Ap Stats Review Inference Procedures

AP STATS REVIEW – INFERENCE PROCEDURES

(2009 #5)

For many years, the medically accepted practice of giving aid to a person experiencing a heart attack was to have the person who placed the emergency call administer chest compression (CC) plus standard mouth-to-mouth resuscitation (MMR) to the heart attack patient until the emergency response team arrived. However, some researchers believed that CC alone would be a more effective approach.

In the 1990s a study was conducted in Seattle in which 518 cases were randomly assigned to treatments: 278 to CC plus standard MMR and 240 to CC alone. A total of 64 patients survived the heart attack: 29 in the group receiving CC plus standard MMR, and 35 in the group receiving CC alone. A test of significance was conducted on the following hypotheses.

Ho: The survival rates for the two treatments are equal.

Ha: The treatment that uses CC alone produces a higher survival rate.

This test resulted in a p-value of 0.0761.

a) Interpret what this p-value measures in the context of this study.

b) Based on this p-value and study design, what conclusion should be drawn in the context of this study? Use a significance level of α = 0.05.

c) Based on your conclusion in part (b), which type of error, Type I or Type II, could have been made? What is one potential consequence of this error?

(2010 #3)

A humane society wanted to estimate with 95 percent confidence the proportion of households in its county that own at least one dog.

a) Interpret the 95 percent confidence level in this context.

The humane society selected a random sample of households in its county and used the sample to estimate the proportion of all households that own at least one dog. The conditions for calculating a 95 percent confidence interval for the proportion of households in this county that own at least one dog were checked and verified, and the resulting confidence interval was 0.417 ± 0.119.

b) A national pet products association claimed that 39 percent of all American households owned at least one dog. Does the humane society’s interval estimate provide evidence that the proportion of dog owners in its county is different from the claimed national proportion? Explain.

c) How many households were selected in the humane society’s sample? Show how you obtained your answer.

(2010 #5)

A large pet store buys the identical species of adult tropical fish from two different suppliers—Buy-Rite Pets and Fish Friends. Several of the managers at the pet store suspect that the lengths of the fish from Fish Friends are consistently greater than the lengths of the fish from Buy-Rite Pets. Random samples of 8 adult fish of the species from Buy-Rite Pets and 10 adult fish of the same species from Fish Friends were selected and the lengths of the fish, in inches, were recorded, as shown in the table below.

Do the data provide convincing evidence that the mean length of the adult fish of the species from Fish Friends is greater than the mean length of the adult fish of the same species from Buy-Rite Pets?

(2011 #4)

High cholesterol levels in people can be reduced by exercise, diet, and medication. Twenty middle-aged males with cholesterol readings between 220 and 240 milligrams per deciliter (mg/dL) of blood were randomly selected from the population of such male patients at a large local hospital. Ten of the 20 males were randomly assigned to group A, advised on appropriate exercise and diet, and also received a placebo. The other 10 males were assigned to group B, received the same advice on appropriate exercise and diet, but received a drug intended to reduce cholesterol instead of a placebo. After three months, posttreatment cholesterol readings were taken for all 20 males and compared to pretreatment cholesterol readings. The tables below give the reduction in cholesterol level (pretreatment reading minus posttreatment reading) for each male in the study.

Do the data provide convincing evidence, at the α = 0.01 level, that the cholesterol drug is effective in producing a reduction in mean cholesterol level beyond that produced by exercise and diet?

(2011B #4)

A parent advisory board for a certain university was concerned about the effect of part-time jobs on the academic achievement of students attending the university. To obtain some information, the advisory board surveyed a simple random sample of 200 of the more than 20,000 students attending the university. Each student reported the average number of hours spent working part-time each week and his or her perception of the effect of part-time work on academic achievement. The data in the table below summarize the students’ responses by average number of hours worked per week (less than 11, 11 to 20, more than 20) and perception of the effect of part-time work on academic achievement (positive, no effect, negative).

