Question

A simple random sample of size n=37 is obtained from a population with μ=65 and σ=15.

​(a) What must be true regarding the distribution of the population in order to use the normal model to compute probabilities involving the sample​ mean? Assuming that this condition is​ true, describe the sampling distribution of overbarx.

​(b) Assuming the normal model can be​ used, determine ​P(overbar x < 69.4).

​(c) Assuming the normal model can be​ used, determine ​P(overbar x ≥ 66.7​).

Answer 1

a)

Sample size must be large ( >= 30) .

Since n = 37 >= 30 , We use normal model to compute probabilities.  

Sampling distribution of -

Mean = = 65

Standard deviation = / sqrt(n)

= 15 / sqrt(37)

= 2.46598

b)

Using central limit theorem,

P( < x) = P( Z < x - / )

so,

P( < 69.4) = P( Z < 69.4 - 65 / 2.46598)

= P( Z < 1.7843)

= 0.9628

c)

P( >= 66.8 ) = P( Z >= 66.8 - 65 / 2.46598)

= P( Z >= 0.7299)

= 0.2327