Question

A random sample is drawn from a population with mean μ = 68 and standard deviation σ = 5.7. [You may find it useful to reference the z table.]

a. Is the sampling distribution of the sample mean with n = 16 and n = 41 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 = 16 will have a normal distribution.

• No, only the sample mean with n = 41 will have a normal distribution.

b. Calculate the probability that the sample mean falls between 68 and 71 for n = 41. (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,

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

mean = = 68

standard deviation = = 5.7

n = 16

=   = 68

= / n = 5.7 / 16 = 1.425

n = 41

=   = 68

= / n = 5.7 / 41 = 0.8902

b) P( 68 <   < 71)  

= P[(68 -68) /0.8902< ( - ) / < (71 -68 ) /0.8902 )]

= P( 0< Z < 3.37 )

= P(Z < 3.37 ) - P(Z < 0)

= 0.9996 - 0.5 = 0.4996