When Y,.Y,.Y, is a random sample from N(u,o), we learned in class that thestatistic T =-Y-uhas t distribution with n-1 degrees of freedom, where Y and S aren-1the sample mean and sample standard deviation, respectively. Write out the form of Tand show algebraically that T can be written as the ratio of two independent xrandomvariables, each divided by their respective degrees of freedom. What is, therefore, thedistribution of T ?,

From the given information, Y1, Y2...Yn is a random sample from Nμ,σ2 T=y¯-μSn-1~tn-1

Answered: When Y,,Y,..Y, is a random sample from…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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If the RVSX1, X2,..., X, all have n .2 the same variance s and he correlation oxil ficient between each pair (X;, X;), i # jof them is p, show that the correlation coefficient 2n between -1 X; and Li=n+1 X; is %3D1 пр 1+(n-1)p
An urn contains 7 black and 18 white balls. A ball is randomly drawn from the urn. Let X = 1 if a white ball is drawn and let X = -1 if a black ball is drawn. Find the following a) mean of X b) variance of X Please provide your working and round your answers to 2 decimal places
Let x¡, denote a series of n numbers {x;: i=1,2,..,n}: Assume that x; is drawn from a distribution that is N(ux, ož). Show that the sample mean x = E, xị has a variance of ož/n carefully stating any required assumptions i. 1 at each step. ii. Show that the sample mean is an unbiased estimator of ux.
7. Let (z,.,) be a random sample from a normal distribution with mean ja and a variance at. Alo let S.. = , - z)* be the sum of squares of the randou sample. State without proof the sampling distribution of the following; (a) t(z) - ạ + b i-, 1,. n, as e positive eal constants. (- ) (b) t(z) - (E) (x) = (e) ti2) - (g) (r) = (e) t(2) = (d) t(a) Er, - E > ( - P (h) 시)- - -1 (-) (e) t(z)
The weights of beagles (dogs) have approximately a normal distribution with u = 17.1 pounds and a standard deviation of o = 1.8 pounds. Suppose that we randomly select n = 64 beagles and measure their weights in pounds. Let X be the random variable representing the mean weight of the selected beagles. Let Xtot be the random variable representing the total weight of the selected beagles. a) About what proportion of beagles have weights between 17 and 18 pounds? b) About what proportion of beagles have weights greater 20 pounds? c) About how many of the 64 sampled beagles have weights less than 16.5 pounds? (nearest integer) d) About how many of the sampled beagles have weights between 17 and 18 pounds? (nearest integer) e) What is the standard deviation of the distribution of X? f) What is the standard deviation of the distribution of Xtot? g) What is the probability that 17 1100? f) Copy your R script for the above into the text box here.
6) Suppose the m.g.f. of the r.v. X is given below: MCA) = (uM/ R0) +(45 |no) exp (t) A(20/1201eapreb a) Find the mean and variance of X b) Find P(X>1)
5. The distribution of male weight in the United States has a standard deviation of roughly o = 25 lbs. The average weight u varies from state to state. Suppose you randomly select 100 males in Louisiana and ask their weights, and you compute the average weight Xn of the n = 100 males. a) Assuming that the standard deviation of weight in Louisiana is 25 lbs, what does the Central Limit Theorem tell you the (approximate) distribution of X will be?
Let XXN(u. 16). How much should the sample size he for testing HeiH = 24 rs. He = 25 ata = 0.025 and ß = 0.409 (a) 77 (b) 48 (c) 35 (d) 28 (e) None
The National Health and Nutrition Examination Survey reported in the recent year, the mean cholesterol level for U.S. adults was 202 with standard deviation of 41 (unit: mg/dl) A simple random sample of 110 adults are chosen. Let x¯ be the sample mean cholesterol with a) The probability that the sample mean cholesterol level of the sample of 110 is greater than 210 is P(x¯>210) = ["", "", "", ""] b) What is the probability that the sample mean is between 190 and 200: P(190<x¯<200) = ["", "", "", ""]
d,e,f e) This result would be (usual, unusual) so the sample likely came from a population whose mean gestation period is(less, more, equal) to 266 days
The average wait time to get seated at a popular restaurant in the city on a Friday night is 12 minutes. Is the mean wait time greater for men who wear a tie? Wait times for 13 randomly selected men who were wearing a tie are shown below. Assume that the distribution of the population is normal. 13, 13, 11, 13, 13, 13, 13, 10, 12, 13, 13, 10, 13 What can be concluded at the the αα = 0.05 level of significance level of significance? The null and alternative hypotheses would be: H0 = (p,u) (<,>,=,not equal) to _____ H1 = (p,u) (<,>,=,not equal) to _____ The test statistic (t,z) = ___ The p-value = _____ Thus, the final conclusion is that ... The data suggest that the population mean wait time for men who wear a tie is not significantly more than 12 at αα = 0.05, so there is statistically insignificant evidence to conclude that the population mean wait time for men who wear a tie is more than 12. The data suggest the population mean is not significantly more than 12…

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