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AP Statistics Testbank 7
Multiple-Choice Questions
In formulating hypotheses for a statistical test of significance, the null hypothesis is often a) a statement of "no effect" or "no difference." b) the probability of observing the data you actually obtained. c) a statement that the data are all 0. d) a statement that the mean of the data is 0. e) usually stated as a strict inequality.
In their advertisements, the marketers of a new diet program would like to claim that their methods result in a mean weight loss of more than 10 pounds in two weeks. In order to determine if this is a valid claim, they hire an independent testing agency which then selects 25 people to be placed on this
diet. The agency should be testing the null hypothesis H 0 : μ= 10 and the alternative hypothesis
a) H a : μ< 10.
b) H a : μ> 10.
c) Ha : μ> 25.
d) H a : μ≠ 10.
e) H a :μ ≠ 10 ± σ/ n.
Which of the following are true statements? (I) If there is evidence sufficient to reject a null hypothesis at the 10%, then there is sufficient evidence to reject this null hypothesis at the 5% level. (II) Whether to use a one- or a two-sided alternative is typically decided after the data are gathered. (III) If the hypothesis test is conducted at the 1% level, there is a 1% chance of rejecting the null hypothesis. a) I only b) II only c) III only d) I, II, and III e) None are true.
SAS high-school counselor Mr. Williams once told me that our students study mathematics on average 2 hours per day. However, because of the “math addiction” diagnosed at this school, I firmly believe that the average is higher. An appropriate set of hypotheses for this situation would be
a) H 0 : μ < 2 , Ha : μ> 2 ;
b) H 0 : μ ≥ 2 , Ha : μ< 2 ;
c) H 0 : μ = 2 , Ha : μ≠ 2 ;
d) H 0 : μ = 2 , Ha : μ> 2 ;
e) H 0 : μ = 2. 8 , Ha : μ> 2. 8.
- SAS high-school counselor Mr. Williams once told me that our students study mathematics on average 2 hours per day. However, because of the “math addiction” diagnosed at this school, I firmly believe that the average is higher. As a result, I decided to have Kaz interview 18 to determine their average mathematics study time; Kaz’s results yield x = 2. 8
G
and s (^) x = 1. 155. Assume that the underlying population of mathematics study times are approximately normally distributed. In which interval will the significance, or P -value be found? a) P <. 0025 b). 0025 < P <. 005 c). 005 < P <. 01 d). 01 < P <. 05 e). 05 < P
- We continue with the situation in problem 5, above. Suppose that Billy now takes Kaz’s results and uses the TInterval function on the TI-83 to compute a 90% confidence interval, with the resulting interval being 2. 8 ±. 47. Consider the following statements:
(I) Billy’s results are irrelevant to our hypothesis testing.
(II) Billy’s results imply that we can reject the hypothesis H 0 : μ= 2 in favor of the alternative
H a : μ> 2 at the 5% level.
(III) Billy’s results imply that we can reject the hypothesis H 0 : μ= 2 in favor of the alternative
H a : μ> 2 at the 1% level.
a) I only b) II only c) III only d) II and III e) I, II, and III
- A patient claims that he consumes only 2000 calories per day, but a dietician suspects that the actual figure is higher. The dietician plans to check his food intake for 30 days and will reject the patient’s claim if the 30-day mean is more than 2100 calories. Assuming that the patient true standard deviation is
σ =350 calories per day, what is the probability that the dietician will mistakenly reject a patient’s true
claim? a). b). c). d). e).
- Mr. S has an appetite for a certain brand of potato chips, which are typically packaged in 14 oz. bags. However, Mr. S has come to suspect that the net weight of the chips in each bag is significantly less than the advertised 14 ounces, leading him to want to test the hypotheses
H 0 : μ = 14 , Ha : μ< 14.
