Point Estimation: 1. The MSE of an estimator looks a lot like a formula for the variance, although it is not. Explain why the MSE is not the variance - don't simply show that the two formulas differ. Explain what the MSE characterizes, and how this differs from the variance of the estimator in question. 2. Write a couple sentences about how consistency and bias are similar, and how they are different. 3. Write a couple sentences about how consistency and MSE are similar, and how they are different. 4. Write a couple sentences about how bias and MSE are similar, and how they are different. 5. What is a point estimate? How is it different from a point estimator and from a population parameter? 6. What is a point estimator? How is it different from a point estimate and from a population parameter? 7. What is a population parameter? How is it different from a point estimate and from a point estimator? THE NORMAL DISTRIBUTION AND THE CLT 8. There is a stock that, in the past has produced, on average annual return of 3.7%, with a standard deviation of 2.9%. These returns are, you assume, normally distributed. You will invest in this stock for one year. What is the probability that you will lose money (i.e. experience a negative return)? 9. There is a stock that, in the past has produced, on average annual return of 3.7%, with a standard deviation of 2.9%. These returns are, you assume, normally distributed. You will invest $350 in this stock for one year. What is the probability that you will receive a return worth at least $75? 10. Kimmy Schmidt has an investment! Its monthly fluctuation is predicted to be normally distributed! The mean should be $2,000, and it should have about $127 as its standard deviation! Kimmy wants to know what values of her investment will be in the top 5% of this investment! Help her find it! Show your work! BASIC CONFIDENCE INTERVALS 11. When estimating the mean of a probability distribution of some population of interest (variance known), a confidence interval is typically constructed using a normal distribution, even if the underlying population distribution is not normal. Why is this done? 12. When investigating a probability distribution of some population of interest, we frequently explore its parameters by adverting to a different probability distribution. What is this other distribution, and why do we use it? 13. "My 95% confidence interval is (20, 30), so there is a 95% chance that the true mean is somewhere between 20 and 30." Explain what is wrong with this claim. 14. While investigating the probability distribution of some population of interest, you notice that the area of the distribution corresponding to your 95% confidence interval is much less than 95% of the distribution of the population of interest. Explain this. 15. Suppose you take a sample of 25 individuals (from a normally distributed population, when the variance is known) and your confidence interval for the mean at the confidence level .93 is (23, 41). Is the area above the curve from 23 to 41 of the distribution from which you drew these 25 individuals greater than, less than, or equal to .93? Explain your answer (you don't have to calculate anything, just say why). 16. Suppose you take a sample of n individuals (from a normally distributed population, when the variance is known) and your confidence interval for the mean at the confidence level c is (L, U). Is the area above the curve from L to U of the distribution from which you drew these n individuals greater than, less than, or equal to c? Explain your answer (you don't have to calculate anything, just say why). 17. Sometimes a standard deviation is presented as a margin of error. If the estimated standard deviation is 14 from n=20 observations, what is corresponding confidence level c