Deck 12: Tests of Hypotheses: Means

Question 1

In a hypothesis test, $\alpha$ indicates

A) the area of the rejection region for the null hypothesis.
B) the probability of a Type II error.
C) whether the test is one-tailed or two-tailed.
D) the probability of accepting the null hypothesis.

Answer 1

the area of the rejection region for the null hypothesis.

Question 2

In conducting a hypothesis test, the procedure always assumes that

A) the alternative hypothesis is true.
B) the alternative hypothesis is false.
C) the null hypothesis is true.
D) the null hypothesis is false

Answer 2

the null hypothesis is true.

Question 3

If the null and alternative hypotheses are $$\begin{aligned}& H_{0}: \mu=80 \\& \mathrm{H}_{A}: \mu>80\end{aligned}$$
Then the area of rejection for the null hypothesis is

A) to the left.
B) to the right.
C) split into two parts in both directions.
D) cannot be determined from the given information.

Answer 3

to the right.

Question 4

When a null hypothesis cannot be rejected, we conclude that

A) the null hypothesis is true.
B) the null hypothesis may be true.
C) the alternative hypothesis is true.
D) the alternative hypothesis may be true.

Question 5

In a large-sample hypothesis test with an alternative hypothesis of $\mu>60$ , the $p$ -value equals

A) $\alpha$ .
B) $\beta$ .
C) the critical value of $z$ .
D) the area to the right of the obtained $z$ value.

Question 6

A null hypothesis will be rejected if

A) $\alpha=.05$ and $p$ -value $=.15$ .
B) $\alpha=.05$ and $p$ -value $=.03$ .
C) $\alpha=.01$ and $p$ -value $=.03$ .
D) the obtained value of $z$ is closer to the null hypothesis mean than the critical value.

Question 7

In a hypothesis test that a population mean is at least 50, it is a Type I error to conclude that the mean is

A) at least 50 when it is not.
B) at least 50 when it is.
C) less than 50 when it is not.
D) less than 50 when it is.

Question 8

The location of the rejection region for the null hypothesis is determined by

A) the sign in the null hypothesis.
B) the sign in the alternative hypothesis.
C) the size of $\alpha$ .
D) the sample size $n$ .

Question 9

If $\alpha=0.05$ in a two-tailed large-sample hypothesis test for a population mean, then

A) 1.96 is a critical value.
B) 1.96 is a $p$ -value.
C) 1.65 is a $p$ -value.
D) 1.65 is a critical value.

Question 10

Which of the following is a possible null hypothesis for a two-tailed hypothesis test?

A) $\bar{x}=30$
B) $\mu \neq 30$
C) $\mu \geq 30$
D) $\mu=30$

Question 11

A manager of a cafeteria wants to estimate the average time customers wait before being served. A random sample of 49 customers has an average waiting time of 8.4 minutes with a standard deviation of 3.5 minutes.

-With $90 \%$ confidence, what can the manager conclude about the possible size of his error in using 8.4 minutes to estimate the true average waiting time?

Question 12

A manager of a cafeteria wants to estimate the average time customers wait before being served. A random sample of 49 customers has an average waiting time of 8.4 minutes with a standard deviation of 3.5 minutes.

-Find a $90 \%$ confidence interval for the true average customer waiting time.

Question 13

A manager of a cafeteria wants to estimate the average time customers wait before being served. A random sample of 49 customers has an average waiting time of 8.4 minutes with a standard deviation of 3.5 minutes.

-With $98 \%$ confidence, what can the manager conclude about the possible size of his error in using 8.4 minutes to estimate the true average waiting time?

Question 14

A manager of a cafeteria wants to estimate the average time customers wait before being served. A random sample of 49 customers has an average waiting time of 8.4 minutes with a standard deviation of 3.5 minutes.

-Find a $98 \%$ confidence interval for the true average customer waiting time.

Question 15

An employment agency wants to estimate the average number of people who will respond to one of their ads. If $\sigma=40$ people, how large a sample is needed to be able to assert with probability 0.95 that the estimate will be off by at most 10 people?

Question 16

An automobile rustproofing company claims that their methods protect cars for 60 months. This hypothesis is tested against the alternative that the protection lasts for less than 60 months. A random sample of 100 cars produces an average time of 57 months with a standard deviation of 15 months.
-State the hypothesis in symbols.

