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Show that the standard score of the sample mean X, is equal to Y. Show that the mean and variance of the random variable y are 0 and 1, respectively. iii) iii) Show using the moment generating function technique that Y is a standard normal random variable. i) ii)

Show that the standard score of the sample mean X, is equal to Y. Show that the mean and variance of the random variable y are 0 and 1, respectively. iii) iii) Show using the moment generating function technique that Y is a standard normal random variable. i) ii)

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.8: Probability
Problem 32E
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PAGE LAYOUT -AA Font A REFERENCES MAILINGS Aa - aly - A i) 11) ▾ EE i) E-25 E* T AaBbCc AaBbCc AaBbC AaBbCcL AaB AaBbcct AaBb T Normal 1 No Spac... Heading 1 Heading 2 Subtitle Subtle REVIEW Paragraph √n VIEW a) Let X1, X2, X3, Xn be a random sample of size n from population X. Suppose that X~N(0, 1) and y = -√n. Show that the standard score of the sample mean x, is equal to Y. Show that the mean and variance of the random variable y are 0 and 1, respectively. III) iii) Show using the moment generating function technique that Y is a standard normal random variable. Styles Title iv) What is the probability that Y² is between 0.02 and 5.02? b) Let X1, X2, X and Y₁, Y₂,..., Ym be random samples from populations with moment generating functions Mx:(t) = e³t-¹2 and My(t) (2) respectively Find the sampling distribution of the statistic W = X₁ + 2X2 X3 + X+ + Xs. ii) What is the value of the sample size , if PI (X-X)²> 68.3392] = 0.025? iii) What is the value of the sample size if P(Y-ur210) < 0.04? 23
Transcribed Image Text:PAGE LAYOUT -AA Font A REFERENCES MAILINGS Aa - aly - A i) 11) ▾ EE i) E-25 E* T AaBbCc AaBbCc AaBbC AaBbCcL AaB AaBbcct AaBb T Normal 1 No Spac... Heading 1 Heading 2 Subtitle Subtle REVIEW Paragraph √n VIEW a) Let X1, X2, X3, Xn be a random sample of size n from population X. Suppose that X~N(0, 1) and y = -√n. Show that the standard score of the sample mean x, is equal to Y. Show that the mean and variance of the random variable y are 0 and 1, respectively. III) iii) Show using the moment generating function technique that Y is a standard normal random variable. Styles Title iv) What is the probability that Y² is between 0.02 and 5.02? b) Let X1, X2, X and Y₁, Y₂,..., Ym be random samples from populations with moment generating functions Mx:(t) = e³t-¹2 and My(t) (2) respectively Find the sampling distribution of the statistic W = X₁ + 2X2 X3 + X+ + Xs. ii) What is the value of the sample size , if PI (X-X)²> 68.3392] = 0.025? iii) What is the value of the sample size if P(Y-ur210) < 0.04? 23
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