Question 1

You are driving a van packed with camping gear (total weight 3,500 pounds including yourself and family) into a northern wilderness area. You take a "short cut" that turns into a one-lane road, with no room to turn around. After 11 miles you come to a narrow bridge with a faded sign saying, "Safe Up to 2 Tons." About a half-mile ahead, you can see that your road rejoins the main highway. You consider the sign's hypothesis carefully before making a decision. The cost of Type I error is:
A)you pass safely over the bridge and everyone's happy.
B)about $23,900, not including medical bills.
C)you will find out just how cold that river actually is.
D)your kids will think you're a chicken.

Accepted Answer

The answer of You are driving a van packed with...

Question 2

Hypothesis tests for a mean using the critical value method require:
A)using a two-tailed test.
B)sampling a normal population.
C)knowing the true population mean.
D)specifying α in advance.

Accepted Answer

Hypothesis tests for a mean using the critical value method require specifying α in advance. The other options are not necessarily required. It can be a one-tailed or two-tailed test, the population does not have to be normal, and knowing the true population mean is usually not possible.

Question 3

Which is not true of p-values? A)When they are small, we want to reject H₀. B)They measure the probability of an incorrect decision. C)They show the chance of Type I error if we reject H₀. D)They do not require α to be specified a priori.

Accepted Answer

P-values measure the probability of observing data at least as extreme as the observed data, assuming the null hypothesis is true. They do not measure the probability of making an incorrect decision.

Question 4

For a right-tailed test of a hypothesis for a population mean with n = 14, the value of the test statistic was t = 1.863. The p-value is:
A)between .05 and .025.
B)between .10 and .05.
C)greater than .10.
D)less than .01.

Accepted Answer

The p-value is the probability of observing a test statistic as extreme as, or more extreme than, the observed value under the null hypothesis. For a t-test with n = 14 (which means 13 degrees of freedom), a t-value of 1.863 falls between the critical values for p = .05 and p = .025 in the t-distribution table, indicating the p-value is between these two values.

Question 5

Guidelines for the Jolly Blue Giant Health Insurance Company say that the average hospitalization for a triple hernia operation should not exceed 30 hours. A diligent auditor studied records of 16 randomly chosen triple hernia operations at Hackmore Hospital and found a mean hospital stay of 40 hours with a standard deviation of 20 hours. "Aha!" she cried, "the average stay exceeds the guideline." At α = .025, the critical value for a right-tailed test of her hypothesis is:
A)1.753
B)2.131
C)1.645
D)1.960

Accepted Answer

Given the significance level (α) of 0.025 for a right-tailed test and the sample size (n) of 16, which implies degrees of freedom (df) of 15 for t-distribution, the critical value closest to the given α in t-distribution tables is 2.131.

Question 6

Dullco Manufacturing claims that its alkaline batteries last at least 40 hours on average in a certain type of portable CD player. But tests on a random sample of 18 batteries from a day's large production run showed a mean battery life of 37.8 hours with a standard deviation of 5.4 hours. In a left-tailed test at α = .05, which is the most accurate statement?
A)We would strongly reject the claim.
B)We would clearly fail to reject the claim.
C)We would face a rather close decision.
D)We would switch to α = .01 for a more powerful test.

Accepted Answer

To conduct a left-tailed test, the null hypothesis is that the true mean battery life is at least 40 hours, and the alternative hypothesis is that the true mean battery life is less than 40 hours. With a sample mean of 37.8 hours and a standard deviation of 5.4 hours, we can calculate the t-statistic as (37.8 - 40) / (5.4 / sqrt(18)) = -2.21. 
Using a t-distribution table with 17 degrees of freedom and a significance level of .05, the critical value is -1.740. Since our calculated t-statistic (-2.21) is less than the critical value (-1.740), we cannot reject the null hypothesis at the .05 level of significance. 
Therefore, the most accurate statement is that we would face a rather close decision.

Question 7

"My careful physical examination shows no evidence of any serious problem," said Doctor Morpheus. "However, a very costly lab test can be performed to check for the rare condition known as estomalgia fatalis. The test is almost invariably negative for persons with your age and symptoms. My personal hypothesis is that the occasional stomach pain you reported is due to indigestion caused by eating tacos with too much hot sauce. But you must decide for yourself." As you consider your doctor's hypothesis, what would be the consequence of Type I error on your part?
A)It can't be determined without knowing the type of test.
B)Your estomalgia fatalis will go undetected.
C)You will waste money on an unnecessary lab test.
D)Your survivors will enjoy a sizeable malpractice award.

Accepted Answer

A Type I error occurs when a null hypothesis is rejected even though it is true. In this case, the null hypothesis is that the occasional stomach pain is due to indigestion and not estomalgia fatalis. If the patient chooses to undergo the expensive lab test despite the low likelihood of having estomalgia fatalis and the test comes back negative (i.e., the null hypothesis is true), it would result in a Type I error as the patient mistakenly believed they had a rare condition. The consequence of this would be wasting money on an unnecessary lab test. Therefore, C is the best choice.

