Q1. A random sample of 10 economists produced the following forecasts for percentage growth in real domestic product in the next year:
2.2 2.8 3.0 2.5 2.4 2.6 2.5 2.4 2.7 2.6
Use unbiased estimation procedures to find point estimates for the following:
a. The population mean?
b. The population variance?
c. The variance of the sample mean
d. The population proportion of economists predicting growth of at least 2.5% in real gross domestic product
e. The variance of the sample proportion of economists predicting growth of at least 2.5% in real gross domestic product
Q2. A college admissions officer for an MBA program has determined that historically applicants have undergraduate grade point averages that are normally distributed with standard deviation 0.45. From a random sample of 25 applications from the current year, the sample mean grade point average is 2.90.
a. Find a 95% confidence interval for the population mean?
b. Without doing calculations, explain whether a 99% confidence interval for the population mean would be wider than, narrower than, or the same width as found in part
c. Suppose that population standard deviation is 0.56 (instead of 0.45), without doing calculations explain whether a 95% confidence interval for the population mean would be wider than, narrower than, or the same width as found in part (a).
d. Suppose that sample mean that is given in the problem is calculated from a sample of size 40 (instead of 25), without doing calculations explain whether a 95% confidence interval for the population mean would be wider than, narrower than, or the same width as found in part (a).
e. Based on these sample results, a statistician computes for the population mean a confidence interval extending from 2.81 to 2.99. Find the confidence level associated with this interval.
f. Suppose that we know that grade point averages that are normally distributed, but we don't have the population standard deviation. From a random sample of 25 applications from the current year, the sample mean grade point average is 2.90, and sample standard deviation is found to be 0.40. Find a 95% confidence interval for the population mean?
Q3. These two questions are about population proportions.
a. In baseball data set, it was found that out of the 59 baseball players, 15 were lefthanded. Is this odd, since the proportion of left-handed males in America is about 11%?
Answer this question by constructing a 95% approximate confidence interval.
b. Let Y be a Binomial random variable with n = 300, and probability of success p. If the observed value of Y is y = 75, find an approximate 90% confidence interval for p.
Q4. Let ?? be the observed mean of a random sample of size n from a normal distribution having mean μ and variance σ2.
a. Find n so that ?? - σ/4 to ?? + σ/4 is a 95% confidence interval for μ.
b. Now, let us generalize the result in the previous part: Find n so that ?? - kσ to ?? + kσ is a 95% confidence interval for μ (Note that indeed this question is more general because if we take k = 1/4, it reduces to part a).
Q5. A manufacturer is concerned about the variability of the levels of impurity contained in consignments of raw material from a supplier. A random sample of 15 consignments showed a standard deviation of 2.36 in the concentration of impurity levels. Assume normality.
a. Find a 95% confidence interval for the population variance.
b. Suppose that when you calculate the 95% confidence interval in part (a), you mistakenly left an area 0.03 in the lower tail, and 0.02 in the upper tail, instead of distributing α = 0.05 evenly between two tails. How the width of the confidence interval would be effected by that mistake. State without doing any calculation whether it would be wider, narrower than or the same that found in part (a)? Briefly explain your answer.
c. Would a 99% confidence interval for this variance be wider or narrower than that found in part (a)?