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Exercise 3: A researcher was interested in studying the effect of a new supplement on the length on running time in casual runners. He selected a random sample of n = 9 casual runners from a normal population of casual runners with a mean running time of = 100 minutes. He then asked the 9 randomly selected runners to take the new supplement before going on a run and record their running time. These runners reported an average running time of M = 106.

Clearly state all four steps of hypothesis testing as they apply to each problem statement: a) - d) (when stating the null and alternative hypotheses, you can either use scientific notation, words, or both). Sketch the distribution for each, shade in the critical region or regions for each problem statement and include relevant statistics/parameters

. a) If the standard deviation is = 10, is the sample mean sufficiently different from the population mean to concludethat the new supplement has a significant effect on running time? Use a two-tailed test with = .05.

b) Now, assume a one-tailed test with = 0.5. If the standard deviation is = 10, is the sample mean sufficiently greater than the population mean to conclude that the new supplement results in longer running time?

c) If the population standard deviation is = 12, is the sample mean sufficiently different from the population mean to concludethat the new supplement has a significant effect on running time? Use a two-tailed test with = .05.

d) Now, assume a one-tailed test with = 0.5. If the standard deviation is = 12, is the sample mean sufficiently greater than the population mean to conclude that the new supplement results in longer running time?

e) Comparing your answers for parts a) through d), explain how the magnitude of the standard deviation (10 vs. 12) and the number of tails in the hypothesis (one vs. two) influence the outcome of a hypothesis test.