The alcohol content of wine depends on the grape variety, the way in which the wine is produced from the grapes, the weather, and other influences. Here are data on the percent of alcohol in wine produced from the same grape variety in the same year by 48 winemakers in the same region of Italy:

12.86 12.88 12.81 12.70 12.51 12.60 12.25 12.53 13.49 12.84

12.93 13.36 13.52 13.62 12.25 13.16 13.88 12.87 13.32 13.08

13.50 12.79 13.11 13.23 12.58 13.17 13.84 12.45 14.34 13.48

12.36 13.69 12.85 12.96 13.78 13.73 13.45 12.82 13.58 13.40

12.20 12.77 14.16 13.71 13.40 13.27 13.17 14.13

(a) Make a histogram of the data, using class width 0.25. The shape of the distribution is a bit irregular, but there are no outliers or strong skewness. There is no reason to avoid use oft procedures for n = 48.

(b) Give a 95% confidence interval for the mean alcohol content of wine of this type. (c) Based on your confidence interval, is the mean alcohol content significantly different at the a = 0.05 level from 12%? From 13%?

In its 11th edition, the U.S. Report on Carcinogens demoted saccharin from the list of possible human carcinogens. Earlier carcinogenicity experiments had found rare instances of bladder tumors among male rats under very specific circumstances. Female rats never exhibited this sensitivity to saccharin, and neither did male or female mice. The toxicity was later explained by the unique metabolic pathway of certain male rats only. Years of observational studies in humans failed to show any sign of carcinogenicity from the regular consumption of saccharin.

(a) What was the assumption made when saccharin was first placed in the list of possible human carcinogens based on the rat data? Why do you think this assumption was made?

(b) What simple condition is not met when applying animal research to humans? Explain why confirmation of the unique physiology of certain male rats led to the removal of saccharin from the list of possible human carcinogens.

Suppose we know that the aspirin content of aspirin tablets in previous examples follow a Normal distribution with standard deviation a = 5 mg.

(a) You must verify the aspirin content of tablets produced in a day to within +- 1 mg with 99% confidence. How large a sample of aspirin tablets from the daily production do you need?

(b) If you needed to produce only a 95% confidence interval with a margin of error no larger that 1 mg, how large a sample of aspirin tablets from the daily production would you need instead?

Statisticians prefer large samples. Describe briefly the effect of increasing the size of a sample ( or the number of subjects in an experiment) on each of the following. (a) The margin of error of a $95\%$ confidence interval. (b) The P-value of a test when $H_0$ is false and all facts about the population remain unchanged as n increases.

Previous examples gives the IQ test scores of 31 seventh-grade girls in a Midwest school district. IQ scores fo llow a Normal distribution with standard deviation a = 15. Treat these 31 girls as an SRS of all seventh-grade girls in this district. IQ scores in a broad population are supposed to have mean u = 100. Is there evidence that the mean in this district differs from 100? Follow the four-step process in your answer.

Here are the IQ test scores of 31 seventh-grade girls in a Midwest school district:

Treat the 31 girls as SRS of all seventh-grade girls in the school district. Suppose that the standard deviation of IQ scores in this population is known to be $\sigma$=15. Give a 95% confidence interval for the mean IQ score $\mu$ in the population. Use the Inference Toolbox.

Here are the IQ test scores of 31 seventh-grade girls in a midwestern school district:

114 100 104 89 102 91 114 114 103 105

108 130 120 132 111 128 118 119 86 72

111 103 74 112 107 103 98 96 112 112 93

The IQ test scores of 47 seventh-grade boys in the same district are

111 107 100 107 115 111 97 112 104 106 113

109 113 128 128 118 113 124 127 136 106 123

124 126 116 127 119 97 102 110 120 103 115

93 123 79 119 110 110 107 105 105 110 77

90 114 106

(a) Make stemplots or histograms of both sets of data. Because the distributions are reasonably symmetric with no extreme outliers, the t procedures will work well.

(b) Treat these data as SRSs from all seventh-grade students in the district. Is there good evidence that girls and boys differ in their mean IQ scores?