Question 17

An automobile rustproofing company claims that their methods protect cars for 60 months. This hypothesis is tested against the alternative that the protection lasts for less than 60 months. A random sample of 100 cars produces an average time of 57 months with a standard deviation of 15 months.

-Test the hypothesis using the $5 \%$ significance level.

Question 18

An automobile rustproofing company claims that their methods protect cars for 60 months. This hypothesis is tested against the alternative that the protection lasts for less than 60 months. A random sample of 100 cars produces an average time of 57 months with a standard deviation of 15 months.
-Find the value of $\bar{x}$ necessary to reject the null hypothesis.

Question 19

Two supermarket owners each claim that more customers enter their store than enter the other's store. A survey was taken and it was found that in 60 days an average of 810 people per day entered the first store with a standard deviation of 40 people. A corresponding survey found that in 80 days an average of 800 people per day entered the second store with a standard deviation of 30 people.
-For the above situation, state the hypotheses if this is a two-tailed test.

Question 20

Two supermarket owners each claim that more customers enter their store than enter the other's store. A survey was taken and it was found that in 60 days an average of 810 people per day entered the first store with a standard deviation of 40 people. A corresponding survey found that in 80 days an average of 800 people per day entered the second store with a standard deviation of 30 people.

-For the above situation, conduct the two-tailed test with $\alpha=0.05$ .

Question 21

Two supermarket owners each claim that more customers enter their store than enter the other's store. A survey was taken and it was found that in 60 days an average of 810 people per day entered the first store with a standard deviation of 40 people. A corresponding survey found that in 80 days an average of 800 people per day entered the second store with a standard deviation of 30 people.

-For the above situation, test the hypothesis that $\mu_{1}-\mu_{2}=5$ against a two-tailed alternative.

Question 22

Two supermarket owners each claim that more customers enter their store than enter the other's store. A survey was taken and it was found that in 60 days an average of 810 people per day entered the first store with a standard deviation of 40 people. A corresponding survey found that in 80 days an average of 800 people per day entered the second store with a standard deviation of 30 people.

-For the above situation, explain under what condition a Type I error would be committed.

Question 23

Two supermarket owners each claim that more customers enter their store than enter the other's store. A survey was taken and it was found that in 60 days an average of 810 people per day entered the first store with a standard deviation of 40 people. A corresponding survey found that in 80 days an average of 800 people per day entered the second store with a standard deviation of 30 people.

-For the above situation, explain under what condition a Type II error would be committed.

Question 24

An automobile manufacturer has claimed that his car averages at least $30 \mathrm{~m}$ .p.g. on the highway. A random sample of 16 such cars finds an average of $25 \mathrm{~m}$ . p.g. with a standard deviation of $8 \mathrm{~m} . p . g$ .
-State the hypotheses symbolically for the one-tailed test.

Question 25

An automobile manufacturer has claimed that his car averages at least $30 \mathrm{~m}$ .p.g. on the highway. A random sample of 16 such cars finds an average of $25 \mathrm{~m}$ . p.g. with a standard deviation of $8 \mathrm{~m} . p . g$ .

-Conduct the hypothesis test if $\alpha=0.01$ .

Question 26

A bank would like to evaluate whether there is a difference in the effectiveness in the methods they use to teach their management trainees. For 10 employees that are taught by method 1, the average score on an evaluation examination was 86 with a standard deviation of 6 . For 12 employees taught by method 2 , the average score was 81 with a standard deviation of 4.
-State the hypotheses for the two-tailed test in symbols.

Question 27

A bank would like to evaluate whether there is a difference in the effectiveness in the methods they use to teach their management trainees. For 10 employees that are taught by method 1, the average score on an evaluation examination was 86 with a standard deviation of 6 . For 12 employees taught by method 2 , the average score was 81 with a standard deviation of 4.

-Conduct the hypothesis test using $\alpha=0.05$ .

Question 28

A housing developer suspects that housing prices in a geographical area are higher than the national average.
-If the developer decides to build a housing community only if a survey of housing prices in the area confirms his suspicions, what hypothesis and alternative should he set up?