He sends Erica out to perform some measurements, who returns with the sample mean of x = 13. 82 oz. and the sample deviation of oz. for 16 bags of these chips. Based on Erica’s measurements, and assuming that the weights are approximately normally distributed, we would
sx = 0. 24
a) reject H 0 at significance level 0.10 but not at 0.05. b) reject H 0 at significance level 0.05 but not at 0.025. c) reject H 0 at significance level 0.025 but not at 0.01. d) reject H 0 at significance level 0.01. e) not reject H 0 at any level of significance.
Suppose that you would like to test a hypothesis about the mean of a population using a significance level of 0.05. Suppose further that you would like to use the t -statistic even though you suspect that the population is slightly skewed (and therefore not normal). Which of the following is correct? a) You should not the t -statistic since the population does not have a normal distribution. b) You may use the t -statistic provided that your sample size is large―say, at least 50. c) You may use the t -statistic, but you should probably only claim the significance level is 0.10. d) You may not use the t -statistic in this situation only for confidence intervals but not for tests of hypotheses. e) None of the above is correct.
Suppose that Mr. S sends two students, Tommy and Myung-Soo, out to test a hypothesis about a population proportion. Tommy returns with a measurement with P -value of 0.03 and Myung-Soo returns with a measurement with P -value of 0.022.
(I) Tommy’s results are more significant than Myung-Soo’s. (II) Myung-Soo’s results are more significant than Tommy’s. (III) The null hypothesis can be rejected at the 5% level of significance.
a) I only b) II only c) III only d) I and III e) II and III
A service station advertises that its mechanics can change a muffler in only 15 minutes. A consumers’ group doubts this claim and runs a hypothesis test using an SRS or 60 cars needing new mufflers. In this sample the mean changing time is 16.25 minutes with a standard deviation of 3.5 minutes. Is this strong evidence against the 15-minute claim? a) Yes, because the P -value is only.. b) No, because the P -value is .0028. c) Yes, because the P -value is .28. d) No, because 15 is within 16.25 ±3.5. e) Yes, because 16.25 is larger than the claimed 15 minutes.
In leaving for school on an overcast April morning you make a judgement on the null hypothesis: H 0 :Theweather willremain dry. What would the results be of Type I and Type II errors? a) Type I error: get drenched Type II error: needlessly carry around an umbrella b) Type I error: needlessly carry around an umbrella Type II error: get drenched c) Type I error: carry an umbrella, and it rains Type II error: carry no umbrella, but the weather remains dry d) Type I error: get drenched Type II error: carry no umbrella, but the weather remains dry e) Type I error: get drenched Type II error: carry an umbrella, and it rains
An assembly-line machine is supposed to turn out ball bearings with a diameter of 1.25 centimeters. Each morning the first 30 ball bearings produced are pulled and measured. If their mean diameter is under 1.23 centimeters or over 1.27 centimeters, 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 : μ = 1. 25 , Ha : μ≠ 1. 25.
The engineer is asked to make adjustments when the null hypothesis is rejected. In test terminology, what would be the result of a Type II error? a) A warranted halt in production to adjust the machinery b) An unnecessary stoppage of the production process c) Continued production of wrong size ball bearings d) Continued production of proper size ball bearings e) Continued production of ball bearings that randomly are the right or wrong size
- A piece of medical equipment is not functioning properly, however, in running operational checks, a lab technician does not find evidence of the malfunction. The lab technician has committed a) a Type I error b) a Type II error c) both a Type I and a Type II error d) neither a Type I nor a Type II error e) A random sampling error
Free-Response Questions
- When a law firm represents a group of people in a class action lawsuit and wins that lawsuit, the firm receives a percentage of the group’s monetary settlement. That settlement amount is based on the total number of people in the group⎯the larger the group and the larger the settlement, the more money the firm will receive.
A law firm is trying to decide whether to represent car owners in a class action lawsuit against the manufacturer of a certain make and model for a particular defect. If 5 percent or less of the cars of this make and model have the defect, the firm will not recover its expenses. Therefore, the firm will handle the lawsuit only if it is convinced that more than 5 percent of cars of this make and model have the defect. The firm plans to take a random sample of 1,000 people who bought this car and ask them if they experienced this defect in their cars.
a) Define the parameter of interest and state the null and alternative hypotheses that the law firm should test.
b) In the context of this situation, describe Type I and Type II errors and describe the consequences of each type of error for the law firm.
