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AP Statistics - TEP: Type I/II Errors & Power

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1 Statistical Inference > Type I/II errors, power · Level 3
An industry spokesperson claims that the mean income of entry-level employees is \$29,500 with a standard deviation of $2500. A reporter plans to test this claim through interviews with a random sample of 40 entry-level employees. If the reporter finds a sample mean more than \$500 less than the claimed \$29,500, she will dispute the spokesperson's claim. What is the probability that the reporter will commit a Type I error?
A
\(P\left(t < \dfrac{29000-29500}{\dfrac{2500}{\sqrt{40}}}\right)\) with df = 40−1
B
\(P\left(z < \dfrac{29000-29500}{\dfrac{2500}{\sqrt{40}}}\right)\)
C
\(P\left(t < \dfrac{29000-29500}{2500}\right)\) with df = 40−1
D
\(P\left(z < \dfrac{29000-29500}{2500}\right)\)
E
This cannot be calculated without knowing the sample standard deviation.
2 Statistical Inference > Type I/II errors, power · Level 3
When leaving for school on an overcast morning, you make a judgment on the null hypothesis: The weather will remain dry. What would the results be of Type I and Type II errors?
A
Type I error: carry an umbrella and it rains Type II error: carry no umbrella, but weather remains dry
B
Type I error: get drenched Type II error: carry no umbrella, but weather remains dry
C
Type I error: get drenched Type II error: carry an umbrella, and it rains
D
Type I error: get drenched Type II error: needlessly carry around an umbrella
E
Type I error: needlessly carry around an umbrella Type II error: get drenched
3 Statistical Inference > Type I/II errors, power · Level 3
Consider a hypothesis test with \(H_0: \mu = 127\) and \(H_a: \mu < 127\). Which of the following choices of significance level and sample size results in the greatest power of the test when \(\mu = 125\)?
A
\(\alpha = 0.05\), \(n = 30\)
B
\(\alpha = 0.01\), \(n = 30\)
C
\(\alpha = 0.05\), \(n = 45\)
D
\(\alpha = 0.01\), \(n = 45\)
E
There is no way of answering without knowing the strength of the given power.
4 Statistical Inference > Type I/II errors, power · Level 3
An assembly line machine is supposed to turn out fidget spinners with spin times of 4 minutes. Each day, a simple random sample (SRS) of five fidget spinners are pulled and their spin times measured. If their mean is under 3.5 minutes, the machinery is stopped and an engineer is called to make adjustments before production is resumed. The quality control procedure may be viewed as a hypothesis test with \(H_0: \mu = 4.0\) and \(H_a: \mu < 4.0\). What would a Type II error result in?
A
A warranted halt in production to adjust the machinery
B
An unnecessary stoppage of the production process
C
Continued production of fidget spinners with low spin times
D
Continued production of fidget spinners with the proper spin times
E
Continued production of fidget spinners that may or may not have the proper spin times
5 Statistical Inference > Type I/II errors, power · Level 3
Given an experiment with \(H_0: \mu = 31\), \(H_a: \mu > 31\), and a possible correct value of 32, which of the following will increase with an increase in the sample size \(n\)?
A
The probability of a Type I error
B
The probability of a Type II error
C
The power of the test
D
The significance level \(\alpha\)
E
\(1 - \text{power}\)
6 Statistical Inference > Type I/II errors, power · Level 3
A quality inspector plans to test a sample of 50 bottles of water with a claimed pH level of 6.0. If she finds the mean pH to be less than 5.5, she will have the bottling machinery stopped and the water sources inspected. From previous analysis, it is known that the standard deviation in pH level among bottles is 0.75. What is the probability that the inspector will commit a Type I error and mistakenly stop a machine when the pH level of the water really is the claimed 6.0?
A
\(P\left(z < \dfrac{5.5-6.0}{0.75}\right)\)
B
\(P\left(z < \dfrac{5.5-6.0}{\dfrac{0.75}{\sqrt{50}}}\right)\)
C
\(P\left(t < \dfrac{5.5-6.0}{0.75}\right)\) with df \(= 50-1\)
D
\(P\left(t < \dfrac{5.5-6.0}{\dfrac{0.75}{\sqrt{50}}}\right)\) with df \(= 50-1\)
E
\(P\left(t < \dfrac{5.5-6.0}{\dfrac{0.75}{\sqrt{50}}}\right)\) with df \(= 50\)
7 Statistical Inference > Type I/II errors, power · Level 3
Which of the following is incorrect?
