Question 1

The Boston public school district has had difficulty maintaining on-time bus service for its students ("A Year Later, School Buses Still Late," Boston Globe, October 5). Suppose the district develops a new bus schedule to help combat chronic lateness on a particularly woeful route. Historically, the bus service on the route has been, on average, 12 minutes late. After the schedule adjustment, the first 36 runs were an average of eight minutes late. As a result, the Boston public school district claimed that the schedule adjustment was an improvement-students were not as late. Assume a population standard deviation for bus arrival time of 12 minutes. The value of the test statistic is ________.

Accepted Answer

Since we are given a population standard deviation, we can use a z-test. The null hypothesis is that the average bus arrival time is still 12 minutes late, while the alternative hypothesis is that the average bus arrival time is less than 12 minutes late (improved). Using the sample mean, sample size, and population standard deviation, we can calculate the test statistic: z = (8-12)/(12/sqrt(36)) = -2.00. Therefore, the correct answer is B.

Question 2

A schoolteacher is worried that the concentration of dangerous, cancer-causing radon gas in her classroom is greater than the safe level of 4pCi/L. The school samples the air for 36 days and finds an average concentration of 4.4pCi/L with a standard deviation of 1pCi/L. The value of the test statistic is ________.

Accepted Answer

The test statistic is calculated using the t-distribution because the sample size is small (n=36). The formula for the t-statistic is (sample mean - population mean) / (sample standard deviation / sqrt(n)). Plugging in the values: (4.4 - 4) / (1 / sqrt(36)) = 2.40.

Question 3

A car dealer who sells only late-model luxury cars recently hired a new salesperson and believes that this salesperson is selling at lower markups. He knows that the long-run average markup in his lot is $5,600. He takes a random sample of 16 of the new salesperson's sales and finds an average markup of $5,000 and a standard deviation of $800. Assume the markups are normally distributed. What is the value of an appropriate test statistic for the car dealer to use to test his claim?

Accepted Answer

The appropriate test statistic is a t-test because the population standard deviation is unknown and the sample size is relatively small. The test statistic is calculated using the formula: $t = \frac{\bar{x} - \mu}{(s/\sqrt{n})}$, where $\bar{x}$ is the sample mean, $\mu$ is the population mean, $s$ is the sample standard deviation, and $n$ is the sample size. Plugging in the given values: $t = \frac{5000 - 5600}{(800/\sqrt{16})} = \frac{-600}{200} = -3$.

Question 4

A 99% confidence interval for the population mean yields the following results: [−3.79, 5.86]. At the 1% significance level, what decision should be made regarding the following hypothesis test with H₀: μ = 0, H_A: μ ≠ 0?

Accepted Answer

Since the confidence interval contains zero, we cannot reject the null hypothesis at the 1% significance level. Therefore, we cannot conclude that the mean differs from zero. The correct choice is D, "Do not reject H₀; we cannot conclude that the mean differs from zero."

Question 5

A university interested in tracking its honors program believes that the proportion of graduates with a GPA of 3.00 or below is less than 0.20. In a sample of 200 graduates, 30 students have a GPA of 3.00 or below. In testing the university's belief, how does one define the population parameter of interest?

Accepted Answer

The population parameter of interest is the proportion of graduates with a GPA of 3.00 or below. This is because the university is interested in testing whether the proportion in the population is less than 0.20. Therefore, the parameter of interest is the proportion, not the mean, number, or standard deviation.

Question 6

The Boston public school district has had difficulty maintaining on-time bus service for its students ("A Year Later, School Buses Still Late," Boston Globe, October 5). Suppose the district develops a new bus schedule to help combat chronic lateness on a particularly woeful route. Historically, the bus service on the route has been, on average, 12 minutes late. After the schedule adjustment, the first 36 runs were an average of eight minutes late. As a result, the Boston public school district claimed that the schedule adjustment was an improvement-students were not as late. Assume a population standard deviation for bus arrival time of 12 minutes. Which of the following can be used to determine whether the schedule adjustment reduced the average lateness time of 12 minutes?

Accepted Answer

The answer of The Boston public school district has had...

Question 7

To test if the mean IQ of employees in an organization is greater than 100, a sample of 30 employees is taken and the value of the test statistic is computed as t₂₉ = 2.42 If we choose a 5% significance level, we ________.

Accepted Answer

The null hypothesis H0 states that the mean IQ of employees in the organization is equal to 100. The alternative hypothesis Ha states that the mean IQ is greater than 100. 
At a 5% significance level, the critical value of the t-distribution with 29 degrees of freedom (sample size -1) is 1.699. Since the computed t-statistic of 2.42 is greater than the critical value, we reject the null hypothesis. Therefore, we conclude that the mean IQ of employees in the organization is greater than 100.

