1. A (hypothetical) study is conducted to see if the Investment firm of Dewey, Cheatum, and Howe (who also work for Car Talk) provides sound investment advice. An analysis of all stocks traded on the NYSE for a particular year showed that returns varied according to a normal distribution with a mean of 7.5% and an SD of 22%. (A negative return represents a loss.) A portfolio prepared by DC&H had these annual returns for the same time period:
29.8, 31.7, 33.4, -16.1, -1.8, -5.2, -55.9, -6.9, 39.5, -8.4
a) Sketch the distribution of the population of NYSE returns.
b) What's the average of this portfolio?
c) Find the median and IQR of this portfolio.
d) What's the probability that a randomly selected stock from the NYSE will have a return less than 0%?
e) What's the probability that a randomly selected stock from this portfolio will have an annual return less than 0%?
f) What's the SE for an average of 10 randomly selected stocks from the NYSE?
g) What's the probability that the average of 10 randomly selected stocks from the NYSE will be less than 0%?
h) How many SE's above or below 0 is the average of this portfolio?
i) Use this sample to find a 90% Confidence Interval for the mean return.
j) Do these data support DC&H's claim to be a "good" financial investment firm? By "good", I mean does it look like they can select stocks better than, say, a random process? Set up a hypothesis test to prove your point. Explicitly state each step of the test. Don't forget to give a conclusion!
2. In the back of a national health magazine there is an ad for magnets that claims to relieve muscle aches if the magnets are worn around the wrists while exercising. To support this claim, the company ran ads in magazines and newspapers asking for volunteers interested in a study to determine the power of magnets to relieve muscle pain. Volunteers had to be regular exercisers (as determined by their answers to a questionnaire) who suffer from occaisonal muscle aches and pains. Out of the 1000 eligible respondents, 100 were selected at random to participate in the study. At the beginning of the study, the subjects were given a detailed medical questionnaire designed to assess their pain level. These scores range from 0 to 200, with scores of 200 representing extreme pain and 0 representing no pain. The subjects were then given magnets and told to wear them whenever they performed any exercise or sporting activities. At the end of 6 months, the subjects were again given the pain questionnaire. A "change" score was calculated: pain score after minus pain score before. The 100 pain scores had an average of -25 with an SD of 50.
1. The study gave a 90% Confidence Interval for the mean pain relief as:
(-31.45, -18.55). They claim in their ad that this means that 90% of all subjects experienced pain relief in this range. Is this a valid interpretation of the Confidence Interval? Explain.
2. Do you agree with the claim? Explain in detail. Your answer should include either a confidence interval or a hypothesis test, as well as a full examination of the data as given above.
3. How would you design a study to test this
claim? Explain each decision and feature you include.