Confidence Interval for a Single Proportion Chapter 19 Confidence Interval for a Single Proportion

Proportions – Categorical Data Remember that when data is categorical we have a population parameter p and a sample statistic . In this situation we are trying to estimate p so its value is unknown. We only have

Confidence tells how “confident” we are that our calculations captured the true proportion - lends credibility to our inferences. A Confidence Interval in General: statistic  (critical value)·(standard deviation)

A Confidence Interval for a Proportion: z* is the critical value – we look it up on the table.

Conditions-same as for sampling distributions!! 1) Randomization. 2) Independence. 3) Success/Failure. To insure that the sample size is large enough to approximate normal, np  10 and n(1 – p)  10

Two key phrases when making statements: Phrase 1: Interprets a single confidence interval. This is a statement about one interval that was calculated from a single sample: We are #% confident that the true proportion of ____(context) lies in the interval ……

Two key phrases when making statements: Phrase 2: Interprets the confidence level. It is a general statement that is not about one specific interval but the overall process: Saying that we are “95% confident” means that with this data, if many intervals were constructed in this manner, we would expect approximately 95% of them to contain the true proportion of ____(context).

Reducing Margin of Error The more observations we have (n) the more we reduce our variability. Data collection can be difficult or costly however. We must balance our desire for a small margin of error with practical judgment.

Reducing Margin of Error There is also a relationship between the confidence level and the margin of error. As our confidence increases so does our margin of error. A great deal of confidence is not very helpful when it makes the margin of error so large that the interval tells us nothing.

Remember that this is inference. It is not a promise or a certainty. **When trying to achieve a certain margin of error, if p-hat has not been established yet, then you can estimate it to get an approximate sample size. If no good estimate is available, p-hat = .5 is the most conservative estimate of p-hat and will insure a margin or error smaller than that desired.

Example 1 Suppose a new treatment for a certain disease is given to a random sample of 200 patients with the disease. The treatment was successful for 166 of the patients. A) Construct and interpret a 99% confidence interval for the proportion of patients with this disease who were successfully treated. B) In the context of this situation, explain what it means to be 99% confident in any interval. C) If the traditional treatment for this disease has a success rate of about 70%, does this interval give evidence that the new treatment is better? Explain.

Example 2 The Princeton Metro Times 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 as consuming 5-6 drinks in 1 sitting for men and 4-5 drinks in 1 sitting for women. Construct and interpret a 90% confidence interval for the proportion of students at the College of New Jersey who are binge drinkers.

Example 3 An automobile manufacturer would like to know what proportion of its customers are not satisfied with the service provided by their local dealer. The customer relations department will survey a random sample of customers and compute a 95% confidence interval for the proportion who are not satisfied. From past studies they believe that this proportion will be about 0.2. Find the sample size needed if the margin of error of this confidence interval is to be about 0.03.

Example 4 Girls younger than 18 seeking services at Planned Parenthood family planning clinics in Wisconsin were surveyed to determine whether mandatory parental notification would cause them to stop using sexual health services. Of the 118 girls surveyed, 55 indicated they would stop using all services if their parents were informed they were seeking prescription contraceptives. Construct a 95% confidence interval for the proportion who would stop using services if parents were informed. Be sure to interpret your interval with the correct population that it represents.