Question

The heights of North American women are nor-mally distributed with a mean of 64 inches and a standard deviation of 2 inches. a. b. c. What is the probability that a randomly selected woman is taller than 66 inches? A random sample of four women is selected. What is the probability that the sample mean height is greater than 66 inches? What is the probability that the mean height of a random sample of 100 women is greater than

Answer 1

Complete Question

The complete question is shown on the first uploaded image

Answer:

a

`P(X > 66) = P(Z> 1 ) = 0.15866`

b

`P(\= X > 66) = P(Z> 2 ) = 0.02275`

c

`P(\= X > 66) = P(Z> 10 ) = 0`

Explanation:

From the question we are told that

The population mean is
`\mu = 64 \ inches`

The standard deviation is
`\sigma = 2 \ inches`

The probability that a randomly selected woman is taller than 66 inches is mathematically represented as

`P(X > 66) = P((X - \mu )/(\sigma ) > ( 66 - \mu )/(\sigma) )`

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

So

`P(X > 66) = P(Z> ( 66 - 64 )/( 2) )`

`P(X > 66) = P(Z> 1 )`

From the z-table the value of
`P(Z > 1 ) = 0.15866`

So

`P(X > 66) = P(Z> 1 ) = 0.15866`

Considering b

sample mean is n = 4

Generally the standard error of mean is mathematically represented as

`\sigma _(\= x) = (\sigma )/(√(4) )`

=>
`\sigma _(\= x) = (2 )/(√(4) )`

=>
`\sigma _(\= x) = 1`

The probability that the sample mean height is greater than 66 inches

`P(\= X > 66) = P((X - \mu )/(\sigma_(\= x ) ) > ( 66 - \mu )/(\sigma_(\= x )) )`

=>
`P(\= X > 66) = P(Z > ( 66 - 64 )/(1) )`

=>
`P(\= X > 66) = P(Z> 2 )`

From the z-table the value of
`P(Z > 2 ) = 0.02275`

=>
`P(\= X > 66) = P(Z> 2 ) = 0.02275`

Considering b

sample mean is n = 100

Generally the standard error of mean is mathematically represented as

`\sigma _(\= x) = (2 )/(√(100) )`

=>
`\sigma _(\= x) = 0.2`

The probability that the sample mean height is greater than 66 inches

`P(\= X > 66) = P(Z > ( 66 - 64 )/(0.2) )`

=>
`P(\= X > 66) = P(Z> 10 )`

From the z-table the value of
`P(Z > 10 ) = 0`

`P(\= X > 66) = P(Z> 10 ) = 0`