# Introductory Statistics

## 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.

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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.

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Problem 3

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

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Problem 4

The mean entry level salary of an employee at a company is $58,000. You believe it is higher for IT professionals in the company. State the null and alternative hypotheses. Check back soon! Problem 5 A sociologist claims the probability that a person picked at random in Times Square in New York City is visiting the area is 0.83. You want to test to see if the proportion is actually less. What is the random variable? Describe in words. Check back soon! Problem 6 A sociologist claims the probability that a person picked at random in Times Square in New York City is visiting the area is 0.83. You want to test to see if the claim is correct. State the null and alternative hypotheses. Check back soon! Problem 7 In a population of fish, approximately 42% are female. A test is conducted to see if, in fact, the proportion is less. State the null and alternative hypotheses. Check back soon! Problem 8 Suppose that a recent article stated that the mean time spent in jail by a first–time convicted burglar is 2.5 years. A study was then done to see if the mean time has increased in the new century. A random sample of 26 first-time convicted burglars in a recent year was picked. The mean length of time in jail from the survey was 3 years with a standard deviation of 1.8 years. Suppose that it is somehow known that the population standard deviation is 1.5. If you were conducting a hypothesis test to determine if the mean length of jail time has increased, what would the null and alternative hypotheses be? The distribution of the population is normal. a.$H_{0} :$_________ b.$H_{a} :$_________ Check back soon! Problem 9 A random survey of 75 death row inmates revealed that the mean length of time on death row is 17.4 years with a standard deviation of 6.3 years. If you were conducting a hypothesis test to determine if the population mean time on death row could likely be 15 years, what would the null and alternative hypotheses be? a.$H_{0} :$______ b.$H_{a} :$______ Check back soon! Problem 10 The National Institute of Mental Health published an article stating that in any one-year period, approximately 9.5 percent of American adults suffer from depression or a depressive illness. Suppose that in a survey of 100 people in a certain town, seven of them suffered from depression or a depressive illness. If you were conducting a hypothesis test to determine if the true proportion of people in that town suffering from depression or a depressive illness is lower than the percent in the general adult American population, what would the null and alternative hypotheses be? a.$H_{0} :$______ b.$H_{a} :$______ Check back soon! Problem 11 The mean price of mid-sized cars in a region is$32,000. A test is conducted to see if the claim is true. State the Type I and Type II errors in complete sentences.

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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.

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Problem 13

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

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Problem 14

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

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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.

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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?

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Problem 17

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

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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.

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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?

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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?

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Problem 21

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

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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.

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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.

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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.

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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?

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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?

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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?

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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?

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Problem 29

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

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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?

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Problem 31

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

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Problem 32

When do you reject the null hypothesis?

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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?

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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?

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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.

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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.

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Problem 37

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

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Problem 38

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

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Problem 39

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

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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.

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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?

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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?

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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.

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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?

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Problem 45

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

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Problem 46

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

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Problem 47

State the distribution to use for the hypothesis test.

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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):

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Problem 49

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

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Problem 50

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

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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?

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Problem 52

Draw the general graph of a left-tailed test.

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Problem 53

Draw the graph of a two-tailed test.

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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?

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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?

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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?

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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?

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Problem 58

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

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Problem 59

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

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Problem 60

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

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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?

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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.

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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. $p > 0.30$

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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. $p \leq 0.20$

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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. $H_{0} : \mu=4.5, H a : \mu > 4.5$

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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.

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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.

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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.

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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$\% .$

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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.

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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

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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 $X \sim$ _________
$$\begin{array}{l}{\text { a. } \quad N\left(7.24, \frac{1.93}{\sqrt{22}}\right)} \\ {\text { b. } N(7.24,1.93)} \\ {\text { C. } t 22} \\ {\text { d. } t_{21}}\end{array}$$

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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):

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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?

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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?

