Question

A random sample is drawn from a normally distributed population with mean μ = 18 and standard deviation σ = 2.3.

a. Are the sampling distributions of the sample mean with n = 26 and n = 52 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 = 26 will have a normal distribution.

• No, only the sample mean with n = 52 will have a normal distribution.

b. Calculate the probabilities that the sample mean is less than 18.6 for both sample sizes. (Round answers to 4 decimal places.)

n Probability

26 ( _________)

52 ( _________)

Answer 1

Solution :

Given that ,

mean = = 18

standard deviation = = 23

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

b) n = 26

   =   = 18

= / n = 23 / 26= 4.5107

P( < 18.6) = P(( - ) / < (18.6 - 18) /4.5107 )

= P(z < 0.13 )

= 0.5517

probability = 0.5517

n =52

=   = 18

= / n = 23 / 52= 3.1895.

P( < 18.6 ) = P(( - ) / < (18.6-18) /3.1895 )

= P(z < 0.19 )

= 0.5753

probability = 0.5753