A chi-square test was used to determine if there is an association between the effect of part-time work on academic achievement and the average number of hours per week that students work. Computer output that resulted from performing this test is shown below.

a) State the null and alternative hypotheses for this test.

b) Discuss whether the conditions for a chi-square inference procedure are met for these data.

c) Given the results from the chi-square test, what should the advisory board conclude?

d) Based on your conclusion in part (c), which type of error (Type I or Type II) might the advisory board have made? Describe this error in the context of the question.

(2011B #5)

During a flu vaccine shortage in the United States, it was believed that 45 percent of vaccine-eligible people received flu vaccine. The results of a survey given to a random sample of 2,350 vaccine-eligible people indicated that 978 of the 2,350 people had received flu vaccine.

a) Construct a 99 percent confidence interval for the proportion of vaccine-eligible people who had received flu vaccine. Use your confidence interval to comment on the belief that 45 percent of the vaccine-eligible people had received flu vaccine.

b) Suppose a similar survey will be given to vaccine-eligible people in Canada by Canadian health officials. A 99 percent confidence interval for the proportion of people who will have received flu vaccine is to be constructed. What is the smallest sample size that can be used to guarantee that the margin of error will be less than or equal to 0.02 ?

MULTIPLE CHOICE:

CHAPTER 10

1. You want to compute a 96% confidence interval for a population mean. Assume that the population standard deviation is known to be 10 and the sample size is 50. The value of z* to be used in this calculation is

(a)1.960 (b) 1.645 (c) 1.7507 (d) 2.0537 (e) None. The answer is .

2. You want to estimate the mean SAT score for a population of students with a 90% confidence interval. Assume that the population standard deviation is s = 100. If you want the margin of error to be approximately 10, you will need a sample size of

(a) 16 (b) 271 (c) 38 (d) 1476 (e) None. The answer is .

3. A significance test gives a P-value of 0.04. From this we can

(a) Reject H0 at the 1% significance level

(b) Reject H0 at the 5% significance level

(d) Say that the probability that H0 is true is 0.04

(e) None of the above. The answer is .

4. A significance test was performed to test the null hypothesis H0: µ = 2 versus the alternative Ha: µ 2. The test statistic is z = 1.40. The P-value for this test is approximately

(a) 0.16 (b) 0.08 (c) 0.003 (d) 0.92 (e) 0.70 (f) None. The answer is .

5. You have measured the systolic blood pressure of a random sample of 25 employees of a

company located near you. A 95% confidence interval for the mean systolic blood pressure for the employees of this company is (122, 138). Which of the following statements gives a valid interpretation of this interval?

(a) Ninety-five percent of the sample of employees has a systolic blood pressure between 122 and 138.

(b) Ninety-five percent of the population of employees has a systolic blood pressure between 122 and 138.

(c) If the procedure were repeated many times, 95% of the resulting confidence intervals would contain the population mean systolic blood pressure.

(d) The probability that the population mean blood pressure is between 122 and 138 is .95.

(e) If the procedure were repeated many times, 95% of the sample means would be between 122 and 138.

(f) None of the above. The answer is .

6. An analyst, using a random sample of n = 500 families, obtained a 90% confidence interval for mean monthly family income for a large population: ($600, $800). If the analyst had used a 99% confidence coefficient instead, the confidence interval would be:

(a) Narrower and would involve a larger risk of being incorrect

(b) Wider and would involve a smaller risk of being incorrect

(d) Wider and would involve a larger risk of being incorrect

(e) Wider but it cannot be determined whether the risk of being incorrect would be larger or smaller

7. To assess the accuracy of a laboratory scale, a standard weight that is known to weigh 1 gram is repeatedly weighed a total of n times and the mean of the weighings is computed. Suppose the scale readings are normally distributed with unknown mean and standard deviation

= 0.01 g. How large should n be so that a 95% confidence interval for has a margin of error of ± 0.0001?

