As the principal of a private high school, you are interested in finding out how the training in mathematics at your school compares with that of the public schools in your area. For the last 5 years, the public schools have given all graduating seniors a mathematics proficiency test. The distribution has a mean of 75. You give all the graduating seniors in your school the same mathematics proficiency test. The results show a distribution of 26 scores, with a mean of 82 and a standard deviation of 13.

Using α = 0.05_{2 tail}, what do you conclude?
A student conducted an experiment on 25 schizophrenic patients to test the effect of a new technique on the amount of time schizophrenics need to stay institutionalized. The results showed that under the new treatment, the 25 schizophrenic patients stayed a mean duration of 78 weeks, with a standard deviation of 20 weeks. Previously collected data on a large number of schizophrenic patients showed a normal distribution of scores, with a mean of 85 weeks and a standard deviation of 15 weeks. These data were evaluated using α = 0.05_{2 tail}. The results showed a significant effect. Assume that the standard deviation of the population is unknown. Using α = 0.05_{2 tail}, what do you conclude about the new technique?

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A sample set of 29 scores has a mean of 76 and a standard deviation of 7. Can we accept the hypothesis that the sample is a random sample from a population with a mean greater than 72? Use α = 0.01_{1 tail} in making your decision

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A local business school claims that their graduating seniors get higherpaying jobs than the national average for business school graduates. Last year’s figures for salaries paid to all business school graduates on their first job showed a mean of $10.20 per hour. A random sample of 10 graduates from last year’s class of the local business school showed the following hourly salaries for their first job: $9.40, $10.30, $11.20, $10.80, $10.40, $9.70, $9.80, $10.60, $10.70, $10.90. You are skeptical of the business school claim and decide to evaluate the salary of the business school graduates using α = 0.05_{2 tail}.

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Is it reasonable to consider a sample with N = 22, X_{obt} = 42, and s = 9 to be a random sample from a normal population with μ = 38? Use α = 0.05_{1 tail} in making your decision. Assume X_{obt} is in the right direction.
A developmental psychologist is interested in whether tense parents tend to have tense children. A study is done involving one parent for each of 12 families and the oldest child in each family, measuring tension in each pair. Pearsonr = 0.596. Using α = 0.05_{2 tail}, is the relationship significant?

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A cognitive psychologist believes that a particular drug improves shortterm memory. The drug is safe, with no side effects. An experiment is conducted in which 8 randomly selected subjects are given the drug and then given a short time to memorize a list of 10 words. The subjects are then tested for retention 15 minutes after the memorization period. The number of words correctly recalled by each subject is as follows: 7, 8, 12, 5, 10, 6, 8, 7. Over the past few years, the psychologist has collected a lot of data using this task with similar subjects. Although he has lost the original data, he remembers that the mean was 5 words correctly recalled and that the data were normally distributed.
A physician employed by a large corporation believes that due to an increase in sedentary life in the past decade, middleage men have become fatter. In 1995, the corporation measured the percentage of fat in their employees. For the middleage men, the scores were normally distributed, with a mean of 22%. To test her hypothesis, the physician measures the fat percentage in a random sample of 12 middleage men currently employed by the corporation. The fat percentages found were as follows: 24, 40, 29, 32, 33, 25, 15, 22, 18, 25, 16, 27. On the basis of these data, can we conclude that middleage men employed by the corporation have become fatter? Assume a directional H_{1} is legitimate and use α = 0.05_{1 tail} in making your decision.

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A sample set of 30 scores has a mean of 15.0 and a standard deviation of 6.0. Is it reasonable to assume that the sample is a random sample from a population set of scores with a mean of 18.0? In drawing your conclusion, assume the population is normally distributed and use α = 0.05_{2 tail}.

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The chairperson of a psychology department believes the Ph.D. program of her department is particularly good; that it attracts exceptional graduate students, and consequently, the Ph.D. students of her department earn their Ph.D.’s in significantly less time than do their counter parts in other psychology departments. To test this belief, she compares the time it took students to earn their Ph.D.’s in her department over the past three years, with national data for the same time period. In the past three years, this department has awarded 35 Ph.D.’s; the mean time to obtain the Ph.D. was 62.0 months, with a standard deviation of 6.2 months. National data for psychology departments throughout the United States compiled for that time showed that the mean time to obtain a Ph.D. was 78.4 months, with an unknown standard deviation, and the distribution of these scores was normally distributed.

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A sample set of 20 scores has a mean equal to 81 and a standard deviation of 11. Can we reject the hypothesis that this sample is a random sample from a normal population with μ = 77? Use α = 0.01_{2 tail} in making your decision.

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A college counselor wants to determine the average amount of time firstyear students spend studying. He randomly samples 25 students from the freshman class and asks them how many hours a week they study. The mean of the resulting scores is 23 hours, and the standard deviation is 6.4 hours.
(a) Construct the 95% confidence interval for the population mean.