1. Section 7.8 Confidence Intervals
Special Topics

2. A 95% Confidence Interval

3. Now, in plain English…
What this is saying is that a 95% confidence interval is an interval created to estimate what (where) the true parameter might be.
It is centered at p-hat, the sample statistic. From there, it extends out 2 standard deviations.
Since the p-hat value at the center of the interval isn’t the same from sample to sample, not all the intervals will successfully estimate where the parameter p is.
With 95% confidence, we would expect about 95 out of 100 intervals to accurately estimate p. About 5 will miss!

4. Now, the math behind all this…
Here is the formula for a 95% confidence interval for proportions:
The first on the left is the sample estimate we get from our survey or experimental data.
This part is called the “margin or error”:

5. The math part continued….
Now, let’s examine the margin of error portion more closely:
The multiplier “2” is there because of the % rule. About 95% of the sample proportions fall within 2 standard deviations of the center of the sampling distribution.
We use p-hat instead of p in the standard deviation part because we are estimating p, and we don’t know it before we estimate it.

6. Example

7. In-Class Practice
The Princeton Metro Times (Sept. 25, 1999) reported that 48% of a random sample of 369 students at The College of New Jersey indicated that they were “binge drinkers”. Binge drinking was defined to be consuming five to six drinks in one sitting for men and four to five drinks for women. Construct and interpret a 95% confidence interval for p, the proportion of students at The College of New Jersey who are binge drinkers.

8. Another Example….
In an article titled “Fluoridation Brushed off by Utah”, it was reported that a small but vocal minority in Utah has been successful in keeping fluoride out of Utah water supplies in spite of the fact that a clear majority of Utah residents favor fluoridation. To support this statement, the article included the result of a survey of a random sample of 150 Utah residents that found 65% to be in favor of fluoridation. Construct and interpret a 95% confidence interval for p, the true proportion of Utah residents who favor fluoridation. Is this interval consistent with the statement that fluoridation is favored by a clear majority of residents?

9. Theory Behind a Confidence Interval
We get our 95% confidence intervals by using a formula that catches the true unknown population proportion in 95% of all samples.
The red dots below the curve represent sample results.
The lines they are on show plus or minus two SD’s to either side.
All but one successfully “caught” p.

10. Homework
Worksheet 7.8