Full practice problem set due Sunday, October 30th. 1) Variable x is YU statistics students' ratings of how interested they are in their statistics course, on a 1 to 5 scale where higher ratings mean more interested. You wish to estimate the population mean and provide confidence intervals to express error and uncertainty. You draw a sample of size 80 and find a sample mean of 3.05 and sample-based estimate of population SD of 1.04. Compute 95% and 99% confidence intervals. Explain what each is in words. Note that in a T distribution with df=79, the cutoffs for the 2.5% tails are +/- 1.99 and the cutoffs for the 0.5% tails are +/- 2.64. 2) Study the following and then write down answers. What is a confidence interval? What happens to a confidence interval as the confidence level is increased (e.g., from 95% to 99%), and why? What happens to a confidence interval as the sample size is increased, and why? 3) Variable X is self-rated self-esteem on a 1 to 7 scale. You want to test the hypothesis that women have higher self-esteem than men, but you are also willing to entertain the opposite hypothesis (that men have higher self-esteem than women). You assume equal variability in self-esteem across the populations of men and women. You draw a random sample of size 157 women and find a sample mean of 5.2 and samplebased estimate of population standard deviation of 1.6. You draw a sample of size 153 men and find a sample mean of 5.5 and a sample based estimate of population standard deviation of 1.3. Carry out an independent samples t-test using a two-tailed .05 significance level: figure out the mean of the sampling distribution (DDM under the null), the estimated standard error of the sampling distribution, the tstatistic for the difference between sample means; and then make a decision about whether to uphold or reject the null hypothesis, and, if the latter, indicate what alternative hypothesis you have obtained support for. Note that the critical t-values associated with the 2.5% tails in a t-distribution with df = 308 are +/- 1.97