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# Hypothesis Testing of Mean

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Leave out questions 1, 3, 11 and 12.

1. 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: Increasing the difference between the sample mean and the original population mean, increasing the population standard deviation, and increasing the number of scores in a sample.

2. A sample of n=25 third graders is selected to participate in a standardized test. Scores on the normal distribution forms a mean of 150 and standard deviation 25. The mean sample is M=158

a. What is the independent and dependent variables for this study.
b. Assuming a two tailed test, state the null hypothesis in a sentence that includes the indep. And dep. variable
c. Using symbols, state the hypotheses for the two tailed test
d. Sketch the appropriate distribution and locate the critical region for alpha=0.05
e. Calculate the test stat (s score) for the sample
f. What decision should be made about the effect of the program

3. A sample n=50 male college students is obtained and each student is given a series of three endurance tasks and asked to consume 4 ounces of the drink during each break between tasks. The overall endurance score for this sample is M=53. For the general population, without any sports drink, the scores for this task average 50 with stand. Dev 12

a. Can the researcher conclude that endurance scores with sports drink are higher than scores without? Use one tail test with alpha = 0.05
b. Can he conclude that endurance scores with sports drink are significantly different than scores without the drink? Use two tail test with alpha = 0.05
c. Why do the two tests lead to different conclusions?

4. Researchers have noted higher rates of a baseball batter getting hit when weather gets hot. Suppose that over the past 30 years, during any given week of the season, an average mean of 12 players are hit by wild pitches. Assume the dist, is nearly normal with stand. Dev 3. For a sample of n=4 weeks in which daily temp. was extremely hot, the weekly average of hit by pitch players was M=15.5. Are players most likely to get hit during hot weeks? Set alpha to 0.05 for a one tailed test.

5. Explain why t dist. Tend to be flatter and more spread out than normal distributions.

6. To evaluate the effect of a treatment, a sample of n=9 is obtained from a population with a mean of 40. And the treatment is administered to the individuals in the sample. After treatment, the sample mean in found to be M=33

a. If the sample has a standard dev of s= 9, are the data sufficient to conclude that the treatment has a significant effect using a two tailed test with alpha 0.05?
b. If stand. Dev is s = 15, are the data sufficient enough to conclude that the treatment has a sig. effect using two tailed test at alpha 0.05
c. Compare your answers. How does the variability of the scores in the sample influence the outcome of a hypothesis test?

7. Many animals tend to avoid direct eye contact and even patterns that look like eyes. In the study, the birds were tested in a box with two chambers and were free to move from one to another. In once chamber, two large eye spots were painted and the other chamber was plain. The researcher recorded the amount of time each bird spent in the plain chamber during a 60 min session. Suppose the study produced a mean of M=37 minutes in plain chamber with SS=288 for a sample of n=9 birds. If the eye spots have no effect, then the birds should spend an average of 30 minutes in each chamber.

a. Is this sample sufficient to conclude that the eye spots have a sig. influence on the birds' behavior? Using a two tailed test with alpha 0.05, find the answer
b. Compute the estimated Cohen's d to measure the size of the treatment effect.
c. Write a sentence that demonstrates how the outcome of the hypothesis test and the measure of effect size would appear in a research report.

8. If other factors are held constant, how does increasing the sample variance affect the value of the indep. Measures t statistic and the likelihood of rejecting the null hypothesis?

9. One sample has SS = 70 and the second sample has SS =42

a. If n=8 for both samples, find each of the sample variances, and calculate the pooled variance.
b. Now assume that n=8 for the first sample and n=4 for the second. Calculate the two sample variances and the pooled variance. You should find that the pooled variance is closer to the variance for the larger sample

10. When two population means are equal, the estimated standard error for the indep. Measures t test provides a measure of how much difference to expect between two sample means. For each of the following situations, assume that mean one = mean two and calculate how much difference should be expected between the two sample means.

a. One sample has n=8 scores with SS=45 and the second sample has n=4 scores with SS=15
b. One sample has n=8 scores with SS=150 and the second sample has n=4 with SS =90
c. How does large variability affect the size of the standard error for the sample mean difference?

11. For a hypothesis test, what happens to the boundaries for the critical region when the alpha level is lowered - for example, from .05 to .01? Also, what happens to the probability of a Type I error when the alpha level is lowered?

12. Discuss the errors that can be made in hypothesis testing.

a) What is a Type I error? Why might it occur?

b) What is a Type II error? How does it happen?

13. A psychologist has developed a standardized test for measuring vocabulary skills for 4-year-old children. The scores on the test form a normal distribution with µ = 70 and &#963; = 20. A researcher would like to use the test to investigate the hypothesis that children who grow up as an only child develop vocabulary skills at a faster rate than children in larger families. A sample of n = 25 only children is obtained and the mean test score for the sample is M = 77.

a) Is this sample sufficient to conclude that vocabulary skills for only children are significantly higher than those for the general population? Use a one-tailed test with &#945; = .05.

b) Is this sample sufficient to conclude that vocabulary skills for only children are significantly different from those for the general population? Use a two-tailed test with &#945; = .05.

c) Explain why the two tests lead to different conclusions.

14. (Chapter 9) A sample of n = 16 scores has a mean of M = 45 and SS = 960.

a) Calculate the sample standard deviation (s) and the estimated standard error for the sample mean ( ).

b) Describe what is measured by the standard deviation and what is measured by the estimated standard error.

15. A sample is randomly selected from a population with a mean of µ = 50 and a treatment is administered to the individuals in the sample. After treatment, the sample is found to have a mean of M = 56 with a sample standard deviation of s = 8.

a) If there are n = 4 individuals in the sample, are the data sufficient to reject and conclude that the treatment has a significant effect using a two-tailed test with &#945; = .05?

b) If there are n = 16 individuals in the sample, are the data sufficient to reject and conclude that the treatment has a significant effect? Again, assume a two-tailed test with &#945; = .05?