Using the sample information given in Exercises 22-23, give...

Using the sample information given in Exercises $22-23,$ give the best point estimate for the binomial proportion $p$ and calculate the margin of error. Suppose you are writing a questionnaire for a sample survey involving $n=100$ individuals, which will generate estimates for several different binomial proportions. If you want to report a single margin of error for the survey, what margin of error should you use?

Using the sample information given in Exercises $22-23,$ give the best point estimate for the binomial proportion $p$ and calculate the margin of error.
Suppose you are writing a questionnaire for a sample survey involving $n=100$ individuals, which will generate estimates for several different binomial proportions. If you want to report a single margin of error for the survey, what margin of error should you use?

Key Concepts

Binomial Proportion

A binomial proportion represents the probability of success in a yes/no experiment or trial, where each trial is independent and has only two possible outcomes. It is a fundamental concept in statistics, particularly in analyzing binary data where the proportion is estimated from the number of successes in a fixed number of trials.

Point Estimate

The point estimate is a single value derived from the sample data that serves as the best guess or estimate of the true population parameter. For a binomial proportion, the sample proportion (often denoted as p-hat), calculated as the number of successes divided by the sample size, is used as the point estimate.

Margin of Error

The margin of error quantifies the uncertainty associated with the point estimate. In the context of a binomial proportion, it is typically calculated as the critical value from the appropriate probability distribution (often the normal distribution, via the central limit theorem) multiplied by the standard error of the proportion. The standard error is derived as the square root of the product of the point estimate and its complement divided by the sample size.

Worst-Case Margin of Error

When reporting a single margin of error for a survey that estimates several binomial proportions, it is common to use the worst-case scenario. The margin of error is maximized when the true proportion is 0.5 because this maximizes the product p(1 ? p) in the standard error formula. Therefore, calculating the margin of error under the assumption p = 0.5 yields a conservative estimate that ensures all proportions are estimated within at least that range.

Transcript

00:01 So we're doing margin of error here.

00:03 I've gone ahead and written out the formula for margin of error.

00:05 Margin of error is equal to 2 times the square root of p, which is going to be your percent, times the quantity of 1 minus that same p, divided by n, which is your sample size.

00:22 Very nice of them on this problem.

00:24 They are very straightforward.

00:27 They tell us specifically that p is equal to 81%.

00:31 And then they also tell us that n is equal to 100 so we don't have to go searching through a problem that just tells us that that's awesome one key thing here do recognize that they give us p as a percent which is what they're supposed to do it is up to us to adjust that before we plug it into the formula because we do not plug percents into formulas we have to change them into decimal form first percent is a term that means divide by a hundred so if you want to to change a percent to a decimal you just divide by a hundred eighty one divided by a hundred is point eight one that's what we will be using for this formula okay so now to plug in sorry now to plug in we are going to have our margin of error is going to be equal to two times the square root of our p which we have found as point eight one if we're going to plug it in the correct fashion, then we're going to take that times the quantity of 1 minus p which is 1 minus 0 .81 piece the same in both cases right and we're going to divide that by n our sample size which is 100.

01:51 That would be what you'd want your formula to look like before you start plugging into your calculator...

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