Lab: Exploring the Effects of Confidence Level, Sample Size, and Shape I

In this activity, you will use an applet to generate confidence intervals for the population proportion. You will use the results to understand what is meant by the confidence level and the role of sample size.

Open the Confidence Intervals for a Proportion with p = 0.3 Applet (Go to StatCrunch>Applets > Confidence intervals > for a proportion>Compute!. Click Reset.

1. Exploring the role of level of confidence:

a. Construct 1,000 confidence intervals with n = 100, p = 0.3. What proportion of the 95% confidence intervals included the population proportion, 0.3? (We simulate obtaining 1000 different random sample of size n=100 from a population with p=0.3) Reset and Construct another 1,000 confidence intervals with n = 100, p = 0.3. What proportion of the 95% confidence intervals included the population proportion, 0.3? Did the same proportion of intervals include the population proportion each time? What proportion did you expect to include the population proportion?

b. Construct 1,000 confidence intervals with n = 100, p = 0.3. What proportion of the 99% confidence intervals included the population proportion, 0.3? Reset and Construct another 1,000 confidence intervals with n = 100, p = 0.3. What proportion of the 99% confidence intervals included the population proportion, 0.3? Did the same proportion of intervals include the population proportion each time? What proportion did you expect to include the population proportion in each part?

2. Exploring the role of sample size:

a. Construct 1,000 confidence intervals with n = 10, p = 0.3. What proportion of the 95% confidence intervals included the population proportion, 0.3?

b. Construct 1,000 confidence intervals with n = 40, p = 0.3. What proportion of the 95% confidence intervals included the population proportion, 0.3?

c. Construct 1,000 confidence intervals with n = 100, p = 0.3. What proportion of the 95% confidence intervals included the population proportion, 0.3?

d. Did the same proportion of intervals include the population proportion in parts (a), (b), and (c)? What proportion did you expect to include the population proportion in each part? What happens to the proportion of intervals that include the population proportion as the sample size increases? Why, what changes?

3. Further exploration:

a. Construct 100 confidence intervals with p = 0.3, level of confidence = 0.95, and sample size = 100. Select one of the intervals (denoted with a red line) that does not include the population proportion that is to the left of the population proportion, 0.3. If there are no such intervals, repeat until there are. Scroll the mouse cursor over the interval. In the popup window, notice the value of the sample proportion. Determine the number of standard errors the sample proportion is from the population proportion, 0.3, by computing = 0.3 (1) 100 , where the standard error is = (1) .

b. From the same 100 confidence intervals constructed in part a, select one of the intervals that does not include the population proportion (red line) that is to the right of the population proportion. In the pop-up window, notice the value of the sample proportion. Determine the number of standard errors the sample proportion is from the population proportion, 0.3.

c. Are each of the sample proportions from parts a and b more than 1.96 standard errors from the population proportion? Explain why any sample proportion that is more than 1.96 standard errors from the population proportion will result in an interval that does not include the population proportion