Question 1

How many tissues should the Kimberly Clark Corporation package of Kleenex contain? Researchers determined that 60 tissues is the mean number of tissues used during a cold. Suppose a random sample of 100 Kleenex users yielded the following data on the number of tissues used during a cold: X = 52, S = 22. Suppose the alternative you wanted to test was H₁: μ < 60. State the correct rejection region for α = 0.05. A) Reject H₀ if t > 1.6604. B) Reject H₀ if t < -1.6604. C) Reject H₀ if t > 1.9842 or Z < -1.9842. D) Reject H₀ if t < -1.9842.

Accepted Answer

For a one-tailed t-test with a sample size of 100, a significance level of 0.05, and the alternative hypothesis of μ < 60, the rejection region would be t < -1.6604. This can be found using a t-distribution table or calculator. The option B is the closest to this rejection region. The other options either have incorrect values or include a Z-score instead of a t-score.

Question 2

How many tissues should the Kimberly Clark Corporation package of Kleenex contain? Researchers determined that 60 tissues is the mean number of tissues used during a cold. Suppose a random sample of 100 Kleenex users yielded the following data on the number of tissues used during a cold: X = 52, S = 22. Suppose the test statistic does fall in the rejection region at α = 0.05. Which of the following decision is correct?
A) At α = 0.05, you do not reject H₀.
B) At α = 0.05, you reject H₀.
C) At α = 0.10, you do not reject H₀.
D) At α = 0.10, you reject H₀.

Accepted Answer

We need to perform a one-sample t-test with the following hypotheses:
H₀: μ = 60 (the true mean number of tissues used during a cold is 60)
H₁: μ ≠ 60 (the true mean number of tissues used during a cold is not 60)

The sample mean is X̄ = 52 and the sample standard deviation is S = 22. The sample size is n = 100 and the degrees of freedom are df = n - 1 = 99.

The t-score is calculated as:
t = (X̄ - μ) / (S / sqrt(n)) = (52 - 60) / (22 / sqrt(100)) = -3.64

At α = 0.05, the critical values for a two-tailed test with 99 degrees of freedom are ±1.984. Since the absolute value of the t-score (3.64) is greater than the critical value (1.984), we reject the null hypothesis.

Therefore, the correct decision is to reject H₀ at α = 0.05, which is option B.

Question 3

The value that separates a rejection region from a non-rejection region is called the ________.

Accepted Answer

The answer of The value that separates a rejection region...

Question 4

How many tissues should the Kimberly Clark Corporation package of Kleenex contain? Researchers determined that 60 tissues is the mean number of tissues used during a cold. Suppose a random sample of 100 Kleenex users yielded the following data on the number of tissues used during a cold: X = 52, S = 22. Give the null and alternative hypotheses to determine if the number of tissues used during a cold is less than 60.
A) H₀: μ ≤ 60 and H₁: μ > 60.
B) H₀: μ ≥ 60 and H₁: μ < 60.
C) H₀: X ≥ 60 and H₁: X < 60.
D) H₀: X = 52 and H₁: X ≠ 52.

Accepted Answer

The null hypothesis should state that the mean number of tissues used during a cold is greater than or equal to 60, while the alternative hypothesis should state that the mean is less than 60, as we are testing whether the number of tissues used is less than the reported mean of 60. Thus, option B is the correct choice. Option A is incorrect because it assumes that the mean is already less than 60. Option C is incorrect because it uses the sample mean, rather than the population mean, as the statistic of interest. Option D is incorrect because it is testing whether the sample mean is different from 52, rather than testing whether the population mean is less than 60.

Question 5

The power of a statistical test is
A) the probability of not rejecting H₀ when it is false.
B) the probability of rejecting H₀ when it is true.
C) the probability of not rejecting H₀ when it is true.
D) the probability of rejecting H₀ when it is false.

Accepted Answer

The power of a statistical test is the probability of correctly rejecting the null hypothesis (H₀) when it is false.

Question 6

How many tissues should the Kimberly Clark Corporation package of Kleenex contain? Researchers determined that 60 tissues is the mean number of tissues used during a cold. Suppose a random sample of 100 Kleenex users yielded the following data on the number of tissues used during a cold: X = 52, S = 22. Using the sample information provided, calculate the value of the test statistic.
A) t = (52 - 60)/22
B) t = (52 - 60)/(22/100)
C) t = (52 - 60)/(22/100²)
D) t = (52 - 60)/(22/10)

Accepted Answer

The test statistic is calculated using the formula t = (X - μ) / (S/√n), where X is the sample mean, μ is the population mean, S is the sample standard deviation, and n is the sample size. Here, t = (52 - 60) / (22/√100) = (52 - 60) / (22/10).

Question 7

Suppose we wish to test H₀: μ ≤ 47 versus H₁: μ > 47. What will result if we conclude that the mean is greater than 47 when its true value is really 52?
A) We have made a Type I error.
B) We have made a Type II error.
C) We have made a correct decision.
D) None of the above is correct.

