Question 1

Hypothesis tests for a mean using the critical value method require:
A) knowing the true value of σ.
B) sampling a normal population.
C) specifying β in advance.
D) specifying α in advance.

Accepted Answer

Hypothesis tests for a mean using the critical value method require specifying the level of significance, or α, in advance.

Question 2

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

In a test of a mean, we reject H₀ when the calculated p-value is less than the chosen level of significance (usually 0.05 or 0.01). We do not compare the p-value to the critical value of the test statistic. The critical value is used to determine the rejection region and is based on the chosen level of significance and the degrees of freedom of the test statistic. Therefore, Choice A is incorrect.

Question 3

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 not the chance of accepting a true null hypothesis. It is the probability of rejecting a true null hypothesis, which is equivalent to the likelihood of making a Type I error.

Question 4

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

We need to conduct a one-tailed hypothesis test with the null hypothesis Ho: µ≥40 (claim by Dullco Manufacturing) and the alternative hypothesis Ha: µ<40 (since it is a left-tailed test). We will use a t-test since the sample size is less than 30 and the population standard deviation is unknown. With a sample size of 18, we have 17 degrees of freedom (df=n-1). The t-test statistic is calculated as: t=(x̄-µ)/(s/√n)=(37.8-40)/(5.4/√18)=-1.67. The critical t-value at α=.05 and df=17 is -1.740. Since -1.67>-1.740, we fail to reject the null hypothesis. Therefore, we do not have sufficient evidence to support the claim by Dullco Manufacturing. The decision is rather close because the t-test statistic is only slightly less than the critical value. We cannot switch to α=.01 unless we have a stronger reason to reduce the significance level.

Question 5

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 α (the level of significance) makes it easier to reject the null hypothesis, so it makes it more likely that we will reject H₀. The other statements are incorrect: B) Doubling the sample size increases the precision of our estimation but it does not necessarily double the test statistic. C) A higher standard deviation decreases the power of a test for a mean because it increases the uncertainty in our estimate of the population mean. D) The p-value is not the probability that the null hypothesis is false, but rather the probability of obtaining a test statistic as extreme as the observed test statistic, assuming the null hypothesis is true.

Question 6

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 7

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

We need to conduct a hypothesis test to determine whether the sample data supports the claim that the batteries last at least 40 hours on average. The null hypothesis is that the true mean battery life is 40 hours, and the alternative hypothesis is that the true mean battery life is less than 40 hours. We can use a one-sample t-test with a t-distribution and 17 degrees of freedom (since we are estimating the population standard deviation from the sample standard deviation). Using a t-table, we find that the critical value for a one-tailed test with α = .05 and 17 degrees of freedom is -1.740.

To calculate the test statistic, we use the formula t = (x̄ - μ)/(s/√n), where x̄ is the sample mean, μ is the hypothesized population mean (40), s is the sample standard deviation, and n is the sample size. Plugging in the values, we get t = (37.8 - 40)/(5.4/√18) = -1.77.

The p-value is the probability of observing a test statistic as extreme as -1.77 (or more extreme) under the null hypothesis. From a t-table or using software, we find that the p-value is approximately .049. Since this value is slightly greater than .05 (the significance level), we do not reject the null hypothesis. This means that there is not enough evidence to conclude that the true mean battery life is less than 40 hours. Therefore, the answer is choice C (slightly greater than .05).

Question 8

"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 engine 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 Type I error is the cost of rejecting the null hypothesis when it is actually true. In this case, the null hypothesis is that the engine is sound. If the pilot rejects this hypothesis and has the engine torn down, and it turns out that the engine is actually sound, then the pilot will have wasted $1,500.

Question 9

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 10

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 answer of Guidelines for the Jolly Blue Giant Health...

Question 11

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 12

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 13

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

The pilot's hypothesis assumes that the gear is actually down and locked, but the light is simply not working. If this assumption is wrong and the gear is not down and locked, the plane could potentially crash upon landing. Therefore, if the landing is delayed unnecessarily to check the gear, it is better than risking a serious accident due to a Type I error.

Question 14

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 is a threshold value that separates the acceptance and rejection regions in a hypothesis test. It is usually determined by the level of significance, which is typically set at .05 or .01, and depends on the degrees of freedom or sample size. The value of the test statistic is compared to the critical value to determine if the null hypothesis should be rejected or not. Therefore, the critical value is not calculated from the sample data and depends on the specific statistical test being conducted.

Question 15

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 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 means that the test shows a positive result for cancer even if the person does not have it. In this case, the study showed that the test had a 50 percent rate of false positives, which means that about half of the tests showed a cancer that did not exist.

Question 17

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

Choosing the z distribution for a hypothesis test of a mean is most appropriate when the value of σ is known or when the sample size is very large. Additionally, if the population is known to be normal, the z distribution may also be used. However, if the value of σ is very large, it may be more appropriate to use a t distribution for the hypothesis test.

Question 18

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 19

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 20

"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 would occur if you were to undergo the costly lab test and it came back positive for estomalgia fatalis, even though you do not actually have the condition. This would result in unnecessary medical procedures and expenses, as well as anxiety and stress. Choosing to undergo the test when it is not truly necessary would be a costly mistake.

Quiz 9: One-Sample Hypothesis Tests

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

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

The level of significance is not:

Which of the following statements is correct?

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?

Which is not true of p-values?

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?

The critical value in a hypothesis test:

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

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