Question

A population has a mean of 128 and a standard deviation of 32. Suppose a sample of size 64 is selected and

x

is used to estimate μ. (Round your answers to four decimal places.)

(a)

What is the probability that the sample mean will be within ±5 of the population mean?

(b)

What is the probability that the sample mean will be within ±10 of the population mean?

Answer 1

Solution :

Given that ,

a) mean = = 128

standard deviation = = 32

n = 64

=   = 128

= / n = 32/ 64 = 4

a ) within 5 = 128  ± 5 = 123, 133

P(123 < < 133)  

= P[(123 - 128) / 4 < ( - ) / < (133 - 128) / 4 )]

= P(-1.25< Z < 1.25 )

= P(Z < 1.25) - P(Z < -1.25 )

Using z table,  

= 0.8944 - 0.1056

= 0.7888

Probability =0.7888

b ) within 10 = 128  ± 10 = 118, 138

P(118 < < 138)  

= P[(118 - 128) / 4 < ( - ) / < (138 - 128) /4 )]

= P(-2.5< Z < 2.5 )

= P(Z < 2.5) - P(Z < -2.5 )

Using z table,  

= 0.9938 - 0.0062

= 0.9876

Probability= 0.9876