Accepted Answer

We have made a correct decision because the true mean (52) is indeed greater than 47, which aligns with the alternative hypothesis H₁: μ > 47.

Question 8

The owner of a local nightclub has recently surveyed a random sample of n = 250 customers of the club. She would now like to determine whether or not the mean age of her customers is greater than 30. If so, she plans to alter the entertainment to appeal to an older crowd. If not, no entertainment changes will be made. If she wants to have a level of significance at 0.01, what rejection region should she use? A) Reject H₀ if t < -2.3263. B) Reject H₀ if t < -2.5758. C) Reject H₀ if t > 2.3263. D) Reject H₀ if t > 2.5758.

Accepted Answer

Since the owner wants to test if the mean age is greater than 30, a one-tailed test is appropriate. At a 0.01 significance level, the critical t-value for a one-tailed test is approximately 2.3263. Therefore, the rejection region is t > 2.3263.

Question 9

Which of the following statements is not true about the level of significance in a hypothesis test?
A) The larger the level of significance, the more likely you are to reject the null hypothesis.
B) The level of significance is the maximum risk we are willing to accept in making a Type I error.
C) The significance level is also called the α level.
D) The significance level is another name for Type II error.

Accepted Answer

The significance level is not another name for Type II error. The level of significance is specifically associated with Type I error, which is the probability of rejecting a true null hypothesis. Type II error is the probability of failing to reject a false null hypothesis, and is denoted by β.

Question 10

If, as a result of a hypothesis test, you reject the null hypothesis when it is false, then you have committed
A) a Type II error.
B) a Type I error.
C) no error.
D) an acceptance error.

Accepted Answer

Rejecting the null hypothesis when it is false means you have made the correct decision, hence no error has been committed.

Question 11

You have created a 95% confidence interval for μ with the result 10 ≤ μ ≤ 15. What decision will you make if we test H₀: μ = 16 versus H₁: μ ≠ 16 at α = 0.01?
A) Reject H₀ in favor of H₁.
B) Do not reject H₀ in favor of H₁.
C) Fail to reject H₀ in favor of H₁.
D) You cannot tell what our decision will be from the information given.

Accepted Answer

The answer of You have created a 95% confidence interval...

Question 12

The owner of a local nightclub has recently surveyed a random sample of n = 250 customers of the club. She would now like to determine whether or not the mean age of her customers is greater than 30. If so, she plans to alter the entertainment to appeal to an older crowd. If not, no entertainment changes will be made. The appropriate hypotheses to test are: A) H₀: μ ≥ 30 versus H₁: μ < 30. B) H₀: μ ≤ 30 versus H₁: μ > 30. C) H₀: X ≥ 30 versus H₁: X < 30. D) H₀: X ≤ 30 versus H₁: X > 30.

Accepted Answer

The null hypothesis (H₀) should always assume no effect, in this case that the mean age is less than or equal to 30. The alternative hypothesis (H₁) should assume an effect in the direction of interest, in this case that the mean age is greater than 30. Therefore, the appropriate hypotheses to test are H₀: μ ≤ 30 (no effect) versus H₁: μ > 30 (effect in the direction of interest). Choice B is the only option that matches this format.

Question 13

The symbol for the power of a statistical test is
A) α.
B) 1 - α.
C) β.
D) 1 - β.

Accepted Answer

The power of a statistical test is defined as 1 - β, where β is the probability of making a Type II error (failing to reject a false null hypothesis).

Question 14

A ________ is a numerical quantity computed from the data of a sample and is used in reaching a decision on whether or not to reject the null hypothesis.
A) significance level
B) critical value
C) test statistic
D) parameter

Accepted Answer

A test statistic is a numerical quantity calculated from sample data that is used to determine whether or not to reject the null hypothesis. The test statistic is compared to a critical value, which is derived based on the significance level and the degrees of freedom, to make this decision. A parameter is a numerical quantity that describes a population, not a sample. The significance level is a predetermined level of risk of rejecting the null hypothesis when it is actually true.

Question 15

It is possible to directly compare the results of a confidence interval estimate to the results obtained by testing a null hypothesis if
A) a two-tail test for μ is used.
B) a one-tail test for μ is used.
C) Both of the previous statements are true.
D) None of the previous statements is true.

Accepted Answer

The answer of It is possible to directly compare the...

Question 16

Suppose you want to test H₀: μ ≥ 30 versus H₁: μ < 30. Which of the following possible sample results based on a sample of size 36 gives the strongest evidence to reject H₀ in favor of H₁?
A) X = 28, S = 6
B) X = 27, S = 4
C) X = 32, S = 2
D) X = 26, S = 9

Accepted Answer

The sample with the lowest mean (X = 27) and the lowest standard deviation (S = 4) provides the strongest evidence to reject H₀ in favor of H₁.

