1. Suppose X ~ N. That is, X has a normal distribution with μ=30 and σ2=144.
1a. Find a transformation of X that will give it a mean of zero and a variance of one(ie., standardize X).
1b. Find the probability that 18< x<36.
1c. Supposing Y = 10 + 5X, find the mean of Y.
1d. Find the variance of Y.
2. A bank has been receiving complaints from real estate agents that their customers have been waiting too long for mortgage confirmations. The bank prides itself on its mortgage application process and decides to investigate the claims. The bank manager takes a random sample of 20 customers whose mortgage applications have been processed in the last 6 months and finds the following wait times (in days):
5, 7, 22, 4, 12, 9, 9, 14, 3, 6, 5, 8, 10, 17, 12, 10,9, 4, 3, 13
Assume that the random variable measures the number of days a customer waits for mortgage processing at this bank, and assume that is normally distributed.
2a. Find the sample mean of this data X' .
2b. Find the sample variance of . Find the variance of X'.
For (c), (d), and (e), use the appropriate t-distribution
2c. Find the 90% confidence interval for the population mean (μ).
2d. Test the hypothesis that μis equal to 7 at the 95% confidence level. (Should you do a one-tailed or two-tailed test here?)
2e. What is the approximate p-value of this hypothesis?
3. My nephew was born last summer. He has 19 cousins on his father's side (it's a big family). I wish to know the mean, μx, of the distribution of the ages of my nephew's cousins (which is the variable X). I take a sample of 4, with ages X1, X2, X3 and X4. These are all drawn from the same underlying population. Instead of calculating the sample mean of these four, I do the following calculation to create an estimator of , which I call X.
3a. Explain what it means to say that is unbiased?
3b. Show that X* is unbiased.
3c. If the variance of is 15, what is the variance of X*?
3d. If the variance of is 15, what is the variance of the X'?
3e. Which estimator is more efficient, the sample mean or X*? Explain.