Question

A random sample is drawn from a normally distributed population with mean μ = 19 and standard deviation σ = 1.8. [You may find it useful to reference the z table.]

a. Are the sampling distribution of the sample mean with n = 27 and n = 54 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 = 27 will have a normal distribution.

No, only the sample mean with n = 54 will have a normal distribution.

b. Calculate the probabilities that the sample mean is less than 19.9 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 = = 19

standard deviation = = 1.8

a ) n = 27

= 19

₌ / n = 1.8 27 = 0.3464

P ( < 19 )

P ( - /)  < (19.9 - 19 / 0.3464)

P( z  < 0.9 / 0.3464 )

P ( z < 2.60 )   

Using z table

= 0.9953

Probability = 0.9953

b ) n = 54

= 19

₌ / n = 1.8 54 = 2.4495

P ( < 19 )

P ( - /)  < (19.9 - 19 / 2.4495 )

P( z  < 0 .9/ 2.4495 )

P ( z < 0.37 )   

Using z table

= 0.6443

Probability = 0.6443