Question 1

In the past, the mean running time for a certain type of flashlight battery has been 8.0 hours. The manufacturer has introduced a change in the production method and wants to perform a hypothesis
Test to determine whether the mean running time has increased as a result. The hypotheses are: $$\begin{array} { l } 
\mathrm { H } _ { 0 } : \mu = 8.0 \text { hours } \
\mathrm { H } _ { \mathrm { a } } : \mu > 8.0 \text { hours }
\end{array}$$
Explain the meaning of a Type II error.
A) Concluding that $\mu > 8.0$ hours when in fact $\mu > 8.0$ hours
B) Failing to reject the hypothesis that $\mu = 8.0$ hours when in fact $\mu > 8.0$ hours
C) Concluding that $\mu > 8.0$ hours
D) Failing to reject the hypothesis that $\mu = 8.0$ hours when in fact $\mu = 8.0$ hours 110

Accepted Answer

A Type II error occurs when we fail to reject the null hypothesis ($H_0$) when it is actually false. In this context, it means failing to reject $\mu = 8.0$ hours when in fact $\mu > 8.0$ hours.

Question 2

A skeptical paranormal researcher claims that the proportion of Americans that have seen a UFO, p, is less than 1 in every ten thousand. Assume that a hypothesis test of the claim has been conducted and that the conclusion of the test was to reject the null hypothesis. Identify the population to which the
Results of the test apply.
A) Any sample of Americans
B) Any sample of Americans claiming to have seen a UFO
C) All Americans claiming to have seen a UFO
D) All Americans

Accepted Answer

The population to which the results of the test apply is all Americans, as the hypothesis test was conducted to make a general statement about the proportion of all Americans who have seen a UFO.

Question 3

A two-tailed test is conducted at the 5% significance level. What is the P-value required to reject the null hypothesis?
A) Greater than or equal to 0.10
B) Less than or equal to 0.05
C) Less than or equal to 0.10
D) Greater than or equal to 0.05
E) Less than or equal to 0.025

Accepted Answer

In a two-tailed test at the 5% significance level, the null hypothesis is rejected if the P-value is less than or equal to 0.05.

Question 4

The owner of a football team claims that the average attendance at home games is over 4000, and he is therefore justified in moving the team to a city with a larger stadium. Assume that a hypothesis test of the claim has been conducted and that the conclusion of the test was to reject the null hypothesis.
Identify the population to which the results of the test apply.
A) All games played by the team in question in which the attendance is over 4000
B) All future home games to be played by the team in question
C) All home games played by the team in question
D) None of the populations given are appropriate.

Accepted Answer

The population to which the results of the test apply is all home games played by the team in question, as the hypothesis test was conducted on the average attendance at these games.

Question 5

A consumer advocacy group claims that the mean amount of juice in a 16 ounce bottled drink is not 16 ounces, as stated by the bottler. Determine the conclusion of the hypothesis test assuming that the results of the sampling lead to rejection of the null hypothesis.
A) Conclusion: Support the claim that the mean is equal to 16 ounces.
B) Conclusion: Support the claim that the mean is greater than 16 ounces.
C) Conclusion: Support the claim that the mean is not equal to 16 ounces.
D) Conclusion: Support the claim that the mean is less than 16 ounces.

Accepted Answer

If the null hypothesis is rejected, it supports the claim that the mean is not equal to 16 ounces.

Question 6

A consumer group claims that the mean running time for a certain type of flashlight battery is not the same as the manufacturer's claims. Determine the null and alternative hypotheses for the test
Described. 
A) $ \mathrm{H}_{0}: \mu= $ Manufacturer's claims
$ \mathrm{H}_{\mathrm{a}}: \mu< $ Manufacturer's claims

B) $ \mathrm{H}_{0}: \mu= $ Manufacturer's claims 
$ \mathrm{H}_{\mathrm{a}}: \mu \neq $ Manufacturer's claims

C) $ \mathrm{H}_{0}: \mu= $ Manufacturer's claims
$ \mathrm{H}_{\mathrm{a}}: \mu> $ Manufacturer's claims

D) $ \mathrm{H}_{0}: \mu \neq $ Manufacturer's claims 
$ \mathrm{H}_{\mathrm{a}}: \mu= $ Manufacturer's claims

Accepted Answer

The null hypothesis ($ \mathrm{H}_{0} $) states that the mean running time is equal to the manufacturer's claims, while the alternative hypothesis ($ \mathrm{H}_{a} $) suggests it is not equal, indicating a two-tailed test.

