Cognitive Science 14
Homework Set 4 Questions

1. A random sample of 3,000 income tax returns was taken from a given tax year. Each was examined for the number of exemptions claimed for that year. The mean was found to be 3.78, with a standard deviation of 0.97. Find the 99 % confidence interval for the true mean based on these data.
2. In a study of the effects of a medication on the body temperature of normal adults, a scientist wishes to be 95 % sure that the estimates made from a sample are with 0.01 degrees F of the population mean. The population under study is believed to have a standard deviation in body temperature of 0.07 degrees F. How many subjects should be used in the sample if these conditions are to be met?
3. Suppose that in a certain large community the number of hours that a TV set is turned on in a given home during a given week is approximately normally distributed. A sample of 26 homes was selected, and careful logs were kept of how many hours per week the TV set was on. The mean number of hours per week in the sample turned out to be 36.1, with an unbiased sample standard deviation s of 3.3 hours. Find the 95 % confidence interval for the mean number of hours that TV sets are played in the homes of this community.
4. A psychological test was standardized for the population of 10th grade students in such a way that the mean must be 500 and the standard deviation 100. A sample of 90 12th grade students was selected independently and at random, and each was given the test. The sample mean turned out to be 506.7. On this basis, can one say that the population mean for 12th grade students differs from 10th graders?
5. A nine-hole golf course was supposed to have a par of 30, but over a long period, the population of golfers who played this course had a mean of 38.2, with a standard deviation of 3.3. A designer was called in to extend this course to 18 holes, on the understanding that the last nine should have the same difficulty as the first nine. After the course was finished, a random sample of 121 golfers played the course and produced an average score of 42.6 on the last nine holes. Given that this sample was drawn from the same population that played the first nine holes, test the hypothesis that the two sets of nine holes are truly equal in difficulty. Use .
6. A random sample of 175 American women were asked to record their body temperatures twice a day for a full month. From their records, an average value was found for each woman. The mean of these values was 98.7, with a standard deviation s of 0.95. Test the hypothesis that the mean body temperature of such American women is 98.6 against the alternative that the mean is some other value.
7. There was concern in a public-school system that the method of teaching reading then in use might be inferior to other methods. A standardized test was available, giving national norms on reading achievement. For 5th graders, this test showed a national average of 172, with a standard deviation of 16. A random sample was taken of 250 5th graders taught by the method in question, who were each given the test. Frame the null and the alternative hypothesis, and indicate the regions of rejection. Let the probability of incorrectly rejecting the null hypothesis, , be 0.01.
8. Some 80 rats selected at random were taught to run a maze. All finally succeeded in learning the maze, and the average number of trials to perfect performance was 15.91. However, long experience with a population of rats trained to run a similar maze shows that the average number of trials to success is 15, with a standard deviation of 2. Would you say that the new maze appears to be harder for rats to learn than the older, more extensively used maze?
9. In a study of truth in advertising, a government agency opened 500 boxes selected at random of a well known brand or raisin bran. For each box the actual number of raisins was counted. The mean number of raisins was 32.4, with a standard deviation s=4.1. Evaluate the company's claim that each box contains 34 raisins on the average, against the alternative of fewer raisins than claimed.
10. Suppose that the body weight at birth of normal children (single births) within the United States is approximately normally distributed and has a mean of 115.2 oz. A pediatrician believes that the birth weights of normal children born of mothers who are habitual smokers may be lower on the average. To test this hypothesis, records were taken of the birth weights of a random sample of 20 children from mothers who are still smokers. The mean of this sample is 114.0 oz, with s=4.3. Evaluate the pediatrician's theory.