Calculate the margin of error in estimating a binomial...

Calculate the margin of error in estimating a binomial proportion $p$ using samples of size $n=100$ and the values of p given in Exercises $15-19 .$ What value of p produces the largest margin of error?
$p=.5$

Key Concepts

Margin of Error

Margin of error measures the extent of the potential deviation of an estimated parameter from the true population value due to sampling variability. In the context of estimating a proportion, it is typically computed based on the standard error and a critical value from the appropriate probability distribution.

Binomial Proportion

A binomial proportion represents the probability of success in trials that have two possible outcomes (success or failure). Estimating this proportion using sample data involves understanding the distribution and variability inherent in binary data.

Standard Error

The standard error for a binomial proportion is calculated as the square root of [p(1-p)/n]. It quantifies the dispersion of the estimated proportion from the true proportion and is a fundamental component in determining the margin of error.

Sample Size

Sample size plays a critical role in estimation as it affects the precision of the estimate. A larger sample size generally reduces the standard error, leading to a narrower margin of error and thus a more precise estimate.

Optimization of Margin

The margin of error is maximized when the product p(1-p) is at its largest. For a binomial proportion, this maximum occurs at p = 0.5, since this value yields the highest variability, and hence, the largest margin of error in estimation.

Transcript

00:01 Now we're dealing with proportions, and in general it's a p hat plus or minus some z value with alpha over 2 times the square root of p hat, 1 minus the p hat over n.

00:13 And this part is our margin of air.

00:16 And so we want to find the margin of air for these three particular questions.

00:21 And in the first one, we have n is 350, and we have the p hat is 0 .3, and we have our alpha level is 0 .3 .4.

00:30 So our z value, this is really a 99 % confidence interval, so the z value for 0 .05, i should say, is 2 .576.

00:47 So our margin of error for this question will be 2 .576 times the square root of 0 .3 times 0 .7, divided by the sample size of 3 .550, divided by the sample size of 3 .50.

01:01 And that will be our margin of air.

01:03 So 2 .576 times a square root of .3 times .7 divided by 350.

01:12 Gives us a margin of air as a decimal of .063.

01:19 As a percent, 6 .3%.

01:23 On part b, similar question, just changing things up a bit.

01:28 A little bit smaller sample size.

01:30 Our p hat here is 0 .45, and our alpha level is 0 .05, so that would be a 95 % confidence interval.

01:38 So the upper tail will have 0 .025 in it, and the 0 .025 are for the 95 % confidence interval.

01:48 This will be a 1 .960, and so our margin of air will equal 1 .96 times a square root of 0 .45 times its complement 0 .55 divided by the 275.

02:05 So 1 .96 times square root...