Confidence Intervals for Proportions & Statistical Inference

1. Confidence Intervals for Proportions & Statistical Inference

2. Activity 1, Page 1

3. Since the sample statistic varies from sample to sample, we don’t not know how good an estimate of the unknown population proportion any particular sample proportion is. However, we expect the unknown population proportion to be within some reasonable distance of a particular sample proportion. The purpose of confidence intervals is to use the observed sample statistic (proportion) to construct an interval of values that we can be reasonably confident contains the actual unknown population parameter (proportion).

4. Within Applying the CLT, we see that extending two standard deviations on either side of the sample proportion, p-hat, produces an interval around it that captures the value of the population parameter π, for about 95% of all samples—that is, within 2 standard deviations of p-hat. Problem: In practice, we cannot use this formula because the value of π is usually not known (Indeed, the whole point of finding this interval is to estimate the value of π based on the sample proportion p-hat) A reasonable substitute to use as an estimate for π in the standard deviation formula is p-hat: The Standard Error of p-hat

5. Activity 1, Page 2 Activity 1, handout Page 3

6. Confidence interval (CI) for a population proportion has endpoints: p-hat = sample proportion n= sample size Multiplier z* is called the critical value In practice, you first specify a desired level of confidence (i.e., a particular probability), and then find the critical value (a z-score, from the standard Normal table) that corresponds to that level.

7. Activity 2: Finding Critical Values Before calculating a confidence interval, you must determine the critical value z*. Ex. Suppose we want to find the value of z* for a 98% confidence interval. This is a z* value such that the area under the standard Normal curve between –z* and z* is equal to 0.98. Follow these steps:

8. 1. Sketch the standard Normal density curve and shade the area corresponding to the middle 98% of the distribution. Also indicate roughly where –z* and z* fall in your sketch. 2. Based on your sketch what is the the total area under the curve to the left of the value? Total area to the left of z* is .98 + .01 = .99

9. 3.Use the standard Normal table to find the value (z*) that has this area (0.99) to its left under the standard Normal curve. z* = 2.33

10. Find the critical value z* for a 95% confidence interval 1. Sketch the standard Normal density curve and shade the area corresponding to the middle 95% of the distribution. Also indicate roughly where –z* and z* fall in your sketch. 2. Based on your sketch what is the the total area under the curve to the left of the value? Total area to the left of z* is .95 + .025 = .975

11. 3.Use the standard Normal table to find the value (z*) that has this area (0.975) to its left under the standard Normal curve. z* = 1.97

12. Some commonly used confidence levels and associated critical values

13. Activity 3, class handout (hard copy only)