Question 1

After testing a hypothesis,we decided to reject the null hypothesis.Thus,we are exposed to
A)Type I error.
B)Type II error.
C)either Type I or Type II error.
D)neither Type I nor Type II error.

Accepted Answer

Rejecting a null hypothesis when it's actually true, leads to Type I error.

Question 2

The power of a test is the probability that the test will reject a false null hypothesis.

Accepted Answer

The power of a statistical test is the probability of correctly rejecting the null hypothesis when it is false. Therefore, option A is correct.

Question 3

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)that you pass safely over the bridge and everyone's happy.
B)about $23,900,not including medical bills.
C)that you will find out just how cold that river actually is.
D)that your kids will think you are a chicken.

Accepted Answer

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

Question 4

After testing a hypothesis regarding the mean,we decided not to reject H0.Thus,we are exposed to
A)Type I error.
B)Type II error.
C)either Type I or Type II error.
D)neither Type I nor Type II error.

Accepted Answer

Not rejecting the null hypothesis (H0) when it is actually false exposes us to a Type II error.

Question 5

In hypothesis testing,we are trying to prove the null hypothesis.

Accepted Answer

In hypothesis testing, we are actually trying to find evidence against the null hypothesis and support the alternative hypothesis. We use statistical methods to assess whether the observed data provides enough evidence to reject the null hypothesis and accept the alternative hypothesis.

Question 6

Which of the following is incorrect?
A)The level of significance is the probability of making a Type I error.
B)Lowering both α and β at once will require a larger sample size.
C)The probability of rejecting a true null hypothesis increases as n increases.
D)When Type I error increases,Type II error must decrease,ceteris paribus.

Accepted Answer

As the sample size (n) increases, the probability of rejecting a true null hypothesis (Type I error) does not necessarily increase; instead, the power of the test increases, making it easier to detect a true effect if it exists.

Question 7

In hypothesis testing,we are trying to reject the alternative hypothesis.

Accepted Answer

In hypothesis testing, we are trying to reject the null hypothesis, not the alternative hypothesis. The alternative hypothesis is what we want to support if we reject the null hypothesis.

Question 8

Larger samples lead to increased power,ceteris paribus.

Accepted Answer

Larger samples increase the likelihood of detecting true differences or effects and decrease the likelihood of obtaining false results, which ultimately leads to increased power.

Question 9

For a given sample size,when we increase the probability of a Type I error,the probability of a Type II error
A)remains unchanged.
B)increases.
C)decreases.
D)is impossible to determine without more information.

Accepted Answer

Increasing the probability of a Type I error (rejecting a true null hypothesis) generally leads to a decrease in the probability of a Type II error (failing to reject a false null hypothesis), because the criteria for rejecting the null hypothesis become less stringent.

Question 10

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) increases the probability of rejecting the null hypothesis, assuming all other factors are constant.

Answer: None of the other statements are correct. 
 
B) Doubling the sample size does not necessarily double the test statistic, as the test statistic can also depend on the variability of the data and the effect size being tested. 
C) A higher standard deviation would typically decrease the power of a test for a mean, as it indicates more variability in the data and makes it harder to detect a true difference from the null hypothesis. 
D) The p-value is not the probability that the null hypothesis is false; rather, it is the probability of obtaining a test statistic as extreme or more extreme than the observed one, assuming the null hypothesis is true.

Question 11

"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 Type I error, which is rejecting a true null hypothesis, in this scenario would be the pilot being out $1,500 unnecessarily if they choose to proceed with the engine tear down even though there was nothing wrong with the engine. Option A is not a feasible outcome as the statement doesn't suggest that the airplane won't be functional anymore. Option C is not relevant as the information provided doesn't indicate that the pilot is a regular customer. Option D is not relevant as the question doesn't provide any information about α (level of significance).

Question 12

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 did not exist.
D)about half the women tested actually had no cancer.

Accepted Answer

A false positive means that the test showed a cancer that did not actually exist. Since the mammogram test had a 50 percent rate of false positives, this means that about half the tests showed a cancer that did not exist.

Question 13

When σ is unknown,it is more conservative to use z instead of t for the critical value.

Accepted Answer

When σ is unknown, we use t instead of z for the critical value. The t-distribution takes into account the additional uncertainty introduced when estimating the standard deviation from the sample data.

Question 14

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

The Type I error in this situation would be to run away for no good reason based on the assumption that the bull is actually harmless, which could result in missing out on an opportunity or being seen as overly cautious. The other options do not accurately reflect a Type I error.

Question 15

Which of the following is correct?
A)When the sample size increases,β may decrease and power would increase.
B)Type II error can only occur when you reject H0.
C)Type I error can only occur if you fail to reject H0.
D)The level of significance is the probability of Type II error.

Accepted Answer

As sample size increases, the standard error gets smaller, making it easier to detect a true effect. This, in turn, increases the power of the test. However, as the significance level (α) remains the same, the probability of committing a Type II error (β) decreases.

Question 16

John rejected his null hypothesis in a right-tailed test for a mean at α = .025 because his critical t value was 2.000 and his calculated t value was 2.345.We can be sure that
A)John did not commit a Type I error.
B)John did not commit a Type II error.
C)John committed neither a Type I nor Type II error.
D)John committed both a Type I and a Type II error.

Accepted Answer

John did not commit a Type II error because he rejected the null hypothesis when his calculated t value was greater than the critical t value, indicating that the data provides enough evidence against the null hypothesis. A Type II error would occur if he failed to reject a false null hypothesis, which is not the case here.

Question 17

"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 cannot 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

If you were to undergo the unnecessary expensive lab test despite the low probability of having the rare condition estomalgia fatalis, it would be a Type I error on your part. This means that you would have falsely rejected the null hypothesis (that you do not have the condition) and incurred the cost and inconvenience of the test without any real benefit. Therefore, choosing to undergo the lab test would result in a wastage of money.

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

A smaller probability of Type II error implies higher power of the test.

Accepted Answer

This statement is true. Power of a test is the probability of rejecting a false null hypothesis. Type II error is the probability of failing to reject a false null hypothesis. Therefore, if the probability of Type II error is smaller, it means the probability of correctly rejecting a false null hypothesis (i.e., power of the test) is higher.

Question 20

Which statement about α is not correct?
A)It is the probability of committing a Type I error.
B)It is the test's significance level.
C)It is the probability of rejecting a true H0.
D)It is equal to 1 − β.

Accepted Answer

α is the probability of committing a Type I error, the test's significance level, and the probability of rejecting a true H0. However, it is not equal to 1 − β; that is the power of the test.

Quiz 9: One-Sample Hypothesis Tests

After testing a hypothesis,we decided to reject the null hypothesis.Thus,we are exposed to

The power of a test is the probability that the test will reject a false null hypothesis.

After testing a hypothesis regarding the mean,we decided not to reject H0.Thus,we are exposed to

In hypothesis testing,we are trying to prove the null hypothesis.

Which of the following is incorrect?

In hypothesis testing,we are trying to reject the alternative hypothesis.

Larger samples lead to increased power,ceteris paribus.

For a given sample size,when we increase the probability of a Type I error,the probability of a Type II error

Which of the following statements is correct?

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

When σ is unknown,it is more conservative to use z instead of t for the critical value.

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 of the following is correct?

John rejected his null hypothesis in a right-tailed test for a mean at α = .025 because his critical t value was 2.000 and his calculated t value was 2.345.We can be sure that

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 smaller probability of Type II error implies higher power of the test.

Which statement about α is not correct?