Given a normal distribution with mu 50 and sigma 8, and given you select a sampl

Question

Given a normal distribution with mu 50 and sigma 8, and given you select a sample of n equals 100, complete parts (a) through (d).

a. What is the probability that Upper X overbar is less than 49? P(Upper X overbarless than49)equals 0.1056 (Type an integer or decimal rounded to four decimal places as needed.)

b. What is the probability that Upper X overbar is between 49 and 49.5? P(49less thanUpper X overbarless than49.5)equals 0.1603 (Type an integer or decimal rounded to four decimal places as needed.)

c. What is the probability that Upper X overbar is above 50.3?

Explanation / Answer

Here mean = 50

standard deviation = 8

standard error of sample mean se0 = /sqrt(n) = 8/sqrt(100) = 8/10 = 0.8

Here

Here n = 100

(a) Pr(x < 49) = NORMAL (x < 49 ; 50 ; 0.8)

Z = (49 - 50)/ 0.8 = -1.25

Pr(x < 49) = NORMAL (x < 49 ; 50 ; 0.8) = Pr(Z < -1.25) = 0.10565

(b)Pr(49 < X < 49.5) = Pr(x < 49.5 ; 50 ; 0.8)- (Pr(x < 49 ; 50 ; 0.8) = (Z2) - (Z1)

where is the standard normal cumulative distribution

Z2 = (49.5 - 50)/0.8 = - 0.625

Z1 = (49 -50)/ 0.8 = -1.25

Pr(49 < X < 49.5) = Pr(x < 49.5 ; 50 ; 0.8)- (Pr(x < 49 ; 50 ; 0.8) = (Z2) - (Z1) = (-1.25) - (-0.625) = 0.2660 -0.10565 = 0.1603

(c) Pr(x > 50.3) = 1 - Pr(x < 50.3 ; 50 ; 0.8)

Z = (50.3 - 50)/ 0.8 = 0.375

Pr(x > 50.3) = 1 - Pr(x < 50.3 ; 50 ; 0.8) = 1 - Pr(Z < <0.375) = 1- 0.6462 = 0.3538

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