Unit 7

1. A national poll reported that average wages had reached a record high of $21,300. In order to determine how the city compared to the national mean, the chamber of commerce gathered data from the 25 largest employers using a random process and obtained an average of $22,750 with a standard deviation of $2750. Can the chamber support the claim that the city is significantly different from the national average at the 0.01 level? 2. A sampling of 50 random women and 50 random men in the state of Vermont produced a mean age at death of 75.5 years for women and 67.9 years for men with standard deviations of 16.2 years (women) and 18.3 years (men). At the 0.05 level of significance, is there convincing evidence that women live longer than men? 3. There is discussion in the media of grade inflation; that is, average grades rising over time. To determine if grade averages are really higher now than in the past, Professor Smith gathered records from 90 randomly selected students to compare to the mean of 2.54 that existed 10 years ago. The 90 students produced a grade average of 2.69 with a standard deviation of 0.96. At the 0.05 level of significance, is there convincing evidence that grade averages higher than in the past? 4. A researcher wishes to determine whether the salaries of professional nurses employed by private hospitals are higher than those of nurses employed by government-owned hospitals. She selects a random sample of nurses from each type of hospital and calculates the means and standard deviations of their salaries. Private hospital nurses had a mean salary of $26,800 with a standard deviation of $600 (sample of 10 nurses), while the government-owned hospital nurses had a mean salary of $25,400 with a standard deviation of $450 (sample of 8). At the 0.01 level, can she conclude that the private hospitals pay more than the government hospitals? It is known that population mean of salaries for nurses vary normally. 5. A physical education director claims that by taking 800 international units (IU) of vitamin E, a weightlifter can increase his strength. Eight athletes are randomly selected and given a test of strength, using the standard bench press. After two weeks of regular training, supplemented with vitamin E, the same athletes are tested again. Test the effectiveness of the vitamin E regimen at the 0.05 level of significance. Each value in the data that follow represents the maximum number of pounds the athlete can bench-press Athlete 2 3 4 5 6 7 8 Before 210 230 182 205 262 253 219 216 After 219 236 179 204 270 250 222 216 6. Find a 99% confidence interval of the difference between IQ scores for students in schools in two different ethnic neighborhoods. A random sample of 30 was taken from school A with a mean of 105 and standard deviation of 10.2, while a random sample of 30 was taken from school B with a mean score of 108 and a standard deviation of 12.4. 7. Find a 90% confidence interval that represents the range of plausible values for the effectiveness of the vitamin E regimen described in # 5.A local coach has a theory, based on his experience, that participants in sports which stress individual achievement do better academically than those who participate in team sports. He believes that the skills required for individual success, such as motivation and self-discipline, carry over to academic areas. To test his theory, he collected the following data on cumulative grade-point averages of female athletes: team sport participants sample of 7 with mean 2.56 and standard deviation 0.871, individual sport participants sample of 8 with mean 3.04 and standard deviation 0.621. Is his assumption justified at the 0.05 level of significance? Samples were randomly selected and it is known that grade point averages vary normally. 9. To examine seasonal average in volume of stock trades, in millions of shares, the following data were gathered. Is the average volume of trades greater in the fall, at the 0.05 level of significance? SRS size mean standard deviation Fall 30 45 5.2 Spring 35 38.2 3.9 10. A maker of frozen meals claims that the average caloric content of its meals is 800. A researcher tested an SRS of 12 meals and found that the average number of calories was 873 with a standard deviation of 25. Caloric content varies normally. Is there enough evidence to reject the claim at a = 0.02? 11. A sports shoe manufacturer claims that joggers who wear its brand of shoe will jog faster than those who don't use its product. An SRS of eight joggers is taken and the joggers agree to test the claim on a 1-mile track randomly assigning the order of wearing each type of show. The rates (in minutes) of the joggers while wearing the manufacturer's shoe and while wearing any other brand of shoe are shown. Test the claim at @ = 0.025. Runner 2 8 Manufacturer's 8.2 6.3 9.2 8.6 6.8 8.7 8.0 6.9 Brand Other Brand 7.1 6.8 9.8 8.0 5.8 8.0 7.4 8.0 12. A researcher estimates that high school girls miss more days of school than high school boys. A random sample of 36 girls showed that they missed an average of 3.9 days of school per school year; a random sample of 36 boys showed that they missed an average of 3.6 days of school per year. The standard deviations are 0.6 and 0.8, respectively. At the 1% level of significance, test the claim. 13. The average hemoglobin reading for an SRS of 30 teachers was 16 grams per 100 milliliters, with a sample standard deviation of 2 grams. Find the 99% confidence interval of the true mean. 14. A study of teenagers found that a random sample of 50 boys talked on the phone an average of 21 minutes per conversation with a standard deviation of 2.1. A random sample of 50 girls talked on the phone an average of 18 minutes with a standard deviation of 3.2. Find the 95% confidence level of the true differences in means