time convicted burglar is 2.5 years. A study was then done to see if the mean time has increased in the new century. A random sample of 26 first-time convicted burglars in a recent year was picked. The mean length of time in jail from the survey was three years with a standard deviation of 1.8 years. Suppose that it is somehow known that the population standard deviation is 1.5. Conduct a hypothesis test to determine if the mean length of jail time has increased. Assume the distribution of the jail times is approximately normal.

Is the population standard deviation known and, if so, what is it?

Joe L.

### Problem 45

Calculate the following:
a. $x$|_____
b. $\sigma$_______
c. $s_{x}-$_______
d. $n$________

Joe L.

### Problem 46

since both $\sigma$ and $s_{x}$ are given, which should be used? In one to two complete sentences, explain why.

Joe L.

### Problem 47

State the distribution to use for the hypothesis test.

Joe L.

### Problem 48

A random survey of 75 death row inmates revealed that the mean length row is 17.4 years with a standard deviation of 6.3 years. Conduct a hypothesis test to determine if the population mean time on death row could likely be 15 years.
a. Is this a test of one mean or proportion?
b. State the null and alternative hypotheses.
$$H_{0} : \quad \qquad \qquad H_{a} :$$
c. Is this a right-tailed, left-tailed, or two-tailed test?
d. What symbol represents the random variable for this test?
e. In words, define the random variable for this test.
f. Is the population standard deviation known and, if so, what is it?
g. Calculate the following:
$$\begin{array}{l}{\text { i. } x=} \\ {\text { ii. } s=} \\ {\text { iii. } n=}\end{array}$$
h. Which test should be used?
i. State the distribution to use for the hypothesis test.
j. Find the $p$ -value.
k. At a pre-conceived $\alpha=0.05$ , what is your:
i. Decision:
ii. Reason for the decision:
iil. Conclusion (write out in a complete sentence):

Check back soon!

### Problem 49

Assume $H_{0} : \mu=9$ and $H_{a} : \mu<9 .$ Is this a left-tailed, right-tailed, or two-tailed test?

LE
Lucas E.

### Problem 50

Assume $H_{0} : \mu \leq 6$ and $H_{a} : \mu>6 .$ Is this a left-tailed, right-tailed, or two-tailed test?

LE
Lucas E.

### Problem 51

Assume $H_{0} : p=0.25$ and $H a : p \neq 0.25 .$ Is this a left-tailed, right-tailed, or two-tailed test?

LE
Lucas E.

### Problem 52

Draw the general graph of a left-tailed test.

LE
Lucas E.

### Problem 53

Draw the graph of a two-tailed test.

LE
Lucas E.

### Problem 54

A bottle of water is labeled as containing 16 fluid ounces of water. You believe it is less than that. What type of test would you use?

LE
Lucas E.

### Problem 55

Your friend claims that his mean golf score is 63. You want to show that it is higher than that. What type of test would you use?

LE
Lucas E.

### Problem 56

A bathroom scale claims to be able to identify correctly any weight within a pound. You think that it cannot be that accurate. What type of test would you use?

LE
Lucas E.

### Problem 57

You flip a coin and record whether it shows heads or tails. You know the probability of getting heads is 50%, but you think it is less for this particular coin. What type of test would you use?

LE
Lucas E.

### Problem 58

If the alternative hypothesis has a not equals $( \neq)$ symbol, you know to use which type of test?

LE
Lucas E.

### Problem 59

Assume the null hypothesis states that the mean is at least $18 .$ Is this a left-tailed, or two-tailed test?

LE
Lucas E.

### Problem 60

Assume the null hypothesis states that the mean is at most $12 .$ Is this a left-tailed, or two-tailed test?

LE
Lucas E.

### Problem 61

Assume the null hypothesis states that the mean is equal to 88. The alternative hypothesis states that the mean is not equal to 88. Is this a left-tailed, right-tailed, or two-tailed test?

LE
Lucas E.

