Question

Suppose a simple random sample of size n=81 is obtained from a population with mu=89 and sigma=27.

​(a) Describe the sampling distribution of x overbar.

​(b) What is P(x overbar > 92.9)?

​(c) What is P( x overbar < or equals 82.7)​? ​

(d) What is P(86 < x overbar < 96.35)​?

Answer 1

a) sampling distribution of x overbar follows normal distribution from central limit thoerum as sample size n is greater than 30

here mean of sampling distribution =89

and standard deviation of sampling distribution of means =27/sqrt(81)=3

b)

for normal distribution z score =(X-μ)/σ

probability =

P(X>92.9)

=

P(Z>1.3)=

1-P(Z<1.3)=

1-0.9032=

0.0968

c)

probability =

P(X<82.7)

=

P(Z<-2.1)=

0.0179

d)

probability =

P(86<X<96.35)

=

P(-0.3<Z<0.74)=

0.7704-0.3821=

0.3883

( please try 0.3892 if this comes wrong)