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Problem 76

The cost of a daily newspaper varies from city to city. However, the variation among prices remains steady with a standard deviation of 20¢. A study was done to test the claim that the mean cost of a daily newspaper is $1.00. Twelve costs yield a mean cost of 95¢ with a standard deviation of 18¢. Do the data support the claim at the 1% level? Check back soon! Problem 77 An article in the San Jose Mercury News stated that students in the California state university system take 4.5 years, on average, to finish their undergraduate degrees. Suppose you believe that the mean 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? Check back soon! Problem 78 The mean number of sick days an employee takes per year is believed to be about ten. Members of a personnel department do not believe this figure. They randomly survey eight employees. The number of sick days they took for the past year are as follows: 12; 4; 15; 3; 11; 8; 6; 8. Let x = the number of sick days they took for the past year. Should the personnel team believe that the mean number is ten? Check back soon! Problem 79 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 mean work week has increased. 81 women were surveyed with the following results. The sample mean was 83; the sample standard deviation was ten. Does it appear that the mean work week has increased for women at the 5% level? Check back soon! Problem 80 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? Check back soon! Problem 81 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 brown trout’s mean I.Q. is greater than four. 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. Check back soon! Problem 82 Refer to Exercise 9.119. Conduct a hypothesis test to see if your decision and conclusion would change if your belief were that the brown trout’s mean I.Q. is not four. Check back soon! Problem 83 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? Check back soon! Problem 84 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 two 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. Meijie L. Numerade Educator Problem 85 The mean 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 ten engineering friends in start-ups for the lengths of their mean work weeks. Based on the results that follow, should she count on the mean work week to be shorter than 60 hours? Check back soon! Problem 86 Use the “Lap time” data for Lap 4 (see Appendix C) to test the claim that Terri finishes Lap 4, on average, in less than 129 seconds. Use all twenty races given. Check back soon! Problem 87 Use the “Initial Public Offering” data (see Appendix C) to test the claim that the mean offer price was$18 per share. Do not use all the data. Use your random number generator to randomly survey 15 prices.

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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?

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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.

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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

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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?