### Problem 62

Some of the following statements refer to the null hypothesis, some to the alternate hypothesis.
State the null hypothesis, Ho, and the hypothesis. $H_{a},$ in terms of the appropriate parameter $(\mu \text { or } p)$
a. The mean number of years Americans work before retiring is $34 .$
b. At most 60$\%$ of Americans vote in presidential elections.
c. The mean starting salary for San Jose State University graduates is at least $\$ 100,000$per year. d. Twenty-nine percent of high school seniors get drunk each month. e. Fewer than 5$\%$of adults ride the bus to work in Los Angeles. f. The mean number of cars a person owns in her lifetime is not more than ten. g. About half of Americans prefer to live away from cities, given the choice. h. Europeans have a mean paid vacation each year of six weeks. i. The chance of developing breast cancer is under 11$\%$for women. j. Private universities' mean tuition cost is more than$\$20,000$ per year.

Bon Z.

### Problem 63

Over the past few decades, public health officials have link between weight concerns and teen girls' smoking. Ressearchers 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 said they smoked to stay thin. Is there good evidence that more than thirty percent of the teen girls smoke to stay thin? The alternative hypothesis is:
a. $p<0.30$
b. $p \leq 0.30$
c. $p \geq 0.30$
d. $p > 0.30$

Bon Z.

### Problem 64

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 attended the midnight showing. An appropriate alternative hypothesis is:
a. $p=0.20$
b. $p > 0.20$
c. $p < 0.20$
d. $p \leq 0.20$

Bon Z.

### Problem 65

Previously, an organization reported that teenagers spent 4.5 hours per week, on average, on the phone. The organization thinks that, currently, the mean is higher. Fifteen randomly chosen teenagers were asked how many hours per week they spend on the phone. The sample mean was 4.75 hours with a sample standard deviation of 2.0. Conduct a hypothesis test. The null and alternative hypotheses are:
a. $H_{0} : x=4.5, H_{a} : \quad x > 4.5$
b. $H_{0} : \mu \geq 4.5, H_{a} : \mu < 4.5$
c. $H_{0} : \mu=4.75, H_{a} : \mu > 4.75$
d. $H_{0} : \mu=4.5, H a : \mu > 4.5$

Bon Z.

### Problem 66

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

Check back soon!

### Problem 67

For statements a-j in Exercise 9.109 , answer the following in complete sentences.
a. State a consequence of committing a Type I error.
b. State a consequence of committing a Type II error.

Check back soon!

### Problem 68

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 conclude the drug is safe when in, fact, it is unsafe.
b. Not to conclude the drug is safe when, in fact, it is safe.
c. To conclude the drug is safe when, in fact, it is safe.
d. Not to conclude the drug is unsafe when, in fact, it is unsafe.

Robin C.

### Problem 69

A statistics instructor believes that fewer than 20$\%$ of Evergreen Valley College (EVC) students attended the opening midnight showing of the latest Harry Potter movie. She surveys 84 of her students and finds that 11 of them attended the midnight showing. The Type Ierror is to conclude 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$\% .$

Robin C.

### Problem 70

It is believed that Lake Tahoe Community College (ITC) Intermediate Algebra students get less than seven hours of sleep per night, on average. A survey of 22 LTCC Intermediate Algebra students students a mean 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 seven hours of sleep per night, on average?
The Type II error is not to reject that the mean number of sleep LTCC students get per night is at lest seven when, in fact, the mean number of hours
a. is more than seven hours.
b. is at most seven hours.
c. is at least seven hours.
d. is less than seven hours.

Robin C.

### Problem 71

Previously, an organization reported that teenagers spent 4.5 hours per week, on average, on the phone. The organization thinks that, currently, the mean is higher. Fifteen randomly chosen teenagers were asked how many hours per week they spend on the phone. The sample mean was 4.75 hours with a sample standard deviation of 2.0. Conduct a hypothesis test, the Type I error is:
a. to conclude that the current mean hours per week is han $4.5,$ when in fact, it is higher
b. to conclude that the current mean hours per week is higher than $4.5,$ when in fact, it is the same
c. to conclude that the mean hours per week currently is $4.5,$ when in fact, it is higher
d. to conclude that the mean hours per week currently is no higher than $4.5,$ when in fact, it is not higher

Robin C.

