1. In a survey of 700 likely voters, 373 responded that they would vote for the incumbent and 327 responded that they would vote for the challenger. Let ρ denote the fraction of all likely voters who preferred the incumbent at the time of the survey, and let ρ be the fraction of survey respondents who preferred the incumbent.
a. Use the survey results to estimate ρ.
b. Use the estimator of the variance of ??, ??(1 - ??)/??, to calculate the standard error of your estimator.
c. What is the p-value for the test H0: ?? = 0.5 vs. H1: ?? ≠ 0.5?
d. What is the p-value for the test H0: ?? = 0.5 vs. H1: ?? > 0.5?
e. Why do the results from (c) and (d) differ?
f. Did the survey contain statistically significant evidence that the incumbent was ahead of the incumbent at the time of the survey? Explain?
g. Give a 95% confidence interval for p.
h. Give a 95% confidence interval for the difference between the fraction who preferred the incumbent and the fraction who preferred the challenger at the time of the survey.
2. A new version of the SAT is given to 1600 randomly selected high school seniors. The sample mean test score is 1175 and the sample standard deviation is 160.
a. Construct a 95% confidence interval for the population mean test score for high school seniors.
b. Test the hypothesis that the population mean test score for high school seniors on the new version is equal to the known population average mean SAT score of 1165 on the old version.
3. The daily openings and closings of the NASDAQ and the Dow Jones Industrial Average (DJIA).
a. Construct a variable ρCHANGE = CLOSING - OPENING / OPENING for each series.
b. Test whether the average ρCHANGE for each series is significantly different from zero.
c. Compute the correlation and covariance between variables for the DJIA and NASDAQ.
d. Comment on your findings in b & c (one short paragraph).