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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:
Price per box of Mac and Cheese: I could see that the cost varied but I had to sit down to figure out whether or not I was right. If it does turn out that this mouth-watering dish is at most $1, then I'll throw a big cheesy party in our next statistics lab, with enough macaroni and cheese for just me. (After all, as a poor starving student I can't be expected to feed our class of animals!) Check back soon! Problem 93 "William Shakespeare: The Tragedy of Hamlet, Prince of Denmark," by Jacqueline Ghodsi THE CHARACTERS (in order of appearance):$\cdot$HAMLET, Prince of Denmark and student of Statistics POLONIUS, Hamlet's tutor$\cdot$HOROTIO, friend to Hamlet and fellow student Scene: The great library of the castle, in which Hamlet does his lessons Act I (The day is fair, but the face of Hamlet is clouded. He paces the large room. His tutor, Polonius, is reprimanding Hamlet regarding the latter’s recent experience. Horatio is seated at the large table at right stage.) POLONIUS: My Lord, how cans’t thou admit that thou hast seen a ghost! It is but a figment of your imagination! HAMLET: I beg to differ; I know of a certainty that five-and-seventy in one hundred of us, condemned to the whips and scorns of time as we are, have gazed upon a spirit of health, or goblin damn’d, be their intents wicked or charitable. POLONIUS If thou doest insist upon thy wretched vision then let me invest your time; be true to thy work and speak to me through the reason of the null and alternate hypotheses. (He turns to Horatio.) Did not Hamlet himself say, “What piece of work is man, how noble in reason, how infinite in faculties? Then let not this foolishness persist. Go, Horatio, make a survey of three-and-sixty and discover what the true proportion be. For my part, I will never succumb to this fantasy, but deem man to be devoid of all reason should thy proposal of at least five-and-seventy in one hundred hold true. HORATIO (to Hamlet): What should we do, my Lord? HAMLET: Go thy purpose, Horatio. HORATIO: To what end, my Lord? HAMLET: That you must teach me. But let me conjure you by the rights of our fellowship, by the consonance of our youth, but the obligation of our ever-preserved love, be even and direct with me, whether I am right or no. (Horatio exits, followed by Polonius, leaving Hamlet to ponder alone.) Act II (The next day, Hamlet awaits anxiously the presence of his friend, Horatio. Polonius enters and places some books upon the table just a moment before Horatio enters.) POLONIUS: So, Horatio, what is it thou didst reveal through thy deliberations? HORATIO: In a random survey, for which purpose thou thyself sent me forth, I did discover that one-and-forty believe fervently that the spirits of the dead walk with us. Before my God, I might not this believe, without the sensible and true avouch of mine own eyes. POLONIUS: Give thine own thoughts no tongue, Horatio. (Polonius turns to Hamlet.) But look to’t I charge you, my Lord. Come Horatio, let us go together, for this is not our test. (Horatio and Polonius leave together.) HAMLET: To reject, or not reject, that is the question: whether ‘tis nobler in the mind to suffer the slings and arrows of outrageous statistics, or to take arms against a sea of data, and, by opposing, end them. (Hamlet resignedly attends to his task.) Check back soon! Problem 94 "Untitled," by Stephen Chen I've often wondered how software is released and sold to the public. Ironically, I work for a company that sells products with known problems. Unfortunately, most of the problems are difficult to create, which makes them difficult to fix. I usually use the test program X, which tests the product, to try to create a specific problem. When the test program is run to make an error occur, the likelihood of generating an error is 1%. So, armed with this knowledge, I wrote a new test program Y that will generate the same error that test program X creates, but more often. To find out if my test program is better than the original, so that I can convince the management that I'm right, I ran my test program to find out how often I can generate the same error. When I ran my test program 50 times, I generated the error twice. While this may not seem much better, I think that I can convince the management to use my test program instead of the original test program. Am I right? Check back soon! Problem 95 "Japanese Girls’ Names" by Kumi Furuichi It used to be very typical for Japanese girls’ names to end with “ko.” (The trend might have started around my grandmothers’ generation and its peak might have been around my mother’s generation.) “Ko” means “child” in Chinese characters. Parents would name their daughters with “ko” attaching to other Chinese characters which have meanings that they want their daughters to become, such as Sachiko—happy child, Yoshiko—a good child, Yasuko—a healthy child, and so on. However, I noticed recently that only two out of nine of my Japanese girlfriends at this school have names which end with “ko.” More and more, parents seem to have become creative, modernized, and, sometimes, westernized in naming their children. I have a feeling that, while 70 percent or more of my mother’s generation would have names with “ko” at the end, the proportion has dropped among my peers. I wrote down all my Japanese friends’, ex-classmates’, co-workers, and acquaintances’ names that I could remember. Following are the names. (Some are repeats.) Test to see if the proportion has dropped for this