Question 8

Guidelines for the Jolly Blue Giant Health Insurance Company say that the average hospitalization for a triple hernia operation should not exceed 30 hours. A diligent auditor studied records of 16 randomly chosen triple hernia operations at Hackmore Hospital, and found a mean hospital stay of 40 hours with a standard deviation of 20 hours. "Aha!" she cried, "the average stay exceeds the guideline." The p-value for a right-tailed test of her hypothesis is:
A)between .05 and .10.
B)between .025 and .05.
C)between .01 and .025.
D)less than .01.

Accepted Answer

The p-value is calculated using a t-test for the sample mean. With a sample size of 16, the t-statistic is calculated and compared to a t-distribution with 15 degrees of freedom. The calculated t-value falls between the critical values for the significance levels of 0.025 and 0.05, indicating the p-value is between these two values.

Question 9

Dullco Manufacturing claims that its alkaline batteries last at least 40 hours on average in a certain type of portable CD player. But tests on a random sample of 18 batteries from a day's large production run showed a mean battery life of 37.8 hours with a standard deviation of 5.4 hours. To test DullCo's hypothesis, the test statistic is:
A)-1.980
B)-1.728
C)-2.101
D)-1.960

Accepted Answer

The test statistic for this hypothesis test is calculated using the formula for a t-test because the population standard deviation is unknown and the sample size is small (n < 30). The formula is: $ 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{37.8 - 40}{(5.4/\sqrt{18})} = -1.728 $.

Question 10

As you are crossing a field at the farm, your country cousin Jake assures you, "Don't worry about that old bull coming toward us. He's harmless." As you consider Jake's hypothesis, what would be Type I error on your part?
A)You will soon feel the bull's horns.
B)You will run away for no good reason.
C)Jake will not have any more visits from you.

Accepted Answer

Type I error refers to a false positive, where the null hypothesis (in this case, the bull being harmless) is rejected when it is actually true. Choosing option B (running away for no good reason) would be a Type I error because the bull may actually be harmless and running away would be an unnecessary response. Option A is not a Type I error as feeling the bull's horns would mean the null hypothesis was false (i.e., the bull is not harmless). Option C is not related to the concept of Type I error.

Question 11

The level of significance is not:
A)the probability of a "false rejection."
B)a value between 0 and 1.
C)the likelihood of rejecting the null hypothesis when it is true.
D)the chance of accepting a true null hypothesis.

Accepted Answer

The level of significance is the chosen probability of rejecting the null hypothesis when it is true. Therefore, D (the chance of accepting a true null hypothesis) is not a correct statement about the level of significance.

Question 12

Which is not a likely reason to choose the z distribution for a hypothesis test of a mean?
A)The value of σ is known.
B)The sample size n is very large.
C)The population is normal.
D)The value of σ is very large.

Accepted Answer

If the value of σ is very large, it indicates that the population variability is large, and in such cases, the t-distribution is a better choice. The z-distribution assumes that the population standard deviation is known, and in real-life scenarios, the population standard deviation is usually estimated using the sample standard deviation when calculating the test statistic. Hence, option D is not a likely reason to choose the z-distribution. The other three options, A, B, and C, are likely reasons to choose the z-distribution.

Question 13

Guidelines for the Jolly Blue Giant Health Insurance Company say that the average hospitalization for a triple hernia operation should not exceed 30 hours. A diligent auditor studied records of 16 randomly chosen triple hernia operations at Hackmore Hospital and found a mean hospital stay of 40 hours with a standard deviation of 20 hours. "Aha!" she cried, "the average stay exceeds the guideline." The value of the test statistic for her hypothesis is:
A)2.080
B)0.481
C)1.866
D)2.000

Accepted Answer

The test statistic for this hypothesis can be calculated using the formula for a one-sample t-test: $t = \frac{\bar{x} - \mu}{(s/\sqrt{n})}$, 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: $t = \frac{40 - 30}{(20/\sqrt{16})} = \frac{10}{5} = 2.000$.

Question 14

"I believe your airplane's engine is sound," states the mechanic. "I've been over it carefully, and can't see anything wrong. I'd be happy to tear the engine down completely for an internal inspection at a cost of $1,500. But I believe that roughness you heard in the engine on your last flight was probably just a bit of water in the fuel, which passed harmlessly through the engine and is now gone." As the pilot considers the mechanic's hypothesis, the cost of Type I error is:
A)the pilot will experience the thrill of no-engine flight.
B)the pilot will be out $1,500 unnecessarily.
C)the mechanic will lose a good customer.
D)impossible to determine without knowing α.

