Question

The mean height of women in a country​ (ages 20​29) is inches. A random sample of women in this age group is selected. What is the probability that the mean height for the sample is greater than ​inches? Assume . The probability that the mean height for the sample is greater than inches is nothing.

Answer 1

Complete Question

The mean height of women in a country (ages 20-29) is 64.4 inches. A random sample of 75 women in this age ground is selected. what is the probability that the mean height for the sample is greater than 65 inches? assume
`\sigma = 2.97`

Answer:

The value is
`P(X > 65) = 0.039715`

Explanation:

From question we are told that

The mean is
`\mu = 64.4 \ inches`

The sample size is
`n = 75`

The probability that the mean height for the sample is greater than 65 inches is mathematically represented as

`P(X > 65) = P[(X - \mu )/( \sigma_(\= x) ) > (65 - 64.4 )/( \sigma_(\= x) ) ]`

Where
`\sigma _(\= x )` is the standard error of mean which is evaluated as

`\sigma_(\= x ) = (\sigma)/(√(n) )`

=>
`\sigma_(\= x ) = (2.97)/(√(75) )`

=>
`\sigma_(\= x ) = 0.343`

Generally
`(X - \mu )/( \sigma_(\= x ) ) = Z(The \ standardized \ value \ of \ X )`

`P(X > 65) = P[Z> (65 - 64.4 )/(0.342 ) ]`

So

`P(X > 65) = P[Z >1.754 ]`

From the z-table the value of

`P(X > 65) = P[Z >1.754 ] = 0.039715`

`P(X > 65) = 0.039715`