A professor asked her sophomore students, "How many drinks do you typically have per session?" (A drink is defined as one 12 oz beer, one 4 oz glass of wine, or one 1 oz shot of liquor.) Some of the students didn't drink. Table below gives the responses of the female and male students who did drink. It is likely that some of the students exaggerated a bit. The sample is all students in one large sophomore-level class. The class is popular, so we are tentatively willing to regard its members as an SRS of sophomore students at this college. Do a complete analysi that reports on

(a) the drinking behavior claimed by sophomore women.

(b) the drinking behavior claimed by sophomore men.

(c) a comparison of the behavior of women and men.

Female students

2.5 9 1 3.5 2.5 3 1 3 3 3 3 2.5 2.5

5 3.5 5 1 2 1 7 3 7 4 4 6.5 4

3 6 5 3 8 6 6 3 6 8 3 4 7

4 5 3.5 4 2 1 5 5 3 3 6 4 2

. 7 7 7 5.5 3 2.5 10 5 4 9 8 1 6

2 5 2.5 3 4.5 9 5 4 4 3 4 6 7

4 5 1 5 3 4 10 7 3 4 4 4 4

2 1 2.5 2.5

Male students

7 7.5 8 15 3 4 1 5 11 4.5 6 4 10

16 4 8 5 9 7 7 3 5 6.5 1 12 4

6 8 8 4.5 10.5 8 6 10 1 9 8 7 8

15 3 10 7 4 6 5 2 10 7 9 5 8

7 3 7 6 4 5 2 5 5.5 9 10 10 4

8 4 2 4 12.5 3 15 2 6 3 4 3 10

6 4.5 5

Sharks. Great white sharks are big and hungry. Here are the lengths in feet of 44 great whites:

(a) Examine these data for shape, center, spread, and outliers. The distribution is reasonably Normal except for one outlier in each direction. Because these are not extreme and preserve the symmetry of the distribution, use of the t procedures is safe with 44 observations.

(b) Give a 95% confidence interval for the mean length of great white sharks. Based on this interval, is there significant evidence at the 5% level to reject the claim "Great white sharks average 20 feet in length"?

(c) Before accepting the conclusions of (b), you need more information about the data. What would you like to know?

Radon is a colorless, odorless gas that is naturally released by rocks and soils and may concentrate in tightly closed houses. Because radon is slightly radioactive, there is some concern that it may be a health hazard. Radon detectors are sold to homeowners worried about this risk, but the detectors may be inaccurate. University researchers placed a random sample of 11 detectors in a chamber where they were exposed to 105 picocuries per liter of radon over 3 days. A graph of the radon readings from the 11 detectors shows no strong skewness or outliers. The mean reading is 104.82 and the standard deviation of the readings is 9.54. Is there convincing evidence at the 10% level that the mean reading differs from the true value 105?

Bottles of a popular cola are supposed to contain 300 milliliters (ml) of cola. There is some variation from bottle to bottle because the filling machinery is not perfectly precise. An inspector measures the contents of six randomly selected bottles from a single day's production. The results are 299.4; 297.7; 301.0; 298.9; 300.2; 297.0. Do these data provide convincing evidence that the mean amount of cola in all the bottles filled that day differs from the target value of 300 ml?

A bottling company uses a filling machine to fill plastic bottles with cola. The bottles are supposed to contain 300 milliliters (ml). In fact, the contents vary according to a Normal distribution with mean

$$\mu = 298 ml$$

and standard deviation

$$\sigma = 3 ml.$$

What is the probability that the mean contents of six randomly selected bottles are less than 295 ml? Show your work.

How much difference do a couple of weeks make for birth weight? Late perterm babies are born with 34 to 36 completed weeks of gestation. The distribution of birth weights (in grams) for late preterm babies is approximately N(2750, 560).

(a) What is the probability that a randomly chosen late preterm baby would have a low birth weight (less than 2500 grams)?

(b) What is the probability that a randomly chosen late preterm baby would have a very low birth weight (less than 1500 grams)?

(c) How do these results compare with your findings in the previous exercise?

Because the t procedures are robust, the most important condition for their safe use is that

(a) the population standard deviation a- is known.

(b) the population distribution is exactly Normal.

(c) the data can be regarded as an SRS from the population.