Question

# A random sample is drawn from a population with mean μ = 52 and standard deviation...

A random sample is drawn from a population with mean μ = 52 and standard deviation σ = 4.3.

a. Is the sampling distribution of the sample mean with n = 13 and n = 39 normally distributed?

• Yes, both the sample means will have a normal distribution.

• No, both the sample means will not have a normal distribution.

• No, only the sample mean with n = 13 will have a normal distribution.

• No, only the sample mean with n = 39 will have a normal distribution.

b. Calculate the probability that the sample mean falls between 52 and 54 for n = 39. (Round intermediate calculations to at least 4 decimal places, “z” value to 2 decimal places, and final answer to 4 decimal places.)

a)

Yes, both the sample means will have a normal distribution.

b)

Here, μ = 52, σ = 0.6886, x1 = 52 and x2 = 54. We need to compute P(52<= X <= 54). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z1 = (52 - 52)/0.6886 = 0
z2 = (54 - 52)/0.6886 = 2.9

Therefore, we get
P(52 <= X <= 54) = P((54 - 52)/0.6886) <= z <= (54 - 52)/0.6886)
= P(0 <= z <= 2.9) = P(z <= 2.9) - P(z <= 0)
= 0.9981 - 0.5
= 0.4981