# ECON 306 – Homework 1 1

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ECON 306 – Homework 1 1. 1a. 1b. 1c. 1d. 2. 2a. 2b. 2c. 2d. 2e. 3. Suppose . That is, X has a normal distribution with μ=30 and σ2=144. Find a transformation of that will give it a mean of zero and a variance of one (ie., standardize ). Find the probability that . Supposing 5X, find the mean of . Find the variance of . 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. Find the sample mean of this data ( . Find the sample variance of . Find the variance of . For (c), (d), and (e), use the appropriate t- distribution Find the 90% confidence interval for the population mean (μ). Test the hypothesis that μ is equal to 7 at the 95% confidence level. (Should you do a one- tailed or two- tailed test here?) What is the approximate p- value of this hypothesis? 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, , of the distribution of the ages of my nephew’s cousins (which is the variable X). I take a sample of 4, with ages , , , and . 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 . 3a. 3b. 3c. 3d. 3e. Explain what it means to say that is unbiased? Show that is unbiased. If the variance of is 15, what is the variance of ? If the variance of is 15, what is the variance of the ? Which estimator is more efficient, the sample mean or ? Explain. 4. Suppose , and . 4a. If X and W are uncorrelated, find the mean and variance of . 4b. Find the probability that . Henceforth, suppose that X and W have a correlation coefficient ρ=- .25. 4c. What is the covariance of X and W? 4d. Find the probability that . 4e. Find the probability that

5. Our bank from Question 2 has decided to look more deeply into the matter of customer wait times. In addition to information on the waiting times, the bank has compiled information about the credit scores of the applicants. That is, the bank has 20 observation of the following 2 variables: Observation 1 2 3 4 5 6 7 8 9 10 Wait Time 5 7 22 4 12 9 9 14 3 6 Credit Score 740 730 550 700 650 660 630 600 760 730 Observation 11 12 13 14 15 16 17 18 19 20 Wait Time 5 8 10 17 12 10 9 4 3 13 Credit Score 700 620 600 580 650 670 670 790 750 610 5a. Find the sample mean and variance of the Credit Score variable (you can call this variable Y if you like). 5b. Find the sample covariance and sample correlation coefficient of Wait Times and Credit Scores. 5c. Give a short interpretation of the correlation coefficient for this example. 5d. What story can you tell that would explain the correlation coefficient the bank observes.

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