(a) 100

(b) 196

(d) 10000

(e) 38416

CHAPTER 11

1. You want to compute a 90% confidence interval for the mean of a population with unknown population standard deviation. The sample size is 30. The value of t* you would use for this interval is

(a) 1.96 (b) 1.645 (c) 1.699 (d) 0.90 (e) 1.311 (f) None of the above

2. A 95% confidence interval for the mean reading achievement score for a population of third-grade students is (44.2, 54.2). The margin of error of this interval is

(a) 95% (b) 5 (c) 2.5 (d) 10 (e)The answer cannot be determined from the information given.

3. The effect of acid rain upon the yield of crops is of concern in many places. In order to determine baseline yields, a sample of 13 fields was selected, and the yield of barley (g/400m2) was determined. The output from SAS appears below:

QUANTILES(DEF=4) EXTREMES

N 13 SUM WGTS 13 100% MAX 392 99% 392 LOW HIGH

MEAN 220.231 SUM 2863 75% Q3 234 95% 392 161 225

STD DEV 58.5721 VAR 3430.69 50% MED 221 90% 330 168 232

SKEW 2.21591 KURT 6.61979 25% Q1 174 10% 163 169 236

USS 671689 CSS 41168.3 0% MIN 161 5% 161 179 239

CV 26.5958 STD MEAN 16.245 1% 161 205 392

A 95% confidence interval for the mean yield is:

(a) 220.2 ± 1.96(58.6)

(b) 220.2 ± 1.96(16.2)

(d) 220.2 ± 2.18(16.2)

(e) 220.2 ± 2.16(16.2)

4. To use the two-sample t procedure to perform a significance test on the difference between two means, we assume

(a) The populations’ standard deviations are known

(b) The samples from each population are independent

(d) The sample sizes are large

(e) All of the above

5. We wish to test if a new feed increases the mean weight gain compared to an old feed. At the conclusion of the experiment it was found that the new feed gave a 10 kg bigger gain than the old feed. A two-sample t-test with the proper one-sided alternative was done and the resulting P-value was 0.082. This means:

(a) There is an 8.2% chance the null hypothesis is true.

(b) There was only a 8.2% chance of observing an increase greater than 10 kg (assuming the null hypothesis was true).

(d) There is an 8.2% chance the alternate hypothesis is true.

(e) There is only an 8.2% chance of getting a 10 kg. increase.

6. The water diet requires one to drink two cups of water every half hour from when one gets up until one goes to bed, but otherwise allows one to eat whatever one likes. Four adult volunteers agree to test the diet. They are weighed prior to beginning the diet and after six weeks on the diet. The weights (in pounds) are

Person 1 2 3 4__

Weight before the diet 180 125 240 150

Weight after six weeks 170 130 215 152

For the population of all adults, assume that the weight loss after six weeks on the diet (weight before beginning the diet – weight after six weeks on the diet) is normally distributed with mean µ. To determine if the diet leads to weight loss, we test the hypotheses

H0:m = 0, Ha: m > 0.

Based on these data we conclude that

(a) We would not reject H0 at significance level 0.10.

(b) We would reject H0 at significance level 0.10 but not at 0.05.

(d) We would reject H0 at significance level 0.01.

(e) The sample size is too small to allow use of the t procedures.

The next two questions refer to the following situation: In some mining operations, a byproduct of the processing is mildly radioactive. Of prime concern is the possibility that release of these byproducts into the environment may contaminate the freshwater supply. There are strict regulations for the maximum allowable radioactivity in supplies of drinking water, namely an average of 5 picocuries per liter (pCi/L) or less. However, it is well known that even safe water has occasional hot spots that eventually get diluted, so samples of water are assumed safe unless there is evidence to the contrary. A random sample of 25 specimens of water from a city’s water supply gave a mean of 5.39 pCi/L and a standard deviation of 0.87 pCi/L.