To estimate the difference between two population proportions with a confidence interval, you can use the Central Limit Theorem when the sample sizes are large enough (typically, each at least 30). When a statistical characteristic, such as opinion on an issue (support/don’t support), of the two groups being compared is categorical, people want to report on the differences between the two population proportions — for example, the difference between the proportion of women and men who support a four-day work week. How do you do this? Show
You estimate the difference between two population proportions, p1 – p2, by taking a sample from each population and using the difference of the two sample proportions, plus or minus a margin of error. The result is called a confidence interval for the difference of two population proportions, p1 – p2. The formula for a confidence interval (CI) for the difference between two population proportions is and n1 are the sample proportion and sample size of the first sample, and and n2 are the sample proportion and sample size of the second sample. The value z* is the appropriate value from the standard normal distribution for your desired confidence level. (Refer to the following table for z*-values.) z*-values for Various Confidence LevelsConfidence Levelz*-value80%1.2890%1.645 (by convention)95%1.9698%2.3399%2.58To calculate a CI for the difference between two population proportions, do the following:
The formula shown here for a CI for p1 – p2 is used under the condition that both of the sample sizes are large enough for the Central Limit Theorem to be applied and allow you to use a z*-value; this is true when you are estimating proportions using large scale surveys, for example. For small sample sizes, confidence intervals are beyond the scope of an intro statistics course. Suppose you work for the Las Vegas Chamber of Commerce, and you want to estimate with 95% confidence the difference between the percentage of all females who have ever gone to see an Elvis impersonator and the percentage of all males who have ever gone to see an Elvis impersonator, in order to help determine how you should market your entertainment offerings.
Of course, there are some guys out there that wouldn’t admit they’d ever seen an Elvis impersonator (although they’ve probably pretended to be one doing karaoke at some point). This may create some bias in the results. Notice that you could get a negative value for For example, if you had switched the males and females, you would have gotten –0.19 for this difference. That’s okay, but you can avoid negative differences in the sample proportions by having the group with the larger sample proportion serve as the first group (here, females). However, even if the group with the larger sample proportion serves as the first group, sometimes you will still get negative values in the confidence interval. Suppose in the above example that only 0.43 of women had seen an Elvis impersonator. Thus, the difference in proportions is 0.09, and the upper end of the confidence interval is 0.09 + 0.13 = 0.22 while the lower end is 0.09 – 0.13 = –0.04. This means that the true difference is reasonably anywhere from 22% more women to 4% more men. It’s too close to tell for sure. What is the 95% confidence interval for the difference in population means?Suppose we want to generate a 95% confidence interval estimate for an unknown population mean. This means that there is a 95% probability that the confidence interval will contain the true population mean. Thus, P( [sample mean] - margin of error < μ < [sample mean] + margin of error) = 0.95.
What is the 95% confidence interval for the population proportion?To calculate the confidence interval, we must find p′, q′. p′ = 0.842 is the sample proportion; this is the point estimate of the population proportion. Since the requested confidence level is CL = 0.95, then α = 1 – CL = 1 – 0.95 = 0.05 ( α 2 ) ( α 2 ) = 0.025.
What is the confidence interval estimate of the difference between the two population means?The confidence interval gives us a range of reasonable values for the difference in population means μ1 − μ2. We call this the two-sample T-interval or the confidence interval to estimate a difference in two population means.
When calculating a 95% confidence interval for the difference between two means Which of the following is true?When calculating a 95% confidence interval for the difference between two means, which of the following is true? When the confidence interval ranges from a positive value to a positive value, we find that there is conclusive evidence (at 95% confidence) that both population means are positive.
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