Suppose that and are independent and many samples of size n =are drawn and regressions estimated. suppose that you test the null hypothesis that iszero at the 5% level and construct a 95% confidence interval for .yixi411β1β1in what fraction of the samples would the null hypothesis that is zero at the 5% level be rejected? β1a. the null hypothesis would be rejected in 5% of the samples. b.the null hypothesis would be rejected in 90% of the samples. c.the null hypothesis would be rejected in 10% of the samples. d.the null hypothesis would be rejected in 95% of the samples. in what fraction of the samples would the value = 0 be included in the 95% confidence interval for ? β1β1a.5% of the confidence intervals would contain the value = 0.β1b.95% of the confidence intervals would contain the value = 0.β1c.10% of the confidence intervals would contain the value = 0.β1d.90% of the confidence intervals would contain the value = 0.

Aprisoner is trapped in a cell containing three doors. the first door leads to a tunnel that returns him to his cell after two days of travel. the second leads to a tunnel that returns him to his cell after three days of travel. the third door leads immediately to freedom. (a) assuming that the prisoner will always select doors 1, 2 and 3 with probabili- ties 0.5,0.3,0.2 (respectively), what is the expected number of days until he reaches freedom? (b) assuming that the prisoner is always equally likely to choose among those doors that he has not used, what is the expected number of days until he reaches freedom? (in this version, if the prisoner initially tries door 1, for example, then when he returns to the cell, he will now select only from doors 2 and 3.) (c) for parts (a) and (b), find the variance of the number of days until the prisoner reaches freedom. hint for part (b): define ni to be the number of additional days the prisoner spends after initially choosing door i and returning to his cell.