Question 8

The owner of a large car dealership believes that the financial crisis decreased the number of customers visiting her dealership. The dealership has historically had 800 customers per day. The owner takes a sample of 100 days and finds the average number of customers visiting the dealership per day was 750. Assume that the population standard deviation is 350. At the 5% significance level, the decision is to ________.

Accepted Answer

The answer of The owner of a large car dealership...

Question 9

A university interested in tracking its honors program believes that the proportion of graduates with a GPA of 3.00 or below is less than 0.20. In a sample of 200 graduates, 30 students have a GPA of 3.00 or below. The value of the test statistic and its associated p-value are ________.

Accepted Answer

Since the sample size is large (n=200), we can use a z-test. The test statistic can be calculated as (p_hat - p) / sqrt(p * (1-p) / n), where p_hat is the sample proportion of graduates with a GPA of 3.00 or below, p is the hypothesized proportion (0.20), and n is the sample size. The calculation yields a test statistic of z = -1.77. The p-value can be calculated using a standard normal table or calculator, and is approximately equal to 0.0385. Therefore, the best answer is B.

Question 10

A newly hired basketball coach promised a high-paced attack that will put more points on the board than the team's previously tepid offense historically managed. After a few months, the team owner looks at the data to test the coach's claim. He takes a sample of 36 of the team's games under the new coach and finds that they scored an average of 101 points with a standard deviation of 6 points. Over the past 10 years, the team had averaged 99 points. What is the value of the appropriate test statistic to test the new coach's claim at the 1% significance level?

Accepted Answer

Since we are comparing the sample mean (101) to a known population mean (99), we use a one-sample t-test. The formula for the t-test statistic is:

t = (x̄ - μ) / (s / sqrt(n))

Where x̄ is the sample mean (101), μ is the population mean (99), s is the sample standard deviation (6), and n is the sample size (36).

Plugging in the numbers, we get:

t = (101 - 99) / (6 / sqrt(36)) = 3

At the 1% significance level, with 35 degrees of freedom (n-1), the critical value for a one-tailed test is 2.03. Since our calculated t-value (3) is greater than the critical value (2.03), we reject the null hypothesis and conclude that there is sufficient evidence to support the coach's claim that the new high-paced attack has led to a higher point average than in the past.

Question 11

You want to test if more than 20% of homes in a neighborhood have recently been sold through a short sale, at a foreclosure auction, or by the bank following an unsuccessful foreclosure auction. You take a sample of 60 homes from this neighborhood and find that 14 fit your criteria. The appropriate null and alternative hypotheses are ________.

Accepted Answer

The null hypothesis should state that the proportion is less than or equal to 20%, so H₀: p ≤ 0.20 is appropriate. The alternative hypothesis should state that the proportion is greater than 20%, so H_A: p > 0.20 is also appropriate. Option A has a typo and does not specify the proportion correctly. Option B suggests that the null hypothesis is equal to 0.23, which is not correct. Option D has the same issue as option B, and also uses a point estimate instead of a range.

Question 12

A university interested in tracking its honors program believes that the proportion of graduates with a GPA of 3.00 or below is less than 0.20. In a sample of 200 graduates, 30 students have a GPA of 3.00 or below. In testing the university's belief, the appropriate hypotheses are ________.

Accepted Answer

The answer of A university interested in tracking its honors...

Question 13

The owner of a large car dealership believes that the financial crisis decreased the number of customers visiting her dealership. The dealership has historically had 800 customers per day. The owner takes a sample of 100 days and finds the average number of customers visiting the dealership per day was 750. Assume that the population standard deviation is 350.To determine whether there has been a decrease in the average number of customers visiting the dealership daily, the appropriate hypotheses are ________.

Accepted Answer

Since the owner believes that the financial crisis decreased the number of customers visiting her dealership, we are interested in testing whether the mean number of customers per day is lower than the historical average of 800. Therefore, the appropriate null hypothesis is H₀: μ ≥ 800 (meaning that there is no decrease in the mean number of customers per day) and the alternative hypothesis is H_A: μ < 800 (meaning that there is a decrease in the mean number of customers per day). The appropriate choice is therefore C.

Question 14

A schoolteacher is worried that the concentration of dangerous, cancer-causing radon gas in her classroom is greater than the safe level of 4pCi/L. The school samples the air for 36 days and finds an average concentration of 4.4pCi/L with a standard deviation of 1pCi/L. At a 5% significance level, the decision is to ________.

