Question

In two shipments of fruit, A and B, the proportions of fruit that are apples are and , respectively. Suppose that independent random samples of 50 fruit from A and 100 fruit from B. Let be the sample proportion of apples from shipment A and be the sample proportion of apples from shipment B. What is the mean of the sampling distribution of -

Answer 1

This question is incomplete, the complete question is;

In two shipments of fruit, A and B, the proportions of fruit that are apples are
`P^` and
`P^` , respectively. Suppose that independent random samples of 50 fruit from A and 100 fruit from B. Let
`P^` be the sample proportion of apples from shipment A and
`P^` be the sample proportion of apples from shipment B. What is the mean of the sampling distribution of
`P^` -
`P^` ?

Answer:

the mean of the sampling distribution is
`P^` -
`P^`

Explanation:

Given the data in the question;

let
`X_(A)` be number of apples from the shipment A

and
`X_(B)` be number of apples from shipment B

`P^` =
`X_(A)` /
`n_(A)` =
`X_(A)` / 50 { x ¬ Bin(np )

{
`X_(A)` = 50
`P^` )

`P^` =
`X_(B)` /
`n_(B)` =
`X_(B)` / 100

(
`X_(B)` = 100
`P^` )

E(
`P^` -
`P^` ) = E(
`X_(A)` /
`n_(A)` -
`X_(B)` /
`n_(B)` )

= E(
`X_(A)` / 50 -
`X_(B)` / 100 )

= E(
`X_(A)`) / 50 - E(
`X_(B)`) / 100

= 50
`P^` / 50 - 100
`P^` / 100

=
`P^` -
`P^` { E(X) = np ]

Therefore, the mean of the sampling distribution is
`P^` -
`P^`