- A pharmaceutical company has developed a new drug to reduce cholesterol. A regulatory agency will recommend the new drug for use if there is convincing evidence that the mean reduction in cholesterol level after one month of use is more then 20 milligrams/deciliter (mg/dl). This is because a mean reduction of this magnitude would be greater than the mean reduction for the current most widely used drug.
The pharmaceutical company collected data by giving the new drug to a random sample of 50 people from the population of people with high cholesterol. The reduction in cholesterol level after one month of use was recorded for each individual in the sample, resulting in a sample mean reduction of 24 mg/dl and a standard deviation of 15 mg/dl. a) The regulatory agency decides to use a confidence interval estimate for the population mean reduction in cholesterol level for the new drug. Provide the 95% confidence interval for the mean reduction in cholesterol level. b) Because the 95% confidence interval includes 20, the regulatory agency is not convinced that the new drug is better than the current best seller. The pharmaceutical company tested the following hypotheses:
H 0 : μ = 20 versus Ha : μ> 20 ,
where μ represents the population mean reduction in cholesterol level for the new drug, and where
the null hypothesis would be rejected at a level α =. 05. What conclusion was reached? Be sure to
explain which test you used and why you used it.
c) The test procedure resulted in a z -value of 1.89 and a P -value of 0.03. Because the P -value is less than 0.05, the company believes that there is convincing evidence that the mean reduction in cholesterol level for the new drug is more than 20. Explain why the confidence interval and the hypothesis test led to different conclusions.
- Mr. Corbett and Mr. Surowski both believe that the average mathematically-inclined student at SAS can score significantly higher than 90 on the American Mathematics Examination 12 (AMC 12). This was put to the test two weeks ago when 70 SAS students sat for the AMC 12 and scored an overall average of 91.3 with a standard deviation of 6.54. a) State null and alternative hypotheses suitable for testing Mr. Corbett and Mr. Surowski’s claim. b) Compute the P -value associated with x = 91. 3. Do you feel that the resulting average score on the AMC 12 is significant? Why or why not?
c) If we are to reject your null hypothesis in part a) at a level of significance α =. 05 , what is the
outcome? d) At what confidence level will the confidence interval about x yield the same conclusion as in c)? Compute this confidence interval. e) A non-mathematically inclined person could conceivably argue that 91.3 is not much greater than 90 and hence Mr. Corbett and Mr. Surowski’s claim cannot possibly be true. How would you respond to this?
- The Shanghai Municipal Government would like to claim that significantly fewer than 5% of its residents drive private automobiles to work. a) State null and alternative hypothesis suitable to test this claim.
Next, assume that the Municipal Government commissioned a study to investigate the transportation habits of 813 residents, with the following results:
Regularly drive a private automobile 4.6% Regularly ride in taxis 15.9% Regularly take public transportation 57.2% Regularly use other forms of transportation 22.0% 100%
b) Can a normal distribution reasonably be applied to this situation? Give supporting calculations. c) What is the P -value associated with the above data? Do you feel that this is significant? d) Do the above results justify the Shanghai Municipal Government’s claim? Use the level of
significance of α =. 05.
- Every once in a while I hear someone saying that left-handed people especially capable at mathematics. I’m not sure I believe this, but perhaps this is something worth studying. In order to gather some evidence for this, suppose that we administered a standardized mathematics test to 20 left-handed subjects, where we know that the overall mean on this test is μ = 12. 3. a) State null and alternative hypotheses appropriate to this situation.
Next, assume that the mean score on this test is x = 13. 2 with a standard deviation of s=3.32.
b) Compute the P-value of the mean and comment on its relative significance. c) Would you reject your null hypothesis at any “reasonable” level of significance? Why or why not? d) What hypotheses, if any, did you make in the application of your statistical methods?
