Topics: Arithmetic mean, Standard deviation, Randomness Pages: 5 (2655 words) Published: November 8, 2014
Copyright 2010 Graham Elliott. All Rights Reserved.

We are now putting all of the pieces together. Considering each observation xi as an outcome from a random variable Xi , we have that functions g(x1 ; x2 ; :::; xn ) are draws from the random variable Pn g(X1 ; X2 ; :::; Xn ):

For 120a the function we are interested in is the sample mean — g(x1 ; x2 ; :::; xn ) = n1 i=1 xi : In this chapter we work with this function for distributions with many random variables.


From the Text

Question 1. Problem 6-10, p.206
Question 2: Problems 6-16,6-22.
Question 3: Problem 6-36.


Sampling Questions

Question 1. A study suggests that in a trial of 50 people the average decrease in cholestorol from using a particular drug is 18 points with a standard deviation of 10 points. (a) What is the chance that, given the true e¤ect is a reduction of 20 points, that a particular person has no decrease in cholestorol when they take this drug?

(b) A reporter writes a newspaper article discussing the drug. They report that although the study claims that the reduction is 18 points, they have found a user that actually had no change to their cholestorol. They use this persons story to argue that the results are misleading. What is the chance that, if the real e¤ect was zero, that the average in a trial might be 18 just by chance? (c) Is the counter example found by the reporter useful in understanding the results for the study? Question 2. In the questions on normal probabilities, we examined probabilities of di¤erent ages where people marry. We wish to see if the data really supports a mean of 26 years (and standard deviation of 4 years) for …rst marriage. We collect a random sample of 100 people and record what age they were when they …rst married. We …nd that the mean of our sample is 28 years. (a)

What is the chance that if the mean really were 26 years we would obtain a sample with a mean of 28 years or higher?
Do you believe that the true mean could be 26 years?
Question 3. A supermarket is open for 15 hours a day, and sees a stream of customers all day. Suppose that the mean number of customers entering each hour is 100, with a standard deviation of 30. (a) What is the chance on any day that the supermarket has more than 1680 customers in the day? (b) What is the chance that there are less than 1450 in a given day? Question 4. A recent Gallup poll (conducted March 5-8, 2009. polling 1,012 adults nationwide) found that 77% of respondents were in favor of increased …nancial support and incentives for producing energy from alternative sources such as wind and solar power.

(a) Is it reasonable to claim that only 70% of the population are in favor of increasing incentives? (b) Is it reasonable to claim that three quarters of the population are in favor of increasing incentives? Question 5. You are deciding how to invest your holdings of your 401(k) retirement plan. Each of the 50 stocks you can invest in have a mean return of 10% per year with a standard deviation of 6% of the investment. You have $1000 to invest. We will assume that the returns across stocks are independent. 1

(a) Suppose you invest all of the money in a single stock. Assuming that returns are normally distributed, what is the chance that you loose money in the …rst year?
(b) Instead of investing in a single stock, your …nancial planner suggests putting an equal amount into all 50 stocks. What is the chance that you loose money in the …rst year with this strategy? (c) In your answer for (b), do you need to assume that the returns are normally distributed? Why or why not?

Question 6. A Pew Research poll (Dec. 9-13, 2009. 779 Adults nationwide) recently asked "Looking ahead, what’s your best guess: For the country as a whole, do you think that the next decade will be better or worse than the current decade?". Of the respondents, 59% thought things would be better. (a) To put as rosy a spin on this as possible, you want to say that 65%...
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