Accepted Answer

The cost of a Type I error (rejecting a true null hypothesis) in this context would be taking an unnecessary action based on the belief that there is a problem when there isn't one. Therefore, if the pilot opts for the engine teardown based on the mechanic's offer, despite the engine being sound, the unnecessary cost incurred would be $1,500.

Question 15

The critical value in a hypothesis test:
A)is calculated from the sample data.
B)usually is .05 or .01 in most statistical tests.
C)separates the acceptance and rejection regions.
D)depends on the value of the test statistic.

Accepted Answer

The critical value separates the acceptance and rejection regions and determines whether the null hypothesis is rejected or not. The other options (A, B, D) are not true for all hypothesis tests.

Question 16

A study over a 10-year period showed that a certain mammogram test had a 50 percent rate of false positives. This indicates that:
A)about half the tests indicated cancer.
B)about half the tests missed a cancer that exists.
C)about half the tests showed a cancer that didn't exist.
D)about half the women tested actually had no cancer.

Accepted Answer

A false positive result means that the test showed the presence of cancer when it wasn't actually there. So, in this case, 50 percent of the tests showed a cancer that didn't exist. Therefore choice C is the most appropriate answer.

Question 17

For a test of a mean, which of the following is incorrect? A)H₀ is rejected when the calculated p-value is less than the critical value of the test statistic. B)In a right-tailed test, we reject H₀ when the test statistic exceeds the critical value. C)The critical value is based on the researcher's chosen level of significance. D)If H₀: μ ≤ 100 and H₁: μ > 100, then the test is right-tailed.

Accepted Answer

Option A is incorrect because we reject H₀ when the calculated p-value is less than the chosen level of significance (usually 0.05) and not when it is less than the critical value of the test statistic. The critical value is used to determine the rejection region for the test statistic based on the level of significance and degrees of freedom.

Question 18

After lowering the landing gear, the pilot notices that the "gear down and locked" light is not illuminated. "It's probably just a burned out light bulb," she says, as she proceeds on final approach for landing. Considering the pilot's hypothesis, which is the result of Type I error?
A)The sound of metal scraping on concrete will be heard upon landing.
B)The landing is delayed unnecessarily while the bulb and gear are checked.
C)We cannot be sure without knowing whether or not the bulb is actually faulty.

Accepted Answer

A Type I error occurs when a true null hypothesis is incorrectly rejected. In this scenario, if the pilot's hypothesis (the bulb is just burned out) is true but she acts as if it's false (assuming the gear isn't down and locked), leading to unnecessary checks, it would be a Type I error.

Question 19

Dullco Manufacturing claims that its alkaline batteries last at least 40 hours on average in a certain type of portable CD player. But tests on a random sample of 18 batteries from a day's large production run showed a mean battery life of 37.8 hours with a standard deviation of 5.4 hours. To test DullCo's hypothesis, the p-value is:
A)slightly less than .05.
B)exactly equal to .05.
C)slightly greater than .05.
D)uncertain without knowing α.

Accepted Answer

The answer of Dullco Manufacturing claims that its alkaline batteries...

Question 20

Which of the following statements is correct? A)Increasing α will make it more likely that we will reject H₀, ceteris paribus. B)Doubling the sample size roughly doubles the test statistic, ceteris paribus. C)A higher standard deviation would increase the power of a test for a mean. D)The p-value shows the probability that the null hypothesis is false.

Accepted Answer

Increasing the significance level (α) will increase the rejection region, making it more likely to reject the null hypothesis, ceteris paribus.

Answer: B
 Doubling the sample size will increase the test statistic (assuming the same effect size and variance), but not necessarily double it, ceteris paribus.

Answer: False
 A higher standard deviation would decrease the power of a test for a mean. Higher variance means the data points are spread apart, which makes it harder to detect a difference between the sample mean and population mean.

Answer: False
 The p-value measures the evidence against the null hypothesis, not the probability that the null hypothesis is false. It represents the probability of observing a test statistic as extreme or more extreme than the one observed, assuming the null hypothesis is true.

Quiz 9: One-Sample Hypothesis Tests

Hypothesis tests for a mean using the critical value method require:

Which is not true of p-values?

For a right-tailed test of a hypothesis for a population mean with n = 14, the value of the test statistic was t = 1.863. The p-value is:

As you are crossing a field at the farm, your country cousin Jake assures you, "Don't worry about that old bull coming toward us. He's harmless." As you consider Jake's hypothesis, what would be Type I error on your part?

The level of significance is not:

Which is not a likely reason to choose the z distribution for a hypothesis test of a mean?

The critical value in a hypothesis test:

A study over a 10-year period showed that a certain mammogram test had a 50 percent rate of false positives. This indicates that:

For a test of a mean, which of the following is incorrect?

After lowering the landing gear, the pilot notices that the "gear down and locked" light is not illuminated. "It's probably just a burned out light bulb," she says, as she proceeds on final approach for landing. Considering the pilot's hypothesis, which is the result of Type I error?

Which of the following statements is correct?