### Problem 72

It is believed that Lake Tahoe Community College (LTCC) Intermediate Algebra students geven hours of sleep per night, on average. A survey of 22 LTCC Intermediate Algerated a mean 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 seven hours of sleep per night, on average? The distribution to be used for this test is $X \sim$ _________
$$\begin{array}{l}{\text { a. } \quad N\left(7.24, \frac{1.93}{\sqrt{22}}\right)} \\ {\text { b. } N(7.24,1.93)} \\ {\text { C. } t 22} \\ {\text { d. } t_{21}}\end{array}$$

Bryan M.

### Problem 73

The National Institute of Mental Health published an article stating that in any one-year period, approximately 9.5 percent of American adults suffer from depression or a depressive illness. Suppose that in a survey of 100 people in a certain town, seven of them suffered from depression or a depressive illness. Conduct a hypothesis test to determine if the true proportion of people in that town suffering from depression or a depressive illness is lower than the percent in the general adult American population.
a. Is this a test of one mean or proportion?
b. State the null and alternative hypotheses.
$H_{0} : \qquad \qquad \qquad H_{a} :$
c. Is this a right-tailed, left-tailed, or two-tailed test?
d. What symbol represents the random variable for this test?
e. In words, define the random variable for this test.
f. Calculate the following:
$$\begin{array}{l}{\text { i. } x=} \\ {\text { ii. } \quad n=} \\ {\text { iil. } p^{\prime}=}\end{array}$$
g. Calculate $\sigma_{x}=\qquad .$ Show the formula set-up.
h. State the distribution to use for the hypothesis test.
i. Find the $p$ -value.
j. At a pre-conceived $\alpha=0.05,$ what is your:
i. Decision:
ii. Reason for the decision:
iii. Conclusion (write out in a complete sentence):

Check back soon!

### Problem 74

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 8,000. A survey of owners of that tire design is conducted. From the 28 tires surveyed, the mean lifespan was 46,500 miles with a standard deviation of 9,800 miles. Using alpha 0.05, is the data highly inconsistent with the claim?

CC
Charles C.

### Problem 75

From generation to generation, the mean 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 mean starting age is at least 19. The sample mean was 18.1 with a sample standard deviation of 1.3. Do the data support the claim at the 5% level?

LE
Lucas E.

Check back soon!

### Problem 88

"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?

LE
Lucas E.

### Problem 89

"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.

LE
Lucas E.

### Problem 90

"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

LE
Lucas E.

### Problem 91

"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?

LE
Lucas E.

### Problem 92

"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 mean 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:

LE
Lucas E.

### Problem 115

Instructions: For the following ten exercises, Hypothesis testing: For the following ten exercises, answer each question.
a. State the null and alternate hypothesis.
b. State the p-value.
c. State alpha.
d. What is your decision?
e. Write a conclusion.
f. Answer any other questions asked in the problem.
According to the N.Y. Times Almanac the mean family size in the U.S. is 3.18. A sample of a college math class resulted in the following family sizes: 5; 4; 5; 4; 4; 3; 6; 4; 3; 3; 5; 5; 6; 3; 3; 2; 7; 4; 5; 2; 2; 2; 3; 2 At ? = 0.05 level, is the class’ mean family size greater than the national average? Does the Almanac result remain valid?
Why?

LE
Lucas E.

### Problem 116

Instructions: For the following ten exercises, Hypothesis testing: For the following ten exercises, answer each question.
a. State the null and alternate hypothesis.
b. State the p-value.
c. State alpha.
d. What is your decision?
e. Write a conclusion.
f. Answer any other questions asked in the problem.
The student academic group on a college campus claims that freshman students study at least 2.5 hours per day, on average. One Introduction to Statistics class was skeptical. The class took a random sample of 30 freshman students and found a mean study time of 137 minutes with a standard deviation of 45 minutes. At ? = 0.01 level, is the student academic group’s claim correct?

Callie S.