Discuss the below:
For the following four problems:
1. State the Null Hypothesis and the Alternative Hypothesis
2. Determine the test statistic.
3. Determine the P-value
4. Make a decision regarding the hypotheses based on the P-value and the Level of Significance.
Q1. In a recent year, some professional baseball players complained that umpires were calling more strikes than the average rate of 61% called the previous year. At one point in the season, umpire Dan Morrison called strikes in 2231 of 3581 pitches (based on data from USA Today). Use a 0.05 significance level to test the claim that his strike rate is greater than 61%.
Q2. The health of the bear population in Yellowstone National Park is monitored by periodic measurements taken from anesthetized bears. A sample of 54 bears has a mean weight of 182.9lbs. Assuming that the population standard deviation is known to be 121.8lbs, use a 0.05 level of significance to test the claim the population mean of all such bear weights is less than 200lbs.
Q3. The birth weights (in kilograms) are recorded for a sample of male babies born to mothers taking a special vitamin supplement (based on data from the New York Department of Health). When testing the claim that the mean birth weight for all male babies of mothers given vitamins is equal to 3.39kg, which is the mean weight of the population of all male babies, a sample of 16 babies had a mean of 3.675kg and a standard deviation of 0.657. Based on these results, does the vitamin supplement appear to have any effect on the mean birth weight? Use the 0.01 level of significance.
Q4. Patients with Chronic Fatigue Syndrome were tested, and then retested after being treated with fludrocortisone. The changes in fatigue after treatment were measured (based on data from 'The Relationship between Neurally Mediated Hypotension and the Chronic Fatigue Syndrome; by Bou-Holaigah, Rowe, Kan, and Calkins, Journal of the American Medical Association, Vol.274, No. 12). A standard scale from -7 to +7 was used with positive values representing improvements. The sample of 21 patients had a mean score of 4 on the scale with a sample standard deviation, s = 2.17 Use a 1% level of significance level to test the hypothesis that the mean change is positive(i.e. there is improvement.). Does the treatment appear to be effective?