Question 7

$\mathrm { H } _ { \mathrm { a } } : \mu > 19.9 , \bar { x } = 23.1 , \sigma = 8 , \mathrm { n } = 100$. Without computing a $\mathrm { P }$-value, determine whether the alternate hypothesis is supported and give a reason for your conclusion.
A) $\mathrm { H } _ { \mathrm { a } }$ is not supported : $\bar { x }$ is more than 3 standard deviations above the claimed mean.
B) $\mathrm { H } _ { \mathrm { a } }$ is not supported: $\bar { x }$ is less than 1 standard deviation above the claimed mean.
C) $\mathrm { H } _ { \mathrm { a } }$ is supported ; $\bar { x }$ is more than 3 standard deviations above the claimed mean.
D) $\mathrm { H } _ { \mathrm { a } }$ is supported : $\bar { x }$ is less than 1 standard deviation above the claimed mean.

Accepted Answer

The answer of \(\mathrm { H } _ { \mathrm...

Question 8

A study of a brand of "in the shell peanuts" gives the following results: $$\begin{array} { c | c } 
\begin{array} { l } 
\text { Number of } \
\text { Peanuts per bag }
\end{array} & \text { Probability } \
\hline 25 & 0.006 \
30 & 0.020 \
35 & 0.093 \
40 & 0.150 \
45 & 0.350 \
50 & 0.217 \
55 & 0.170
\end{array}$$ If a fan purchased a bag with 30 peanuts, what is the lowest level at which this would be a significant event?
A) 0.05
B) 0.025
C) 0.01
D) It is not significant at any of the levels given.

Accepted Answer

The answer of A study of a brand of "in...

Question 9

In the past, the mean running time for a certain type of flashlight battery has been 9.8 hours. The manufacturer has introduced a change in the production method and wants to perform a hypothesis test to determine whether the mean running time has increased as a result. The hypotheses are: 
$\mathrm { H } _ { 0 } : \mu = 9.8$ hours
$\mathrm { H } _ { \mathrm { a } } : \mu > 9.8 \text { hours }$
 Suppose that the results of the sampling lead to rejection of the null hypothesis. Classify that conclusion as a Type I error, a Type II error, or a correct decision, if in fact the mean running time has not increased.
A) Type I error
B) Type II error
C) Correct decision

Accepted Answer

Rejecting the null hypothesis when it is actually true is a Type I error.

Question 10

A nationwide study of American homeowners revealed that 65% have one or more lawn mowers. A lawn equipment manufacturer, located in Omaha, feels the estimate is too low for households in omaha. Find the P-value for a test of the claim that the proportion with lawn mowers in Omaha is higher than 65%. Among 497 randomly selected homes in Omaha, 340 had one or more lawn
Mowers. Use Table 5.1 to find the best answer.
A) 0.0559
B) 0.1118
C) 0.0252
D) 0.0505

Accepted Answer

The sample proportion is 340/497 = 0.6843. The null hypothesis is that the proportion is 0.65. Using a one-sample z-test for proportions, the z-score is calculated and then the P-value is found. The P-value for a z-score of approximately 1.60 is about 0.0559.

Question 11

A two-tailed test is conducted at the 5% significance level. What is the right tail percentile required to reject the null hypothesis?
A) 10%
B) 5%
C) 2.5%
D) 95%
E) 97.5%

Accepted Answer

In a two-tailed test at the 5% significance level, each tail has 2.5%. Therefore, the right tail percentile required to reject the null hypothesis is 97.5%.

Question 12

A supplier of DVDs claims that no more than 1% of the DVDs are defective. In a random sample of 600 DVDs, it is found that 3% are defective, but the supplier claims that this is only a sample fluctuation. At the 0.01 level of significance, test the supplier's claim that no more than 1% are defective.

A) Do not reject the null hypothesis and conclude that there is evidence to support the claim that more than 1% of the DVDs are defective.
B) Reject the null hypothesis and conclude that there is insufficient evidence to support the claim that more than 1% of the DVDs are defective.
C) Do not reject the null hypothesis and conclude that there is insufficient evidence to support the claim that more than 1% of the DVDs are defective.
D) Reject the null hypothesis and conclude that there is sufficient evidence to support the claim that more than 1% of the DVDs are defective.

Accepted Answer

The null hypothesis is that the proportion of defective DVDs is 1% or less. With a sample proportion of 3% and a significance level of 0.01, the test statistic will likely fall in the critical region, leading to the rejection of the null hypothesis. Thus, there is sufficient evidence to support the claim that more than 1% of the DVDs are defective.

