Advanced Placement (AP) College

A newspaper editor wants to investigate whether the residents of two
counties support the proposal to build a wind farm. She plans to take an
SRS of 30 residents from the southern county and 40 residents from the
northern county, then look at the difference (southern - northern)
between the proportions of residents that support the proposal in each
sample. Suppose that 65% of the residents of the southern county and
52% of the residents of the northern county approve.
What are the mean and standard deviation of the sampling distribution of
the difference in sample proportions?

Answers

Answer 1

The mean of the sampling distribution of the difference in sample proportions is 13% and the standard deviation is 11.76%.

What is the standard deviation?

The mean of the sampling distribution of the difference in sample proportions can be calculated as the difference between the population proportions of the two counties, as the sample proportions are expected to be unbiased estimates of the population proportions. In this case, the population proportion of residents supporting the proposal in the southern county is 65% and in the northern county is 52%.

Mean of sampling distribution of difference in sample proportions:

= Population proportion of southern county - Population proportion of northern county

= 65% - 52%

= 0.65 - 0.52

= 0.13 or 13%

The standard deviation of the sampling distribution of the difference in sample proportions can be calculated using the following formula:

Standard deviation of sampling distribution of difference in sample proportions:

= √[ (p1 * (1 - p1) / n1) + (p2 * (1 - p2) / n2) ]

where:

p1 and p2 are the population proportions of the two counties,

n1 and n2 are the sample sizes from the southern and northern counties, respectively.

Given that p1 = 0.65, p2 = 0.52, n1 = 30, and n2 = 40, we can plug in these values into the formula to calculate the standard deviation.

Standard deviation of sampling distribution of difference in sample proportions:

= √[ (0.65 * (1 - 0.65) / 30) + (0.52 * (1 - 0.52) / 40) ]

= √[ (0.65 * 0.35 / 30) + (0.52 * 0.48 / 40) ]

= √[ (0.2275 / 30) + (0.2496 / 40) ]

= √[ 0.007583 + 0.00624 ]

= √0.013823

= 0.1176 or 11.76%

So, the mean of the sampling distribution of the difference in sample proportions is 13% and the standard deviation is 11.76%.

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Answers

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