A normally distributed random variable X possesses a mean of

A normally distributed random variable X possesses a mean of ? = 20 and a
standard deviation of ? = 5. A random sample of n = 31 observations is to be selected.
Let X be the sample average.
a) Describe the sampling distribution of X (i.e. describe the distribution of X and give ?x, ?x)
b) Find the z-score of x = 22
c) Find P(X ? 22) =
d) Find P(20 ? X ? 22))

Please show detailed steps

Solution

a) Describe the sampling distribution of X (i.e. describe the distribution of X and give ?x, ?x)

X follows Normal distriubtion with ?x =20 and ?x =s/vn =5/sqrt(31) =0.8980265

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b) Find the z-score of x = 22

Z-score= (X-mean)/s

=(22-20)/0.8980265

=2.23

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c) Find P(X ? 22) = P((X-mean)/s >(22-20)/0.8980265)

=P(Z>2.23) = 0.0129 (from standard normal table)

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d) Find P(20 ? X ? 22)) = P((20-20)/0.8980265<Z<(22-20)/0.8980265)

=P(0<Z<2.23) = 0.4871 (from standard normal table)