Question

Suppose \(x\) has a distribution with \(\mu=51\) and \(σ=2\).

(a) If random samples of size \(n=16\) are selected, can we say anything about the \(\bar{X}\) distribution of sample means?

Q Yes, the \(\bar{x}\) distribution is nomal with mean \(\mu \bar{x}=61\) and \(\sigma \bar{x}=2\).

0 Yes, the \(\bar{x}\) distribution is normal with mean \(\mu \bar{x}=61\) and \(\sigma \bar{x}=0.1\).

9. No, the sample size is too small.

Ye yes, the \(\bar{x}\) distribution is normal with mean \(\mu \bar{x}=61\) and \(\sigma_{x}=0.5\).

(b) If the original \(x\) distribution is normal, can we say anything about the \(\bar{x}\) distribution o random samples of size 16 ?

Yes, the \(\bar{x}\) distribution is normal with mean \(\mu \bar{x}=61\) and \(a \bar{x}=0.5\).

O Yes, the \(\bar{x}\) distribution is normal with mean \(\mu \bar{x}=51\) and \(\sigma \bar{x}=2\).

o No, the sample size is too small.

0. Yes, the \(\bar{x}\) distribution is normal with mean \(\mu \bar{x}=61\) and \(g \bar{x}=0.1\).

Find \(P(57, \bar{x} \leq 62)\), (Round your answer to four decimal places-)

Answer 1

a) The correct option is D

μx̄ = 61

σx̄ = σ / √n = 2 / sqrt(16) = 0.5

b) The correct option is A

μx̄ = 61

σx̄ = σ / √n = 2 / sqrt(16) = 0.5

c)

Since μ = 61 and σ = 2 we have: 57-61 X-μ 62-61 P(57 < X < 62 ) = P (57-61 <X-μ<62-61 ) = P < くー Since Z = z-μ , 57-61--2 and 62-61 = 0.5 we have: P(57 < X < 62 ) = P (-2<Z<0.5) Step 3: Use the standard normal table to conclude that P(-2<Z<0.5)=0.6687