Question

Given a normal population which has a mean of 110 and a standard deviation of 5, find the probability that a random sample of 49 has a mean between 109 and 112. Report your answer to four decimal places.

Answer 1

0.9168

Explanation:

From the data given:

Mean = 110

standard deviation = 5

Let consider a random sample n =49 which have a mean between 109 and 112.

The test statistics can be computed as:

`Z_1 = (x- \bar x)/((\sigma)/(√(n)))`

`Z_1 = (109- 110)/((5)/(√(49)))`

`Z_1= (-1)/((5)/(7))`

`Z_1` = -1.4

`Z_2= (x- \bar x)/((\sigma)/(√(n)))`

`Z_2 = (112- 110)/((5)/(√(49)))`

`Z_2 = (2)/((5)/(7))`

`Z_2 =2.8`

Thus; P(109 <
`\overline x` < 112) = P( - 1.4 < Z < 2.8)

= P(Z < 2.8) - P( Z < -1.4)

= 0.9974 - 0.0806

= 0.9168