Question 29

A housing developer suspects that housing prices in a geographical area are higher than the national average.
-If the developer decides to build the community and the survey of housing prices does not produce an average significantly lower than the national average, what hypothesis and alternative should he set up?

Question 30

A housing developer suspects that housing prices in a geographical area are higher than the national average. Suppose the average national home price is $\$ 64,000$ with a standard deviation of $\$ 5,000$ , and the developer decides to build the community only if, in a random sample of 50 newly -sold houses, the average price is more than $\$ 66,000$ .
-What is the probability that the developer builds the community when the prices in the area are only average?

Question 31

A housing developer suspects that housing prices in a geographical area are higher than the national average. Suppose the average national home price is $\$ 64,000$ with a standard deviation of $\$ 5,000$ , and the developer decides to build the community only if, in a random sample of 50 newly -sold houses, the average price is more than $\$ 66,000$ .
-What is the probability of the developer not building the community if the area price average is $\$ 67,000$ ?

Question 32

We would like to estimate the average amount of time that all students at a certain college spent studying during Spring Weekend. We would like to be $\mathbf{9 0} \%$ confident that our estimate is within .5 hours of the actual population average.
-Find the minimum number of students that need to be sampled if it is known that the maximum and minimum amounts of hours were done by students who studied for 15 hours and 0 hours respectively.

Question 33

We would like to estimate the average amount of time that all students at a certain college spent studying during Spring Weekend. We would like to be $\mathbf{9 0} \%$ confident that our estimate is within .5 hours of the actual population average.
-Find a $90 \%$ confidence interval for the average study time of all students at the college if the sample average was 3.5 hours.

Question 34

The manufacturers of a deodorant claim that the mean drying time of their product is, at most, 15 minutes. A sample consisting of 16 cans of the product was used to test the manufacturer's claim. The experiment yielded a mean drying time of 18 minutes with a standard deviation of 6 minutes. Test the claim at the 0.05 significance level.

Question 35

A college food service wanted to determine whether male students consumed more pizza during a given week than female students. A sample of 60 male students found an average consumption of 48 ounces with a standard deviation of 30 ounces. A sample of 80 female students finds an average consumption of 25 ounces with a standard deviation of 20 ounces. Test the appropriate hypothesis at the 0.01 significance level.

Question 36

If hypotheses are $H_{0}: \mu=50 ; H_{A}: \mu \neq 50$ with a sample size of 36 and $s=4$ , find the critical value of $x$ necessary to reject $H_{0}$ for $\alpha=0.02$ .

Question 37

A motor vehicle bureau claims that the mean time it takes an arriving applicant to obtain a new registration is 30 minutes. If, as suspected, this figure is too low, what null and alternative hypotheses should be used to put this to the test?

Question 38

In the hypothesis tests for large and small sample means and differences between means, the left hand term in the numerator is always a population value.

Question 39

A hypothesis test is always an attempt to make a decision about a sample value.

Question 40

The alternative hypothesis can never contain an equal sign.

Question 41

If we increase the confidence level, then we decrease the width of the interval.

Question 42

The larger the sample size $n$ , the smaller the maximum error of estimate.

Question 43

A confidence interval found by using the sample mean as an estimate is an attempt to provide an interval for the population mean.

Question 44

The level of significance should always be specified before a significance test is performed.

Question 45

Increasing the sample size increases the probability of making a Type I error.

Question 46

If we reject a true null hypothesis, then we are commit ting a Type II error.

Question 47

The given confidence level is substituted into the formula used to obtain the required sample size.

Question 48

In a one-tailed test that a population mean is at least 50, the alternative hypothesis can be written in symbols as __________.

Question 49

The probability of a Type I error is equal to __________.

Question 50

The population value that $\bar{x}_{1}-\bar{x}_{2}$ estimates is __________.

Question 51

The failure to reject a false null hypothesis is called a __________ error.

Question 52

A small-sample confidence interval formula for a population mean differs from the corresponding large-sample formula in that the small-sample formula contains __________.

Question 53

A sample size of 14 is used to find a confidence interval. The number of degrees of freedom needed is __________.

Question 54

The value of $z$ that separates the acceptance region from the rejection region is called __________.

Question 55

In a test that a difference of two population means is equal to 5, the alternative hypothesis can be written in symbols as __________.