1. (4 marks) Simplify the following expressions.
a. Find
b. Find , where
2. (4 marks) The population mean and variance of the random variable X are and , respectively. Prove that, for a sufficiently large sample size n and the sample mean
a.
b.
3. (6 marks) Let X be a random variable with a probability density function (PDF) given by
a. Solve for c.
b. Calculate .
c. Calculate .
4. (2 marks) A random sample of n voters is selected to estimate the proportion of voters who plan to vote for Candidate A in an election, . How large does n need to be so that we can obtain a 90 percent confidence interval with a margin of error of .
5. (10 marks) Let X and Y be two continuous variables with a joint PDF given by
a. Calculate the marginal PDF of X.
b. Calculate the marginal PDF of Y.
c. Briefly explain if X and Y are independent.
d. Calculate .
e. Calculate .
6. (10 marks) The data file assignment.xlsx contains the grades for 33 students on assignment 1 and assignment 2 . Let be normally and independently distributed with a mean and variance of and , respectively.
a. Calculate .
b. Calculate .
c. State the appropriate null and alternative hypotheses to test whether the performance on the assignments does not differ.
d. Briefly explain whether a t- or Z-test is more appropriate.
e. Perform the appropriate test at the 5-percent level of significance and briefly explain your conclusion.
7. (10 marks) The data file fultonfish.dat shows the daily sales of fish (in pounds) for a period of time.
a. Test against at the 5-percent level of significance. Briefly explain your result using a diagram showing the estimated value of the test statistic and the critical value.
b. Calculate the p-value of the test statistic and briefly explain how it can be used to perform the hypothesis test. Show the p-value in the diagram.
Let total weekly sales be given by , where represents sales on weekday i.
c. Calculate.
d. Calculate.
e. Derive the probability distribution of and calculate a 95-percent confidence interval estimate for .
8. (6 marks) A police chief claims that the standard deviation in the length of response times is less than 3.7 minutes. A random sample of 9 response times from a normal population has a standard deviation of 3.0 minutes.
a. State the appropriate null and alternative hypotheses.
b. Briefly explain whether a t- or -statistic is more appropriate.
c. Perform the appropriate test for and briefly explain your conclusion.
10 years ago
Purchase the answer to view it
- simplify_the_following_expressions_a.docx