A
The power of a test concerns its ability to detect a true alternative hypothesis.
B
The significance level of a test is the probability of rejecting a true null hypothesis.
C
The probability of a Type I error plus the probability of a Type II error always equals 1.
D
Power equals 1 minus the probability of failing to reject a false null hypothesis.
E
Anything that makes a null hypothesis harder to reject increases the probability of committing a Type II error.
8 Statistical Inference > Type I/II errors, power · Level 3
Suppose \(H_0: p = 0.4\), \(H_a: p < 0.4\), and against the alternative \(p = 0.3\), the power is 0.8. Which of the following is a valid conclusion?
A
The probability of committing a Type I error is 0.1.
B
If \(p = 0.3\) is true, the probability of failing to reject \(H_0\) is 0.2.
C
The probability of committing a Type II error is 0.7.
D
All of the above are valid conclusions.
E
None of the above are valid conclusions.
9 Statistical Inference > Type I/II errors, power · Level 3
Which of the following is a true statement?
A
A Type I error is a conditional probability.
B
A Type II error results if one incorrectly assumes the data are normally distributed.
C
Types I and II errors are caused by mistakes, however small, by the person conducting the test.
D
The probability of a Type II error does not depend on the probability of a Type I error.
E
In conducting a hypothesis test, it is possible to make both a Type I and Type II error simultaneously.
10 Statistical Inference > Type I/II errors, power · Level 3
Which of the following statements is incorrect?
A
The significance level of a test is the probability of a Type II error.
B
Given a particular alternative, the power of a test against that alternative is 1 minus the probability of the Type II error associated with that alternative.
C
If the significance level remains fixed, increasing the sample size reduces the probability of a Type II error.
D
If the significance level remains fixed, increasing the sample size raises the power.
E
With the sample size held fixed, increasing the significance level decreases the probability of a Type II error.
11 Statistical Inference > Type I/II errors, power · Level 3
A manufacturer of surgical gowns (to protect surgeons from contamination) periodically checks a sample of its product and performs a major shutdown and inspection if any leaks are detected. Similarly, a manufacturer of blankets periodically checks the sizes of its blankets coming off an assembly line and halts production if measurements are sufficiently off target. In both situations, we have the null hypothesis that the production equipment is performing satisfactorily. For each situation, which is the more serious concern, a Type I or Type II error?
A
Surgical gown producer: Type I error Blanket manufacturer: Type I error
B
Surgical gown producer: Type I error Blanket manufacturer: Type II error
C
Surgical gown producer: Type II error Blanket manufacturer: Type I error
D
Surgical gown producer: Type II error Blanket manufacturer: Type II error
E
This is impossible to answer without making an expected value judgment between human life and accurate blanket sizes.
12 Statistical Inference > Type I/II errors, power · Level 3
If all other variables remain constant, which of the following will not increase the power of a hypothesis test?
A
Increasing the sample size
B
Increasing the significance level
C
Increasing the probability of a Type II error
D
Decreased variability in the data
E
An increased difference between the true and the hypothesized parameters
13 Statistical Inference > Type I/II errors, power · Level 3
A company that produces bungee cords continually monitors the strength of the cords. If the mean strength from a sample drops below a specified level, the production process is halted and the machinery inspected. Which of the following would result from a Type I error?
A
Halting the production process when sufficient customer complaints are received
B
Halting the production process when the bungee cord strength is below specifications
C
Halting the production process when the bungee cord strength is within specifications
D
Allowing the production process to continue when the bungee cord strength is below specifications
E
Allowing the production process to continue when the bungee cord strength is within specifications
14 Statistical Inference > Type I/II errors, power · Level 3
Given that the power of a significance test against a particular alternative is 97 percent, which of the following is true?
A
The probability of mistakenly rejecting a true null hypothesis is less than 3 percent.
B
The probability of mistakenly rejecting a true null hypothesis is 3 percent.
C
The probability of mistakenly rejecting a true null hypothesis is greater than 3 percent.
