6.3 Confidence Intervals for Population Proportions

6.3 Confidence Intervals for Population Proportions Statistics Mrs. Spitz Spring 2009

Objectives/Assignment • How to find a sample proportion • How to construct a confidence interval for a population proportion • How to determine a minimum sample size when estimating a population proportion. Assignment: pp. 280-282 #1-27 all

Schedule for coming weeks: • Today – Notes 6.3. Homework due BOC on Friday. • Friday, 1/16/09 – Notes 6.4. Assignment due Tuesday on our return. • Monday – 1/19/09 – No school • Tuesday – 1/20/09 – Chapter Review • Thursday-Chapter Review 6 DUE – Test – Chapter 6 • Friday – 1/23/09 – 7.1 Hypothesis Testing

Sample Proportions • Recall from section 4.2 that the probability of success in a single trial of a binomial experiment is p. This probability is a population proportion. In this section, you will learn how to estimate a population proportion, p using a confidence interval. As with confidence intervals for µ, you will start with a point estimate (6.1)

Definition: • The point estimate for p, the population proportion of successes, is given by the proportion of successes in a sample and is denoted by: • where x is the number of successes in the sample and n is the number in the sample. The point estimate for the number of failures is .The symbols and are read as “p hat” and “q hat”

Ex. 1: Finding a point estimate for p • In a survey of 883 American adults, 380 said that their favorite sport is football. Find a point estimate for the population proportion of adults who say their favorite sport is football. • SOLUTION: Using n =883 and x = 380

Insight • In the first two sections, estimates were made for the quantitative data. In this section, sample proportions are used to make estimates for qualitative data.

Confidence Intervals for a Population P • Constructing a confidence interval for a population proportion p is similar to constructing a confidence interval for a population mean. You start with a point estimate and calculate a maximum error of estimate.

Definition: • A c-confidence interval for the population proportion p is where The probability that the confidence interval contains p is c.

Notes • In section 5.5, you learned that a binomial can be approximated by the normal distribution if np  5 and nq  5. When and , the sampling distribution for is approximately normal with a mean of p = p and a standard error of

Guidelines: Constructing a Confidence Interval for a Population Proportion In words • ID the sample stats, n and x • Find the point estimate • Verify the sampling distribution of p(hat) can be approximated by the normal distribution. • Find the critical zc that corresponds to the given level of confidence, c. • Find the maximum error of estimate, E. • Find the left and the right endpoints and form the confidence interval. Is and is ? Use a standard normal table. Left endpoint: Right endpoint: Interval:

Ex. 2: Constructing a Confidence interval for p • Construct a 95% confidence interval for the proportion of American adults who say that their favorite sport is football. • SOLUTION: Form example 1, , So, . Using n = 883, you can verify that the sampling distribution of can be approximated by the normal distribution. and

Left Endpoint Right Endpoint Ex. 2: Constructing a Confidence interval for p Using zc = 1.96, the maximum error of estimate is: The 95% confidence interval is as follows: So, with 95% confidence, you can say that the proportion of adults who say that footbal is their favorite sport is between 39.7% and 46.3%.

Opinion Polls • The confidence level of 95% used in Example 2 is typical of opinion polls. The result; however, is usually not stated as a confidence interval. Instead the result of Example 2 would usually be stated as 43% with a margin of error of 3.3%.”

Ex. 3: Constructing a Confidence Interval for p • The graph shown below is from a survey of 935 adults. Construct a 99% confidence interval for the proportion of adults who think that airplanes are the safest mode of transportation.

Left Endpoint Right Endpoint Solution: So with 99% confidence, you can say that the proportion of adults who think that airplanes are the safest mode of transportation is between 40.8% and 49.2% • From the graph So, Using these values and the values n = 935 and zc = 2.575, the maximum error of estimate is: The 99% confidence inteval is as follows:

Increasing Sample Size to Increase Precision • One way to increase the precision of the confidence interval without decreasing the level of confidence is to increase the sample size.

Insight – why 0.5? • The reason for using 0.5 as values for p hat and q hat when no preliminary estimate is available is that these values yield a maximum value for the product • In other words, if you don’t estimate thevalues of p hat and q hat, you must pay the penalty of using a larger sample.

Ex. 4: Determining a Minimum Sample Size • You are running a political campaign and wish to estimate with 95% confidence, the proportion of registered voters, who will vote for your candidate. What is the minimum sample size needed if you are to accurately within 3% of the population proportion?

SOLUTION • Because you do not have a preliminary estimate for p, use and . Using zc = 1.96, and E = 0.03, you can solve for n. Because n is a decimal, round up to the nearest whole number. So, at least 1068 registered voters should be included in the sample.

Assignment due Friday BOC. Assignment: pp. 280-282 #1-22 all

6.3 Confidence Intervals for Population Proportions