Answer to Question #196535 in Statistics and Probability for samuel

Question

b.     Given the population 2,1,3,3,5,4,7 find the probability that a random sample of size 25, selected with replacement, will yield a sample mean greater than 3.7, but less than 4.3.

Answer 1

Given population is- (2,1,3,3,5,7,4)

population mean $\mu=\dfrac{2+1+3+3+5+4+7}{7}=\dfrac{25}{7}=3.55$

Sample standard deviation-

$s=\sqrt{\dfrac{(x-\mu)^2}{n}}$

$=\sqrt{\dfrac{(-1.55)^2+(-2.55)^2+(-0.55)^2+(-0.55)^2+(1.45)^2+(0.45)^2+(3.45)^2}{7-1}}$

$=\sqrt{\dfrac{23.7175}{7}}=\sqrt{3.3882}=1.9892$

Required Probability

$P(3.7<x<4.3)=P(\dfrac{3.7-3.77}{1.9892}<z<\dfrac{4.3-3.77}{1.9892})$

$=P(-0.035<z<0.266)\\[9pt]=0.11884$