Question

A simple random sample of size n=64 is obtained from a population with μ=77 σ=24.

​(a) Describe the sampling distribution of x

​(b) What is P (x > 80.75)​?

​(c) What is P (x ≤ 70.25)​?

​(d) What is P (74.6<x<83.3​)?

Answer 1

Solution :

a) According to central limit theorem, if we have a population with mean μ and standard deviation σ and if we take a sample of sufficiently large size (n > 30) from this population, then sampling distribution of sample mean x̅ follows approximately normal distribution with mean μ and standard deviation σ/√n.

i.e. x̅ ~ N(μ, σ²/n)

We have, μ = 77, σ = 24 and sample size (n) = 64

Since, sample size n > 30, therefore sampling distribution of x̅ will be approximately normal with mean  μ = 77 and standard deviation σ/√n = 24/√64 = 3.

b) We have to find P(x̅ > 80.75).

We know that if x̅ ~ N(μ, σ²/n) then,

Using "pnorm" function of R we get, P(Z > 1.25) = 0.1056

c) We have to find P(x̅ < 70.25).

We know that if x̅ ~ N(μ, σ²/n) then,

Using "pnorm" function of R we get, P(Z < -2.25) = 0.0122

d) We have to find P(74.6 < x̅ < 83.3).

P(74.6 < x̅ < 83.3) = P(x̅ < 83.3) - P(x̅ ≤ 74.6)

We know that if x̅ ~ N(μ, σ²/n) then,

Using "pnorm" function of R we get,

P(Z < 2.1) = 0.9821 and P(Z ≤ -0.8) = 0.2118

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