Question 13

A skeptical paranormal researcher claims that the proportion of Americans that have seen a UFO is less than 1 in every one thousand. State the null hypothesis and the alternative hypothesis for a test of Significance. A) $ \mathrm{H}_{0}: \mathrm{p}=0.001 $ $\mathrm{H}_{\mathrm{a}}: \mathrm{p}>0.001$ B) $ \mathrm{H}_{0}: \mathrm{p}=0.001 $ $\mathrm{H}_{\mathrm{a}}: \mathrm{p}<0.001$ C) $ \mathrm{H}_{0}: \mathrm{p}>0.001 $ $\mathrm{H}_{\mathrm{a}}: \mathrm{p}=0.001$ D) $ \mathrm{H}_{0}: \mathrm{p}<0.001 $ $\mathrm{H}_{\mathrm{a}}: \mathrm{p}=0.001$

Accepted Answer

The null hypothesis ($ \mathrm{H}_{0} $) assumes the proportion is equal to 0.001, while the alternative hypothesis ($ \mathrm{H}_{a} $) reflects the researcher's claim that the proportion is less than 0.001.

Question 14

In 1990, the average duration of long-distance telephone calls originating in one town was 9.4 minutes. A long-distance telephone company wants to perform a hypothesis test to determine whether the average duration of long-distance phone calls has changed from the 1990 mean of 9.4 minutes.
The hypotheses are: $$
\begin{array}{l}
\mathrm{H}_{0}: \mu=9.4 \text { minutes } \
\mathrm{H}_{\mathrm{a}}: \mu \neq 9.4 \text { minutes }
\end{array}
$$
Explain the meaning of a correct decision.
A) Failing to reject the hypothesis that $ \mu=9.4 $ minutes when in fact $ \mu=9.4 $ minutes OR concluding that $ \mu>9.4 $ minutes when in fact $ \mu>9.4 $ minutes
B) Failing to reject the hypothesis that $ \mu=9.4 $ minutes when in fact $ \mu=9.4 $ minutes OR concluding that $ \mu \neq 9.4 $ minutes when in fact $ \mu \neq 9.4 $ minutes
C) Failing to reject the hypothesis that $ \mu=9.4 $ minutes when in fact $ \mu \neq 9.4 $ minutes
D) Concluding that $ \mu \neq 9.4 $ minutes when in fact $ \mu=9.4 $ minutes

Accepted Answer

A correct decision in hypothesis testing is either failing to reject the null hypothesis when it is true or rejecting the null hypothesis when the alternative hypothesis is true.

Question 15

$$\mathrm { z } = 1.8 \text { for } \mathrm { H } _ { \mathrm { a } } : \mu > \text { claimed value. What is the P-value for the test? }$$
A) 0.9641
B) 3.59
C) 96.41
D) 0.0359
E) None of the above

Accepted Answer

The P-value for a one-tailed test with z = 1.8 is 1 - 0.9641 = 0.0359.

Question 16

A researcher claims that the amounts of acetaminophen in a certain brand of cold tablets have a mean different from the 600 mg claimed by the manufacturer. Test this claim at the 0.02 level of significance. The mean acetaminophen content for a random sample of n = 41 tablets is 603.3 mg.
Assume that the population standard deviation is 4.9 mg.
A) Since the test statistic is greater than the critical z, there is sufficient evidence to accept the null hypothesis and to support the claim that the mean content of acetaminophen is 600 mg.
B) Since the test statistic is greater than the critical z, there is sufficient evidence to reject the null hypothesis and to support the claim that the mean content of acetaminophen is not 600 mg.
C) Since the test statistic is less than the critical z, there is sufficient evidence to reject the null hypothesis and to support the claim that the mean content of acetaminophen is not 600 mg.
D) Since the test statistic is greater than the critical z, there is insufficient evidence to reject the null hypothesis and to support the claim that the mean content of acetaminophen is not 600 mg.

Accepted Answer

The test statistic is calculated as (603.3 - 600) / (4.9/√41) ≈ 4.27. The critical z-value for a two-tailed test at the 0.02 level of significance is approximately ±2.33. Since 4.27 is greater than 2.33, we reject the null hypothesis, supporting the claim that the mean content is not 600 mg.

Question 17

The owner of a football team claims that the average attendance at home games is over 3000, and he is therefore justified in moving the team to a city with a larger stadium. Assuming that a hypothesis test of the claim has been conducted and that the conclusion is failure to reject the null hypothesis, State the conclusion in non-technical terms.
A) There is sufficient evidence to support the claim that the mean attendance is greater than 3000.
B) There is sufficient evidence to support the claim that the mean attendance is equal to 3000.
C) There is not sufficient evidence to support the claim that the mean attendance is greater than 3000.
D) There is not sufficient evidence to support the claim that the mean attendance is less than 3000.