generation. Ai, Akemi, Akiko, Ayumi, Chiaki, Chie, Eiko, Eri, Eriko, Fumiko, Harumi, Hitomi, Hiroko, Hiroko, Hidemi, Hisako, Hinako, Izumi, Izumi, Junko, Junko, Kana, Kanako, Kanayo, Kayo, Kayoko, Kazumi, Keiko, Keiko, Kei, Kumi, Kumiko, Kyoko, Kyoko, Madoka, Maho, Mai, Maiko, Maki, Miki, Miki, Mikiko, Mina, Minako, Miyako, Momoko, Nana, Naoko, Naoko, Naoko, Noriko, Rieko, Rika, Rika, Rumiko, Rei, Reiko, Reiko, Sachiko, Sachiko, Sachiyo, Saki, Sayaka, Sayoko, Sayuri, Seiko, Shiho, Shizuka, Sumiko, Takako, Takako, Tomoe, Tomoe, Tomoko, Touko, Yasuko, Yasuko, Yasuyo, Yoko, Yoko, Yoko, Yoshiko, Yoshiko, Yoshiko, Yuka, Yuki, Yuki, Yukiko, Yuko, Yuko. Check back soon! Problem 96 "Phillip’s Wish," by Suzanne Osorio My nephew likes to play Chasing the girls makes his day. He asked his mother If it is okay To get his ear pierced. She said, “No way!” To poke a hole through your ear, Is not what I want for you, dear. He argued his point quite well, Says even my macho pal, Mel, Has gotten this done. It’s all just for fun. C’mon please, mom, please, what the hell. Again Phillip complained to his mother, Saying half his friends (including their brothers) Are piercing their ears And they have no fears He wants to be like the others. She said, “I think it’s much less. We must do a hypothesis test. And if you are right, I won’t put up a fight. But, if not, then my case will rest.” We proceeded to call fifty guys To see whose prediction would fly. Nineteen of the fifty Said piercing was nifty And earrings they’d occasionally buy. Then there’s the other thirty-one, Who said they’d never have this done. So now this poem’s finished. Will his hopes be diminished, Or will my nephew have his fun? Check back soon! Problem 97 "The Craven," by Mark Salangsang Once upon a morning dreary In stats class I was weak and weary. Pondering over last night’s homework Whose answers were now on the board This I did and nothing more. While I nodded nearly napping, Suddenly, there came a tapping. As someone gently rapping, Rapping my head as I snore. Quoth the teacher, “Sleep no more.” “In every class you fall asleep,” The teacher said, his voice was deep. “So a tally I’ve begun to keep Of every class you nap and snore. The percentage being forty-four.” “My dear teacher I must confess, While sleeping is what I do best. The percentage, I think, must be less, A percentage less than forty-four.” This I said and nothing more. “We’ll see,” he said and walked away, And fifty classes from that day He counted till the month of May The classes in which I napped and snored. The number he found was twenty-four. At a significance level of 0.05, Please tell me am I still alive? Or did my grade just take a dive Plunging down beneath the floor? Upon thee I hereby implore. Check back soon! Problem 99 Sixty-eight percent of online courses taught at community colleges nationwide were taught by full-time faculty. To test if 68% also represents California’s percent for full-time faculty teaching the online classes, Long Beach City College (LBCC) in California, was randomly selected for comparison. In the same year, 34 of the 44 online courses LBCC offered were taught by full-time faculty. Conduct a hypothesis test to determine if 68% represents California. NOTE: For more accurate results, use more California community colleges and this past year's data. Check back soon! Problem 100 According to an article in Bloomberg Businessweek, New York City's most recent adult smoking rate is 14%. Suppose that a survey is conducted to determine this year’s rate. Nine out of 70 randomly chosen N.Y. City residents reply that they smoke. Conduct a hypothesis test to determine if the rate is still 14% or if it has decreased. Check back soon! Problem 101 The mean age of De Anza College students in a previous term was 26.6 years old. An instructor thinks the mean age for online students is older than 26.6. She randomly surveys 56 online students and finds that the sample mean is 29.4 with a standard deviation of 2.1. Conduct a hypothesis test. Check back soon! Problem 102 Registered nurses earned an average annual salary of$69,110. For that same year, a survey was conducted of 41 California registered nurses to determine if the annual salary is higher than $69,110 for California nurses. The sample average was$71,121 with a sample standard deviation of $7,489. Conduct a hypothesis test. Check back soon! Problem 103 La Leche League International reports that the mean age of weaning a child from breastfeeding is age four to five worldwide. In America, most nursing mothers wean their children much earlier. Suppose a random survey is conducted of 21 U.S. mothers who recently weaned their children. The mean weaning age was nine months (3/4 year) with a standard deviation of 4 months. Conduct a hypothesis test to determine if the mean weaning age in the U.S. is less than four years old. Check back soon! Problem 104 Over the past few decades, public health officials have examined the link between weight concerns and teen girls' smoking. Researchers surveyed a group of 273 randomly selected teen girls living in Massachusetts (between 12 and 15 years old). After four years the girls were surveyed again. Sixty-three said they smoked to stay thin. Is there good evidence that more than thirty percent of the teen girls smoke to stay thin? After conducting the test, your decision and conclusion are a. Reject$H_{0} :$There is sufficient evidence to conclude that more than 30$\%$of teen girls smoke to stay thin. b. Do not reject Ho: There is not sufficient evidence to conclude that less than 30$\%$of teen girls smoke to stay thin. c. Do not reject$H_{0} :$There is not sufficient evidence to conclude that more than 30$\%$of teen girls smoke to stay thin. d. Reject Ho: There is sufficient evidence to conclude that less than 30$\%$of teen girls smoke to stay thin. Check back soon! Problem 105 A statistics instructor believes that fewer than 20% of Evergreen Valley College (EVC) students attended the opening night midnight