Accepted Answer

The sample mean of 4.4pCi/L is higher than the safe level of 4pCi/L, and with a standard deviation of 1pCi/L over 36 days, a t-test can be used to determine if this difference is statistically significant at the 5% level. Given the information, rejecting the null hypothesis (that the mean concentration is not greater than the safe level) is appropriate, indicating the mean concentration of radon gas is statistically significantly greater than the safe level.

Question 15

What type of data would necessitate using a hypothesis test of the population proportion rather than a test of the population mean?

Accepted Answer

A hypothesis test of the population proportion is used when dealing with categorical or qualitative data, such as the proportion of individuals who prefer a specific brand of soda or the proportion of seeds that will germinate under a certain set of conditions. On the other hand, a test of the population mean is used when dealing with quantitative or numerical data, such as the average height of basketball players or the average weight of apples in a basket. Ratio and interval data are also quantitative, so they would not necessitate using a test of the population proportion.

Question 16

A university interested in tracking its honors program believes that the proportion of graduates with a GPA of 3.00 or below is less than 0.20. In a sample of 200 graduates, 30 students have a GPA of 3.00 or below. At a 5% significance level, the decision is to ________.

Accepted Answer

The proportion of graduates with a GPA of 3.00 or below in the sample is 30/200 = 0.15, which is less than the hypothesized proportion of 0.20. At a 5% significance level, if the test statistic calculated from this sample data falls into the critical region, it suggests that the sample provides enough evidence to reject the null hypothesis (that the true proportion is at least 0.20) in favor of the alternative hypothesis (that the true proportion is less than 0.20). Therefore, without the exact p-value or test statistic provided, the description implies that the statistical test led to rejecting the null hypothesis, supporting the conclusion that the proportion is significantly less than 0.20.

Question 17

A schoolteacher is worried that the concentration of dangerous, cancer-causing radon gas in her classroom is greater than the safe level of 4pCi/L. The school samples the air for 36 days and finds an average concentration of 4.4pCi/L with a standard deviation of 1pCi/L. To test whether the average level of radon gas is greater than the safe level, the appropriate hypotheses are ________.

Accepted Answer

The answer of A schoolteacher is worried that the concentration...

Question 18

A recent report claimed that Americans are retiring later in life (U.S. News & World Report, August 17). An economist wishes to determine if the mean retirement age has increased from 62. To conduct the relevant test, she takes a random sample of 38 Americans who have recently retired and computes the value of the test statistic as t₃₇ = 1.92. With α = 0.05, she ________.

Accepted Answer

The null hypothesis is that the mean retirement age is 62, and the alternative hypothesis is that the mean retirement age has increased. The economist is conducting a one-sample t-test with a sample size of 38 and α = 0.05.

Using a t-distribution table with 37 degrees of freedom (n-1), the critical value for a two-tailed test at α = 0.05 is 2.024. Since the calculated t-statistic of 1.92 is less than the critical value, the economist fails to reject the null hypothesis.

Therefore, she cannot conclude that the mean retirement age has increased. However, given the test's limitations, she may want to consider conducting further research.

Question 19

The owner of a large car dealership believes that the financial crisis decreased the number of customers visiting her dealership. The dealership has historically had 800 customers per day. The owner takes a sample of 100 days and finds the average number of customers visiting the dealership per day was 750. Assume that the population standard deviation is 350. The value of the test statistic is ________.

Accepted Answer

The test statistic is calculated using the formula for a z-test because the population standard deviation is known. The formula is $z = \frac{\bar{x} - \mu}{(\sigma/\sqrt{n})}$, where $\bar{x}$ is the sample mean, $\mu$ is the population mean, $\sigma$ is the population standard deviation, and $n$ is the sample size. Plugging in the given values: $z = \frac{750 - 800}{(350/\sqrt{100})} = \frac{-50}{35} = -1.429$.

Question 20

The owner of a large car dealership believes that the financial crisis decreased the number of customers visiting her dealership. The dealership has historically had 800 customers per day. The owner takes a sample of 100 days and finds the average number of customers visiting the dealership per day was 750. Assume that the population standard deviation is 350. The population parameter to be tested is ________.

Accepted Answer

The population parameter being tested is the mean number of customers visiting the dealership per day. This is the central tendency that is being compared to the sample mean of 750 customers per day. The other answer choices are not relevant to this test as they do not relate to the population parameter being compared.

Quiz 9: Hypothesis Testing

A 99% confidence interval for the population mean yields the following results: [−3.79, 5.86]. At the 1% significance level, what decision should be made regarding the following hypothesis test with H0: μ = 0, HA: μ ≠ 0?

To test if the mean IQ of employees in an organization is greater than 100, a sample of 30 employees is taken and the value of the test statistic is computed as t29 = 2.42 If we choose a 5% significance level, we ________.