Question

The heights of female students at a university follows Normal distribution with a mean 66 inches and a standard deviation 3 inches. A researcher randomly selects 36 female students from the university, surveys their heights and calculates a sample mean.

Now suppose that the population standard deviation is unknown. Also, the researcher calculate the sample standard deviation to be 3 inches.

a) What is the probability that the sample mean height is between 65 inches and 67 inches?

b) Instead of 36, suppose the sample size is 64 only for this sub-question. Then what is the probability that the sample mean height is between 65 inches and 67 inches?

Please answer in excel format if possible! And show the function! Thank youuuu

Answer 1

Solution :

Given that,

mean = =  66 inches

standard deviation = = 3 inches

n = 36

= 66

= / n = 3/ 36 = 0.5

P(65 <   < 67 ) = P((65 - 66) /0.5) < ( - ) / < (67 - 66) /0.5 ))

P(65 <   < 67 ) = P( -2 < Z < 2)

P(65 <   < 67 ) = P(Z < 2) - P(Z <-2 ) Using z table,

P(65 <   < 67 ) = 0.9772 - 0.0228

P(65 <   < 67 ) = 0.9544

n = 64

= / n = 3/ 64 = 0.375

P(65 <   < 67 ) = P((65 - 66) /0.375) < ( - ) / < (67 - 66) /0.375))

P(65 <   < 67 ) = P( -2.67 < Z < 2.67)

P(65 <   < 67 ) = P(Z < 2.67) - P(Z <-2.67) Using z table,

P(65 <   < 67 ) = 0.9962 - 0.0038

P(65 <   < 67 ) = 0.9924