showing of the latest Harry Potter movie. She surveys 84 of her students and finds that 11 of them attended the midnight showing. At a 1$\%$level of significance, an appropriate conclusion is: a. There is insufficient evidence to conclude that the percent of EVC students who attended the midnight showing of Harry Potter is less than 20%. b. There is sufficient evidence to conclude that the percent of EVC students who attended the midnight showing of Harry Potter is more than 20%. c. There is sufficient evidence to conclude that the percent of EVC students who attended the midnight showing of Harry Potter is less than 20%. d. There is insufficient evidence to conclude that the percent of EVC students who attended the midnight showing of Harry Potter is at least 20%. Check back soon! Problem 106 Previously, an organization reported that teenagers spent 4.5 hours per week, on average, on the phone. The organization thinks that, currently, the mean is higher. Fifteen randomly chosen teenagers were asked how many hours per week they spend on the phone. The sample mean was 4.75 hours with a sample standard deviation of 2.0. Conduct a hypothesis test. At a significance level of$a=0.05,$what is the correct conclusion? a. There is enough evidence to conclude that the mumber of hours is more than 4.75 b. There is enough evidence to conclude that the mean number of hore than 4.5 c. There is not enough evidence to conclude that the mumber of hours is more than 4.5 d. There is not enough evidence to conclude that the mean number of hours is more than 4.75 Check back soon! Problem 107 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 Center for Disease Control website, in 2011 at least 18% of high school students have smoked a cigarette. An Introduction to Statistics class in Davies County, KY conducted a hypothesis test at the local high school (a medium sized–approximately 1,200 students–small city demographic) to determine if the local high school’s percentage was lower. One hundred fifty students were chosen at random and surveyed. Of the 150 students surveyed, 82 have smoked. Use a significance level of 0.05 and using appropriate statistical evidence, conduct a hypothesis test and state the conclusions. Check back soon! Problem 108 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. A recent survey in the N.Y. Times Almanac indicated that 48.8% of families own stock. A broker wanted to determine if this survey could be valid. He surveyed a random sample of 250 families and found that 142 owned some type of stock. At the 0.05 significance level, can the survey be considered to be accurate? Check back soon! Problem 109 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. Driver error can be listed as the cause of approximately 54% of all fatal auto accidents, according to the American Automobile Association. Thirty randomly selected fatal accidents are examined, and it is determined that 14 were caused by driver error. Using ? = 0.05, is the AAA proportion accurate? Check back soon! Problem 110 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 US Department of Energy reported that 51.7% of homes were heated by natural gas. A random sample of 221 homes in Kentucky found that 115 were heated by natural gas. Does the evidence support the claim for Kentucky at the ? = 0.05 level in Kentucky? Are the results applicable across the country? Why? Check back soon! Problem 111 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. For Americans using library services, the American Library Association claims that at most 67% of patrons borrow books. The library director in Owensboro, Kentucky feels this is not true, so she asked a local college statistic class to conduct a survey. The class randomly selected 100 patrons and found that 82 borrowed books. Did the class demonstrate that the percentage was higher in Owensboro, KY? Use ? = 0.01 level of significance. What is the possible proportion of patrons that do borrow books from the Owensboro Library? Check back soon! Problem 112 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 Weather Underground reported that the mean amount of summer rainfall for the northeastern US is at least 11.52 inches. Ten cities in the northeast are randomly selected and the mean rainfall amount is calculated to be 7.42 inches with a standard deviation of 1.3 inches. At the ? = 0.05 level, can it be concluded that the mean rainfall was below the reported average? What if ? = 0.01? Assume the amount of summer rainfall follows a normal distribution. Check back soon! Problem 113 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. A survey in the N.Y. Times Almanac finds the mean commute time (one way) is 25.4 minutes for the 15 largest US cities. The Austin, TX chamber of commerce feels that Austin’s commute time is less and wants to publicize this fact. The mean for 25 randomly selected commuters is 22.1 minutes with a standard deviation of 5.3 minutes. At the ? = 0.10 level, is the Austin, TX commute significantly less than the mean commute time for the 15 largest US cities? Check back soon! Problem 114 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. A report by the Gallup Poll found that a woman visits her doctor, on average, at most 5.8 times each year. A random sample of 20 women results in these yearly visit totals$3 ; 2 ; 1 ; 3 ; 7 ; 2 ; 9 ; 4 ; 6 ; 6 ; 8 ; 0 ; 5 ; 6 ; 4 ; 2 ; 1 ; 3 ; 4 ; 1$At the$\alpha=0.05\$ level can it be concluded that the sample mean is higher than 5.8 visits per year?

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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.
e. Write a conclusion.
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?

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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.