Consider two independent random samples and use the following results to find the $95 \%$ confidence interval for the true difference between the population proportions.
$n_1=576, \hat{p}_1=0.53 \text { and } n_2=599, \hat{p}_2=0.62$
Step 1 of 3 : Find the point estimate that should be used in constructing the confidence interval.
Answer (2)
The point estimate for the true difference between the population proportions is calculated as the difference between the sample proportions, resulting in − 0.09 . This means that the first group has a lower proportion compared to the second group by 9%. It is important for constructing a confidence interval for the true difference between the population proportions.
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Calculate the difference between the sample proportions: p ^ 1 − p ^ 2 = 0.53 − 0.62 .
The point estimate is the result of the subtraction: − 0.09 .
The point estimate for the difference between the population proportions is − 0.09 .
Explanation
Understand the problem and provided data We are given two independent random samples with the following data:
Sample 1: n 1 = 576 , p ^ 1 = 0.53 Sample 2: n 2 = 599 , p ^ 2 = 0.62
We want to find the point estimate for the difference between the population proportions. The point estimate is simply the difference between the sample proportions.
Calculate the point estimate The point estimate is calculated as follows: p ^ 1 − p ^ 2 = 0.53 − 0.62 = − 0.09 Therefore, the point estimate for the difference between the population proportions is − 0.09 .
State the final answer The point estimate for the difference between the population proportions is the difference between the sample proportions, which is: p ^ 1 − p ^ 2 = 0.53 − 0.62 = − 0.09
So, the point estimate is − 0.09 .
Examples
In marketing, you might want to know the difference in the proportion of customers who prefer your product versus a competitor's. If you survey 576 customers and find that 53% prefer your product, and survey 599 of your competitor's customers and find that 62% prefer their product, the point estimate of the difference in preference is -9%. This means that, based on your samples, your competitor has a 9% higher preference rate. This information is crucial for making strategic decisions about product improvement and marketing efforts.