Question 17

You have created a 95% confidence interval for μ with the result 10 ≤ μ ≤ 15. What decision will you make if you test H₀: μ = 16 versus H₁: μ ≠ 16 at α = 0.05?
A) Reject H₀ in favor of H₁.
B) Do not reject H₀ in favor of H₁.
C) Fail to reject H₀ in favor of H₁.
D) We cannot tell what our decision will be from the information given.

Accepted Answer

The value 16 is not within the 95% confidence interval (10 to 15), so we reject H₀ at the 0.05 significance level.

Question 18

You have created a 95% confidence interval for μ with the result 10 ≤ μ ≤ 15. What decision will you make if we test H₀: μ = 16 versus H₁: μ ≠ 16 at α = 0.10?
A) Reject H₀ in favor of H₁.
B) Do not reject H₀ in favor of H₁.
C) Fail to reject H₀ in favor of H₁.
D) We cannot tell what our decision will be from the information given.

Accepted Answer

The confidence interval does not include 16, so we reject H₀ at the 0.10 significance level.

Question 19

The owner of a local nightclub has recently surveyed a random sample of n = 250 customers of the club. She would now like to determine whether or not the mean age of her customers is greater than 30. If so, she plans to alter the entertainment to appeal to an older crowd. If not, no entertainment changes will be made. Suppose she found that the sample mean was 30.45 years and the sample standard deviation was 5 years. If she wants to have a level of significance at 0.01, what decision should she make?
A) Reject H₀.
B) Reject H₁.
C) Do not reject H₀.
D) We cannot tell what her decision should be from the information given.

Accepted Answer

The null hypothesis, H₀, is that the mean age of customers is not greater than 30. The alternative hypothesis, H₁, is that the mean age is greater than 30.

At a significance level of 0.01, we need to find the critical value for a one-tailed test (since H₁ is one-tailed). Using a t-distribution with 249 degrees of freedom (since n-1 = 249), we find the critical value to be 2.33 (from a t-table or calculator).

Next, we calculate the test statistic:

t = (sample mean - hypothesized mean) / (sample standard deviation / sqrt(sample size)) 
t = (30.45 - 30) / (5 / sqrt(250)) 
t = 1.84

Since 1.84 is not greater than 2.33, we fail to reject the null hypothesis. We do not have sufficient evidence to conclude that the mean age of customers is greater than 30, and so the nightclub owner should not alter the entertainment to appeal to an older crowd.

Question 20

How many tissues should the Kimberly Clark Corporation package of Kleenex contain? Researchers determined that 60 tissues is the mean number of tissues used during a cold. Suppose a random sample of 100 Kleenex users yielded the following data on the number of tissues used during a cold: X = 52, S = 22. Suppose the test statistic does fall in the rejection region at α = 0.05. Which of the following conclusion is correct?
A) At α = 0.05, there is not sufficient evidence to conclude that the mean number of tissues used during a cold is 60 tissues.
B) At α = 0.05, there is sufficient evidence to conclude that the mean number of tissues used during a cold is 60 tissues.
C) At α = 0.05, there is insufficient evidence to conclude that the mean number of tissues used during a cold is not 60 tissues.
D) At α = 0.10, there is sufficient evidence to conclude that the mean number of tissues used during a cold is not 60 tissues.

Accepted Answer

The answer of How many tissues should the Kimberly Clark...

Quiz 9: Fundamentals of Hypothesis Testing: One-Sample Tests

The value that separates a rejection region from a non-rejection region is called the ________.

The power of a statistical test is

Suppose we wish to test H₀: μ ≤ 47 versus H₁: μ > 47. What will result if we conclude that the mean is greater than 47 when its true value is really 52?

Which of the following statements is not true about the level of significance in a hypothesis test?

If, as a result of a hypothesis test, you reject the null hypothesis when it is false, then you have committed

You have created a 95% confidence interval for μ with the result 10 ≤ μ ≤ 15. What decision will you make if we test H₀: μ = 16 versus H₁: μ ≠ 16 at α = 0.01?

The symbol for the power of a statistical test is

A ________ is a numerical quantity computed from the data of a sample and is used in reaching a decision on whether or not to reject the null hypothesis.

It is possible to directly compare the results of a confidence interval estimate to the results obtained by testing a null hypothesis if

Suppose you want to test H₀: μ ≥ 30 versus H₁: μ < 30. Which of the following possible sample results based on a sample of size 36 gives the strongest evidence to reject H₀ in favor of H₁?

You have created a 95% confidence interval for μ with the result 10 ≤ μ ≤ 15. What decision will you make if you test H₀: μ = 16 versus H₁: μ ≠ 16 at α = 0.05?

You have created a 95% confidence interval for μ with the result 10 ≤ μ ≤ 15. What decision will you make if we test H₀: μ = 16 versus H₁: μ ≠ 16 at α = 0.10?