Testing of hypothesis problems
Can you please help me answer these problems. They are within the attachment. Thanks!
1. Evolutionary theories often emphasize that humans have adapted to their physical environment. One such theory hypothesizes that people should spontaneously follow a 24-hour cycle of sleeping and waking?even if they are not exposed to the usual pattern of sunlight. To test this notion, eight paid volunteers were placed (individually) in a room in which there was no light from the outside and no clocks or other indications of time. They could turn the lights on and off as they wished. After a month in the room, each individual tended to develop a steady cycle. Their cycles at the end of the study were as follows: 25, 27, 25, 23, 24, 25, 26, and 25.
Using the 5% level of significance, what should we conclude about the theory that 24 hours is the natural cycle? (That is, does the average cycle length under these conditions differ significantly from 24 hours?) (a) Use the steps of hypothesis testing. (b) Sketch the distributions involved. (c) Explain your answer to someone who has never taken a course in statistics.
2. Twenty students randomly assigned to an experimental group receive an instructional program; 30 in a control group do not. After 6 months, both groups are tested on their knowledge. The experimental group has a mean of 38 on the test (with an estimated population standard deviation of 3); the control group has a mean of 35 (with an estimated population standard deviation of 5). Using the .05 level, what should the experimenter conclude? (a) Use the steps of hypothesis testing, (b) sketch the distributions involved, and (c) explain your answer to someone who is familiar with the t test for a single sample, but not with the t test for independent means.
3. Do students at various colleges differ in how sociable they are? Twenty-five students were randomly selected from each of three colleges in a particular region and were asked to report on the amount of time they spent socializing each day with other students. The results for College X was a mean of 5 hours and an estimated population variance of 2 hours; for College Y, M = 4, S2 = 1.5; and for College Z, M = 6, S2 = 2.5. What should you conclude? Use the .05 level. (a) Use the steps of hypothesis testing, (b) figure the effect size for the study; and (c) explain your answers to (a) and (b) to someone who has never had a course in statistics.
4. What is the power of the effect in the following planned studies using the analysis of variance with p < .05?
Predicted Effect Size Overall Design Numbers of Levels in Effect Participants Per Cell
(a) small 2 x 2 2 30
(b) small 2 x 2 2 50
(c) small 2 x 3 2 30
(d) small 2 x 3 3 30
(e) Medium 2 x 2 2 30
5. About how many participants do you need in each cell for 80% power in each of the following planned studies, using the analysis of variance with p < .05?
Predicted Effect Size Design Effect
(a) Small 2 x 2 Main effect
(b) Small 2 x 2 Interaction effect
(c) Medium 2 x 2 Main effect
(d) Small 2 x 3 Two level main effect
(e) Small 2 x 3 Three level main effect
(f) Small 2 x 3 Interaction effect