Question

A random sample is drawn from a normally distributed population with mean μ = 27 and standard deviation σ = 2.1. [You may find it useful to reference the z table.]

a. Are the sampling distribution of the sample mean with n = 35 and n = 70 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 = 35 will have a normal distribution.

No, only the sample mean with n = 70 will have a normal distribution.

b. Calculate the probabilities that the sample mean is less than 27.6 for both sample sizes. (Round intermediate calculations to at least 4 decimal places, “z” value to 2 decimal places, and final answer to 4 decimal places.)

Answer 1

Solution :

Given that ,

mean = = 27

standard deviation = = 2.1

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

(b)

n = 35

= 27 and

= / n = 2.1 / 35

P( < 27.6) = P(( - ) / < (27.6 - 27) / 2.135 ) = P(z < 1.69)

Using standard normal table,

P( < 27.6) = 0.9545

Probability = 0.9545

(b)

n = 70

= 27 and

= / n = 2.1 / 70

P( < 27.6) = P(( - ) / < (27.6 - 27) / 2.1/70) = P(z < 2.39)

Using standard normal table,

P( < 27.6) = 0.9916

Probability = 0.9916