Decrease of Value of Z-Score

Part 1
1) The value of the z- score that is obtained for a hypothesis test is influenced by several factors. Some factors influence the size of the numerator of the z- score and other factors influence the size of the standard error in the denominator. For each of the following, indicate whether the factor influences the numerator or denominator of the z- score and determine whether the effect would be to increase the value of the z- score (farther from zero) or decrease the value of z (closer to zero). In each case, assume that all components of the z-score remain constant.
a) Increase the sample size.
b) Increase the population standard deviation.
c) Increase the difference between the sample mean and the value of µ specified in the null hypothesis.

2) A researcher would like to test the effectiveness of a newly developed growth hormone. The researcher knows that under normal circumstances laboratory rats reach an average weight of µ = 950 grams at 10 weeks of age. The distribution of weights ism normal with σ = 30. A random sample of n = 25 newborn rats is obtained, and the hormone is given to each rat. When the rats in the sample reach 10 weeks old each rat is weighed. The mean weight for this sample is M = 974.
a) Identify the independent and the dependent variables for this study.
b) Assuming a two- tailed test, state the null hypothesis in a sentence that includes the independent variable and the dependent variable.
c) Using symbols, state the hypothesis (Ho and H1) for the two tailed test.
d) Sketch the appropriate distribution and locate the critical region for α = .05
e) Calculate the test statistic ( z-score) for the sample.
f) What decision should be made about the null hypothesis, and what decision should be made about the effect of the hormone?

3) Under some circumstances a 6 point treatment effect can be very large, and in some circumstances it can be very small. Assume that a sample of n= 16 individuals is from a population with a mean of µ = 70. A treatment is administrated to the sample and, after treatment, the sample mean is found to be M = 76. Notice that the treatment appears to have increased scores by an average of 6 points.
a) If the population standard deviation is σ = 20, is the 6 point effect large enough to be statistically significant? Use a two tailed test with α = .05.
b) If the population standard deviation is σ = 8, is the 6 point effect large enough to be statistically significant? Use the two tailed test with α = .05.

4) A sample of n= 9 scores is obtained from a normal population distribution with σ = 12. The sample mean is M= 60.
a) With a two tailed test and α = .05, use the sample data to test the hypothesis that the population mean is µ = 65.
b) With a two tailed test and α = .05, use the sample data to test the hypothesis that the population mean is µ = 55
c) In parts (a) and (b) of this problem, you should find that µ = 65 and µ = 55 are both acceptable hypothesis. Explain how two different values can both be acceptable.

Part 2
1) What is the relationship between the value for degrees of freedom and the shape of the t distribution? What happens to the critical value of t for a particular alpha level when df increases in value?

2) A sample of n = 25 individuals is randomly selected from a population with a mean of µ = 65, and a treatment is administered to the individuals in the sample. After treatment, the sample mean is found to be M = 70.
a) If the sample standard deviation is s = 10, are the data sufficient to conclude that the treatment has a significant effect using a two tailed test with a = .05?
b) If the sample standard deviation is s = 20, are the data sufficient to conclude that the treatment has a significant effect using a two tailed test with α = .05?

3) The herbal supplement ginkgo bilboa is advertised as producing an increase in physical strength and stamina. To test this claim, a sample of n = 36 adults is obtained and each person is instructed to take the regular daily dose for the herb for a period of 30 days. At the end of 30 day period, each person is tested on a standard treadmill task for which the average, age adjusted score is µ = 15. The individuals in the sample produce a mean score of M = 16.9 with SS = 1260.
a) Are these data sufficient to conclude that the herb has a statistically significant effect using a two tailed test with α = .05?
b) What decision would be made if the researcher used a one tailed test with α = .05? (Assume that the herb is expected to increase scores.)

Part 3
1) Several factors influence the value obtained for the independent measures t statistic. Some factors affect the numerator of the t statistic and others influence the size of the estimated standard error in the denominator. For each of the following, indicate whether the factors influences the numerator or the denominator of the t statistic and determine whether the effect would be to increase the value of t (farther from zero) or decrease the value of t (closer to zero). In each case, assume that all other factors remain constant.
a) Increase the difference between the two sample means.
b) Increase the size of the two samples.
c) Increase the size of the sample variances.

2) One sample has SS = 35 and a second sample has SS = 45
a) Assuming that n = 6 for both samples, calculate each of the sample variances, and then calculate the pooled variance. Because the samples are the same size, you should find that the pooled variance is exactly halfway between the two sample variances.
b) Now assume that n = 6 for the first sample and n = 16 for the second. Again, 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.

3) As noted, when the two population means are equal, the estimated standard error for the independent measures t test provides a measure of how much difference to expect between two sample means. For each of the following situations, assume that µ1 = µ2 and calculate how much difference should be expected between the two sample means.
a) One sample has n = 8 scores with SS = 30 and the second sample has n = 8 scores with SS = 26.
b) One sample has n = 8 scores with SS = 150 and the second sample has n = 4 scores with SS = 90.