# Hypothesis testing and z tests

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Problem Set 1: Chapter 8, problems 2, 4, 6, 10, 12, 22, 24

2. The value of the z- score in a hypothesis test is influenced by a variety of factors. Assuming that all other variables are held constant, explain how the value of z is influenced by each of the following:

a. increasing the difference between the sample mean and the original population mean.

b. Increasing the population standard deviation.

c. Increasing the number of scores in the sample.

4. If the alpha level is changed from a = .05 to a = .01,

a. What happens to the boundaries for the critical regions?

b. What happens to the probability of a type I error?

6. A researcher is investigating the effectiveness of a new study-skills training program for elementary school children. A sample of n= 25 third grade children and each child is given a standardized achievement test at the end of the year. For the regular population of third grade children, scores on the test form a normal distribution with a mean of u = 150 and a standard deviation of o= 25. The mean for the sample is M = 158.

a. Identity the independent and the dependent variables for this study.

b. Assuming two-tailed test, state the hypotheses (Ho and H1 ) for the two-tailed test.

c. Using symbols, state the hypotheses (Ho and H1) for the two-tailed test.

d. Sketch the appropriate distribution, and locate the critical regions for a= .05.

e. Calculate the test statistics (z score) for the sample.

f. What decision should be made about the effect of the program?

10. State College is evaluating a new English composition course for freshmen. A random sample of n = 25 freshmen is obtained and the students are placed in the course during their first semester. One year later, a writing sample is obtained for each student and the writing samples are graded using a standardized evaluation technique. The average score for the sample is M = 76. For the general population of college students, writing scores form a normal distribution with a mean of u = 70.

a. If the writing scores for the population have a standard deviation of o = 20, does the sample provide enough evidence to conclude that the new composition course has a significant effect?

b. If the population standard deviation is o = 10, is the sample sufficient to demonstrate a significant effect? Again, assume a two-tailed test with a= .05.

c. Briefly explain why you reached different conclusions for part (a) and part (b).

12. To test the effectiveness of a treatment, a sample of n = 25 people is selected from a normal population with a mean of u = 60. After the treatment is administered to the individuals in the sample, the sample mean is found to be M = 55.

a. If the population standard deviation is o = 10, can you conclude that the treatment has a significant effect? Use a two-tailed with a = .05.

b. If the population standard deviation is o = 20, can you conclude that the treatment has s significant effect? Use a two-tailed test with a= .05.

c. Compute Cohen's d measure effect size for both tests (n= 4 and n = 36).

d. Briefly describe how sample size influence the outcome of the hypothesis test. How does sample size influence measures of effect size?

22. Explain how the power of a hypothesis test is influenced by each of the following. Assume that all other factors are held constant.

a. Increasing the alpha level from .01 to .05.

b. Changing from a one-tailed test to a two-tailed test.

24. A researcher is evaluating the influence of a treatment using a sample selected from a normally distributed population with a mean of u = 80 and a standard deviation of o= 20. The researcher expects a 12 point treatment effect and plans to use a two-tailed hypothesis test with u a = .05

a. Compute the power of the test if the researcher uses a sample of n = 16 individuals.

b. Compute the power of the test if the researcher uses a sample of n = 25 individuals.

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##### Solution Summary

Explains in detail several of the concepts underlying z tests, hypothesis testing, statistical decisions, power, Type I errors, and effect size. Shows calculations and graphs. Step-by-step solution of several classic problems.

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