AP Stats ~ Lesson 8.2: Estimating a Population Proportion OBJECTIVES: 9STATE and CHECK the Random, 10%, and Large Counts conditions for constructing a confidence interval for a population proportion. 9DETERMINE critical values for calculating a C % confidence interval for a population proportion using a table or technology. 9CONSTRUCT and INTERPRET a confidence interval for a population proportion. 9DETERMINE the sample size required to obtain a C % confidence interval for a population proportion with a specified margin of error.

Pennies Activity: 1. As a class, determine how to select an SRS of pennies from the

container. 2. In groups of 3 or 4, take an SRS of approximately 100 pennies.

Separate the pennies into two piles: those that are more than 10 years old, and those that aren’t. Count the number in each pile.

3. Determine the point estimate (p-hat) for the unknown population proportion of pennies more than 10 years old.

4. Now for the challenge: each team will be given about 10 minutes to find a 95% confidence interval for the parameter p. Make sure that you are considering any conditions that might be required!

5. Compare your results with others in the class. Discuss problems you encountered, and what you did to overcome them.

Class Results for the Penny Activity:

When we did the activity, what conditions did you check? There are 3 conditions you MUST check: 1. RANDOM. The data should come from a well-designed random sample or

randomized experiment, otherwise there is no scope for inference to a population or inference about cause and effect. This also allows us to construct a sampling distribution, and also model the behavior with a probability distribution (which allows us to find a confidence interval).

2. This also leads us to consider the 10% CONDITION. If we’re taking an SRS, we are losing independence, so we have to make sure the 10% condition is met. Remember the formula?

Population < 10(sample size) or sample size < 1/10(Population)

3. LARGE COUNTS. The method that we use for finding confidence intervals requires that the distribution is normal or approximately normal. Since in practice we don’t actually know the value of the population parameter p, we will adjust our Large Counts condition equation to use our sample statistic.

Think about this… if equals the # of successes/n, and we multiply that by n to check our condition, that means that what we’re really looking at is simply “are the number of successes greater than 10?” and “are the number of failures greater than 10?”. Did we meet those conditions in our penny activity?

It may be helpful for you to remember that the three conditions correspond to the three parts of the confidence interval. The Random condition ensures that the statistic (point estimate) is unbiased. The Large Counts condition ensures that we have a normal distribution, so we know that we are using a valid critical value (z*). The 10% condition ensures that we can use the formula for standard deviation. What happens if one of the conditions isn’t met? The actual capture rate of our intervals will not be as large as the confidence level we are stating. For example, we may be using 95% confidence, but only about 90% of our intervals will actually capture the true population parameter.

CONSTRUCTING CONFIDENCE INTERVALS: We can use the general formula from Section 8.1 to construct a confidence interval for an unknown population proportion p: We will use the sampling distribution statistic to estimate p. The formula for the standard deviation of is Unfortunately we don’t know what the value of p is, so we will replace the p with . This gives us what we call the Standard Error (SE) of p. When the standard deviation of a statistic is estimated from data, the results is called the standard error of the statistic.

Finding a Critical Value So now that we know how to find the statistic and the standard deviation, how do we find the critical value? If the Large Counts condition is met, we can use a Normal curve. To find a level C confidence interval, we need to catch the central area C under the standard Normal curve. To find a 95% confidence interval, we use a critical value of 2 based on the 68-95-99.7 rule. Using Table A or a calculator, we can get a more accurate critical value. Note, the critical value z* is actually 1.96 for a 95% confidence level.

Use Table A to find the critical value z* for an 80% confidence interval. Assume that the Large Counts condition is met.

Since we want to capture the central 80% of the standard Normal distribution, we leave out 20%, or 10% in each tail. Search Table A to find the point z* with area 0.1 to its left.

z .07 .08 .09 – 1.3 .0853 .0838 .0823 – 1.2 .1020 .1003 .0985 – 1.1 .1210 .1190 .1170

The closest entry is z = – 1.28

So, the critical value z* for an 80% confidence interval is z* = 1.28.

Once we find the critical value z*, we can find the confidence interval for the population proportion.

Example: Mrs. Barton’s class took an SRS of 102 pennies and discovered that 57 of the pennies were more than 10 years old. a) Calculate and interpret a 99% confidence interval for p=the true proportion of

pennies that are more than 10 years old.

b) Is it plausible that exactly 60% of the pennies in the collection are more than 10 years old? Explain.

Choosing the Sample Size: In planning a study, we may want to choose a sample size that allows us to estimate a population proportion within a given margin of error.

The margin of error (ME) in the confidence interval for p is

ME = z*ˆ p (1 ˆ p )

nz* is the standard Normal critical value for the level of confidence we want.

Because the margin of error involves the sample proportion ˆ p , we have to guess the latter value when choosing n. There are two ways to do this :

• Use a guess for ˆ p based on past experience or a pilot study

• Use ˆ p = 0.5 as the guess. ME is largest when ˆ p = 0.5

In planning a study, we may want to choose a sample size that allows us to estimate a population proportion within a given margin of error.

Example: Suppose that you want to estimate p=the true proportion of students at your school who have a tattoo with a 95% confidence level, and a margin of error of no more than 0.10. Determine how many students should be surveyed to estimate p within 0.10 with 95% confidence.

Homework

p. 504: #27-33, 35-47 odds, 49-52

Read pp. 507-526