Suppose that 60\%60%60, percent of adults in district A support a new ballot measure, while 45\%45%45, percent of adults in district B support the same measure. Pollsters take an SRS of 200200200 adults from district A and a separate SRS of 100100100 adults from district B to see the difference between the sample proportions (\hat{p}_\text{A}-\hat{p}_\text{B})( p ^ A − p ^ B )left parenthesis, p, with, hat, on top, start subscript, start text, A, end text, end subscript, minus, p, with, hat, on top, start subscript, start text, B, end text, end subscript, right parenthesis. What will be the shape of the sampling distribution of \hat{p}_\text{A}-\hat{p}_\text{B} p ^ A − p ^ B p, with, hat, on top, start subscript, start text, A, end text, end subscript, minus, p, with, hat, on top, start subscript, start text, B, end text, end subscript, and why?

The shape of the sampling distribution of [tex]\hat{p}_\text{A}-\hat{p}_\text{B}[/tex] is approximately normal because the number of success in district A is 120

How to determine the shape of the distribution?

The given parameters are:

Proportion of adults in district A = 60%
Proportion of adults in district B = 45%
Sample size of A, n = 200
Sample size of B, n = 100

In the sample size of 200, the number of adults that support a new ballot measure is:

Success = 60% * 200

Success = 120

Those that oppose is

Failure = 200 - 120

Failure = 80

If 45% supports a new ballot measure in district B, then 55% do not support

The difference between those that support and do not support is:

(55% - 45%) * 100 = 10

Hence, the sampling distribution of [tex]\hat{p}_\text{A}-\hat{p}_\text{B}[/tex] is approximately normal

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