Question

Part 1) What sample size is needed to give a margin of error within +-6% in estimating a population proportion with 95% confidence? Use z-values rounded to three decimal places. Round your answer up to the nearest integer.

Part 2) Use StatKey or other technology to generate a bootstrap distribution of sample proportions and find the standard error for that distribution. Compare the result to the standard error given by the Central Limit Theorem, using the sample proportion as an estimate of the population proportion p.

Proportion of home team wins in soccer, with n=129 and p^=0.673.

Round your answer for the bootstrap S⁢E to two decimal places, and your answer for the formula S⁢E to three decimal places.

Part 3) Comparing Normal and Bootstrap Confidence Intervals

Find a 95% confidence interval for the proportion two ways, using StatKey or other technology and percentiles from a bootstrap distribution and using the normal distribution and the formula for standard error.

Proportion of Reese’s Pieces that are orange, using p^=0.48 with n=150

Round your answers to two decimal places.

Answer 1

Part 1)

For margin of error 6% i.e m= 0.05, the sample size is calculated using the formula,

Where,

Since the population proportion is not given, use

Part 2)

StatKey

The bootstrap distribution of sample proportion is generated in StatKey by following these steps,

Step 1: Select: CI for single Proportion. The screenshot is shown below,

Step 2: Now click on Edit Data and fill the values > count = 87, sample size = 129 > OK. The screenshot is shown below,

Step 3: Click on Two-tail and click on Generate 100 samples. The screenshot is shown below,

The result is obtained,

Standard Error = 0.041

Now the standard error value is obtained by applying the central limit theorem,

The results are same for bot methods

Part 3)

StatKey bootstrap confidence interval

Normal confidence interval

Both the confidence interval are same, the sample difference is because the statKey use the count data value that cause the proportion = 0.674 instead of 0.673.