Question

The mean number of children per household in some city is 1.37, and the standard deviation is 1.21.

(a) If we take a random sample of 325 households, what is the probability that the mean number of children per household in the sample will be more than 1.26?

(b) {NO CALCULATION FOR THIS} To answer (a), did you assume that the number of

children in a household is normally distributed? Why or why not?

4.

Answer 1

## We have given :

The mean number of children per household in some city is 1.37, and the standard deviation is 1.21.

ie μ = population mean = 1.37 and σ = population standard deviation = 1.21

here variable follow normal distribution with μ and  σ  

## (a) If we take a random sample of 325 households, what is the probability that the mean number of children per household in the sample will be more than 1.26?

Answer : we have to find out probability for that the mean number of children per households in the sample will be

more than 1.26 . ie P [ x̄ > 1.26 ]

we can use here central limit theorem : z = ( x̄ - μ ) * √n / σ follow normal distribution with mean o and standard deviation 1 ie N( 0 ,1 )

P [ x̄ > 1.26 ] = P [( x̄ - μ ) * √n / σ > ( 1.26 - μ ) * √n / σ ]

P [ z > ( 1.26 - μ ) * √n / σ ]

P [ z > ( 1.26 - 1.37 ) * √ 325  / 1.21 ]

P [ z > -1.6388 ]

now use statistical table :

= P [ z < 1.6388 ]

= 0.9495

probability for that the mean number of children per households in the sample will be

more than 1.26 is  0.9495 .

## (b) {NO CALCULATION FOR THIS} To answer (a), did you assume that the number of

children in a household is normally distributed? Why or why not?

Answer :

children in a household is normally distributed?

yes ,

here we have population mean and standard deviation and it follow normal distribution

and sample size is large that is greater than 30 . hence by central limit theorem if we have mean and standard deviation then sample mean   follow normal distribution with 0 and 1 ( standard normal )  . that is if population follow normal distribution then sample is also follow normal distribution .