D
The probability of mistakenly failing to reject the false null hypothesis is 3 percent.
E
The probability of mistakenly failing to reject the false null hypothesis is different from 3 percent.
15 Statistical Inference > Type I/II errors, power · Level 3
A company will market a new hybrid luxury smartphone only if it can sell the phone for more than \$900 (otherwise there is not enough profit to make the venture worthwhile). The company does a random survey of 80 potential customers and runs a hypothesis test with \(H_0: \mu = 900\) and \(H_a: \mu > 900\). What would be the consequences of Type I and Type II errors?
A
Type I error: produce a nonprofitable smartphone Type II error: fail to produce a profitable smartphone
B
Type I error: fail to produce a profitable smartphone Type II error: produce a nonprofitable smartphone
C
Type I error: fail to produce a nonprofitable smartphone Type II error: produce a profitable smartphone
D
Type I error: fail to produce a profitable smartphone Type II error: produce a profitable smartphone
E
Type I error: produce a nonprofitable smartphone Type II error: fail to produce a nonprofitable smartphone
16 Statistical Inference > Type I/II errors, power · Level 3
A Human Resource manager suspects that employees are abusing the company's sick day policy and plans to investigate this issue. The hypotheses under consideration are the following: \(H_0\): employees are not abusing the company's sick day policy \(H_a\): employees are abusing the policy by taking sick days when they are not actually sick What is the power of this test?
A
The probability of mistakenly accusing employees of abusing the sick day policy when in fact they aren't abusing the policy
B
The probability of mistakenly not accusing employees who are abusing the sick day policy
C
The probability of correctly not accusing employees who are not abusing the sick day policy
D
The probability of correctly accusing employees who are abusing the sick day policy
E
The probability that the test will come to a correct decision on whether or not employees are abusing the sick day policy
17 Statistical Inference > Type I/II errors, power · Level 3
A hypothesis test is to be performed at a significance level of \(\alpha = 0.10\). What is the effect on the probability of committing a Type I error if the sample size is increased?
A
The probability of committing a Type I error decreases.
B
The probability of committing a Type I error increases.
C
The probability of committing a Type I error remains the same.
D
This cannot be determined without knowing the relevant standard deviation.
E
This cannot be determined without knowing if a Type II error is committed.
18 Statistical Inference > Type I/II errors, power · Level 3
A TV manufacturer claims that 75 percent of purchasers understand the setup instructions. The sales department of a TV superstore disagrees and decides to do a large-sample z-test for a population proportion at the 5% significance level. The null hypothesis that the proportion of TV purchasers who understand the instructions is 75 percent is tested against the alternative hypothesis that the proportion of TV purchasers who understand the instructions is less than 75 percent. For which of the following will the power of the test be highest?
A
Sample size \(n = 50\) and 55 percent of purchasers understand the instructions
B
Sample size \(n = 100\) and 55 percent of purchasers understand the instructions
C
Sample size \(n = 50\) and 65 percent of purchasers understand the instructions
D
Sample size \(n = 100\) and 65 percent of purchasers understand the instructions
E
Sample size \(n = 100\) and 95 percent of purchasers understand the instructions
19 Statistical Inference > Type I/II errors, power · Level 3
Dogs have far more sensitive noses than humans, estimated to be up to 100,000 times better than humans. Researchers believe that dogs can sniff out lung cancer from breath samples of sufferers. Using the dogs, the researchers plan to test thousands of samples and prescribe chemotherapy only if the dogs indicate the presence of cancer. Consider the following hypotheses: \(H_0\): The subject does not have cancer. \(H_a\): The subject does have cancer. What would a Type II error be in context?
A
A subject with cancer is correctly diagnosed with cancer.
B
A subject with cancer is mistakenly thought to be cancer free.
C
A cancer-free subject is correctly diagnosed to be cancer free.
D
A cancer-free subject is mistakenly diagnosed with cancer.
E
Correct and mistaken diagnoses of cancer are confounded.
20 Statistical Inference > Type I/II errors, power · Level 3
What is the probability of a Type II error when a hypothesis test is being conducted at the 10% significance level \((\alpha = 0.10)\)?
A
0.05
B
0.10
C
0.90
D
0.95
E
There is insufficient information to answer this question.

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