Discussion Question 1: Why use t-distribution? In Section 6.2-D, you learn that because we don't know the standard deviation 0' for the population of interest, we have to use the standard deviation 3 from the sample to estimate the standard error of the sample means. But unfortunately, if we use the standard error based on s/'ri to standard the sample mean, the distribution is no longer a standard normal distribution but instead it will be a t-distribution with degree of freedom in - 1. In this Discussion Question, we will see why this is necessary as we work through a condence interval example. First let us recall that when we calculate the condence interval of level of condence as, the formula is _ s 3 2|: 5. E, where E is the sample mean, 3 is the sample standard deviation, in is the sample size, and t' is the endpoint chosen from a t-distribution with degree of freedom in. 1 to give the desired level of condence a. Let us see what will go wrong if we use the formula 3 E :I: z\" x/ ' where z' is the endpoint with the percentile a in the center from a standard normal distribution. We will this applet to generate condence intervals for simulation. (1) Go to the applet, under method, choose means, normal and z with 8. Choose your favorite ,u (between 30 to 50) and a (between 5 to 10), set in = 5, and intervals 500. Use condence level 95%. Hit \"Sample\". What is the percentage of the intervals containing at? (Include a screenshot) (2) Change the intervals to 9500 and hit \"Sample\" again so you will get a total of 10000 condence intervals. Include a screenshot. What problem with the \"z with s\" condence interval procedure does this simulation analysis reveal? (3) In order to solve this problem, do we need the intervals to get a bit narrower or wider? (4) Which of the four terms in the formula E, z', s, or n can we alter to produce an interval (either narrower or wider) that will help us x the problem? (5) Do we want to use a larger or smaller multiplier than 2'? (6) Now let us go back to the applet for simulation. Keep everything the same but change \"2 with s\" to t. Resample 10000 condence intervals. What is the percentage of the intervals containing ,u? (Include a screenshot) Is that close to 95%? 2 ACTIVITIES AND ASSIGNMENTS FOR WEEK 7 (7) What happens with larger sample size? Use \"2 with 3\" method with n = 20, n = 40 and then in = 100. What pattern do you see for the coverage success rate? How do them compare with the coverage success rate with the t method? (8) (Optional) Does it method work equally well for other condence interval? (9) (Optional) By the way, why do the widths of these intervals vary from sample to sample