Question

A random sample is drawn from a normally distributed population with mean μ = 26 and standard deviation σ = 2. [You may find it useful to reference the z table.] a. Are the sampling distribution of the sample mean with n = 34 and n = 68 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 = 34 will have a normal distribution. No, only the sample mean with n = 68 will have a normal distribution. b. Calculate the probabilities that the sample mean is less than 26.6 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

From the theory of the central limit theorem, we know that if n>30 then we can alway assume that sample means will have a normal distribution.

Thus, both of the sample means will have a normal distribution.

For, 1st sample mean,

Standard error = 2/sqrt(34)

Z = (26.6 - 26) / (2/sqrt(34) ) = 1.749

Now, from the standard normal distribution table,

P( Z < 1.749) = 0.9598

For the second sample mean,

standard error = 2/sqrt(68)

Z = (26.6 - 26) / (2/sqrt(68) ) = 2.47

P(Z < 2.47) = 0.9933