Accepted Answer

The failure to reject the null hypothesis suggests that there is not enough evidence to support the claim that the average attendance is greater than 3000.

Question 18

In 1990, the average duration of long-distance telephone calls originating in one town was $ 9.4 $ minutes. A long-distance telephone company wants to perform a hypothesis test to determine whether the average duration of long-distance phone calls has changed from the 1990 mean of $ 9.4 $ minutes. The mean duration for a random sample of 50 calls originating in the town was $ 8.6 $ minutes. Does the data provide sufficient evidence to conclude that the mean call duration, $ \mu $, is different from the 1990 mean of $ 9.4 $ minutes? Perform the appropriate hypothesis test using a significance level of $ 0.01 $. Assume that $ \sigma=4.8 $ minutes.
A) With a $ z $ of $ -1.2 $ there is sufficient eridence to conclude that the mean value has changed from the 1990 mean of $ 9.4 $ minutes.
B) With a P-value of $ 0.2302 $ there is not sufficient evidence to conclude that the mean value is less than the 1990 mean of $ 9.4 $ minutes.
C) With a P-value of $ 0.2302 $ there is sufficient evidence to conclude that the mean value is less than the 1990 mean of $ 9.4 $ minutes.
D) With a $ z $ of $ -1.2 $ there is not sufficient evidence to conclude that the mean value has changed from the 1990 mean of $ 9.4 $ minutes.

Accepted Answer

The calculated z-score is -1.2, and the corresponding p-value is 0.2302, which is greater than the significance level of 0.01. Therefore, there is not sufficient evidence to reject the null hypothesis that the mean call duration has changed from 9.4 minutes.

Question 19

A right-tailed test is conducted at the 5% significance level. Which of the following z-scores is the smallest one in absolute value that leads to rejection of the null hypothesis?
A) 1.61
B) 1.85
C) -1.98
D) -2.06

Accepted Answer

For a right-tailed test at the 5% significance level, the critical z-value is approximately 1.645. Any z-score greater than this value will lead to rejection of the null hypothesis. 1.85 is the smallest z-score greater than 1.645.

Question 20

At one school, the mean amount of time that tenth-graders spend watching television each week is 18.4 hours. The principal introduces a campaign to encourage the students to watch less television.
One year later, the principal wants to perform a hypothesis test to determine whether the average amount of time spent watching television per week has decreased. Formulate the null and alternative hypotheses for the study described.

A) $\mathrm { H } _ { 0 } : \mu = 18.4$ hours
$\mathrm { H } _ { \alpha } : \mu \neq 18.4$ hours

B) $H _ { 0 } : \mu = 18.4$ hours
$\mathrm { H } _ { \alpha } : \mu < 18.4$ hours

C) $\mathrm { H } _ { 0 } : \mu \geq 18.4$ hours
$\mathrm { H } _ { \alpha } : \mu < 18.4$ hours

D) $\mathrm { H } _ { 0 } : \mu = 18.4$ hours
$\mathrm { H } _ { \alpha } : \mu > 18.4$ hours

Accepted Answer

The null hypothesis ($H_0$) assumes no change, so the mean time is still 18.4 hours. The alternative hypothesis ($H_\alpha$) tests if the mean time has decreased, so it should be less than 18.4 hours.

Quiz 9: Hypothesis Testing

A two-tailed test is conducted at the 5% significance level. What is the P-value required to reject the null hypothesis?

A consumer advocacy group claims that the mean amount of juice in a 16 ounce bottled drink is not 16 ounces, as stated by the bottler. Determine the conclusion of the hypothesis test assuming that the results of the sampling lead to rejection of the null hypothesis.

A consumer group claims that the mean running time for a certain type of flashlight battery is not the same as the manufacturer's claims. Determine the null and alternative hypotheses for the test Described.

$\mathrm { H } _ { \mathrm { a } } : \mu > 19.9 , \bar { x } = 23.1 , \sigma = 8 , \mathrm { n } = 100$ . Without computing a $\mathrm { P }$ -value, determine whether the alternate hypothesis is supported and give a reason for your conclusion.

A two-tailed test is conducted at the 5% significance level. What is the right tail percentile required to reject the null hypothesis?

A skeptical paranormal researcher claims that the proportion of Americans that have seen a UFO is less than 1 in every one thousand. State the null hypothesis and the alternative hypothesis for a test of Significance.

$\mathrm { z } = 1.8 \text { for } \mathrm { H } _ { \mathrm { a } } : \mu > \text { claimed value. What is the P-value for the test? }$

A right-tailed test is conducted at the 5% significance level. Which of the following z-scores is the smallest one in absolute value that leads to rejection of the null hypothesis?