Here we continue the discussion of Problem 3.10.14 by determining the endpoints of a confidence interval for an unknown median M. In Problem 3.10.14 we calculated the confidence coefficient for a specific confidence interval. That's the reverse of what one usually does. The true power of the confidence interval technique is that you get to choose whatever confidence coefficient provides adequate certainty for your purposes. Then you determine the confidence interval. Afterward, you have an “interval estimate” for your parameter with a predetermined degree of certainty.Suppose you've taken a sample of size n from a probability distribution (or, as a special case, a numeric population) and arranged it in ascending order. So the sample looks like:  These X(i) 's are called the sample order statistics. Note that each one of these is a number calculated from the sample and so, indeed, is a “statistic.” Thus, XlI) is the “first order statistic,” X(2) the “second order statistic,” and so on. In general, X(k) is the kth-order statistic. We'll determine the endpoints of our confidence interval for the median in terms of the order statistics of a sample.First, suppose we want a 90% confidence interval for the median M. If the endpoints of the interval are to be the order statistics X(h) and X(k), then just as in Problem 3.10 .14(e)We assume the sample is drawn from a continuous distribution (or very large population) so there's a zero chance for an observation in the sample to equal M. Thus, we can ignore equal signs in the square brackets. (a) In Problem 3.10.14(a), show that the sample median is x(12)' In that example, what's the value of X(6)?(b) Express the square bracket in part (a) in terms ofY = # observations in the sample which are less than M .(c) Let Zo = 1.645. Show that if n ≥ 10, the endpoints of the 90% confidence interval for Mare X(h) and X(k), where hand k are determined by(d) Give a 90% confidence interval for the median fill of cups from the drink machine in the employee lounge. Here is the fill in ounces for ten cups: 6.7, 6.4, 6.3, 6.4, 6.3, 6.5, 6.4, 6.2, 6.8, 6.5 .(e) What assumption are you making in the analysis of part (d)? (f) Explain the meaning of the 90% confidence coefficientfrom part (d).(g) Determine a 95% confidence interval for the median fill in part (d).(h) Show how to calculate the 99% confidence interval for the median weight of U.S. pennies given the data in Problem 3.10.14(g).(i) How do you interpret the confidence interval of part (h)?Problem 3.10.14:In Chapter 5, we will introduce the idea of a “confidence interval” for an unknown parameter. It's a range of possible values for the parameter together with the probability-the “confidence coeffi-cient”-that the parameter actually falls within that range. Here, we'll look at a special case, a confidence interval for an unknown median.Suppose we've taken a sample of size n from some probability distribution (or, as a special case, from a numeric population). Call the median M and let min and max refer, respectively, to the smallest and largest observations in the sample. Then the interval (min, max) can be thought of as a range of possible values, a confidence interval, for M. In this problem, we'll determine the confidence coefficient for this interval. That is, we'll determine the probability that this interval contains the median. For simplicity, assume a random observation has a zero chance to actually equal M.(a) Identify M, min, max, and the confidence coefficient. Here's the sample:(b) Let Y be the number of observations in the sample which are less than M . What's the model for Y?(c) P(M (d) P(M > min) =?(e) What's the confidence coefficient for the interval (min, max) as a confidence interval for M? (f) What's the confidence coefficient for part (a)? (g) What's the median weight of U.S. pennies? Here are the weights W of 100 newly minted pennies, reported to the nearest 0.02 gram (taken from W.]. Youden's National Bureau of Standards Publication 672, Experimentation and Measurement):(h) In Chapter 4, we'll see that (3.11, 3.13) is a 99% confidence interval for the median weight, M, of U.S. pennies. How do you interpret this confidence interval? To what does the confidence coefficient refer? (i) What's wrong with interpreting the confidence interval in part (h) by saying “ninty-nine percent of the time, M is between 3.11 and 3.13, the rest of the time it's not”? (j) In part (g), what's the relationship between the sample median andM?