Chapter 9 Estimating a Population Proportion Created by Kathy Fritz

Selecting an Estimator

What makes a statistic a good estimator of a population characteristic?

1.Choose a statistic that is unbiased

Point Estimator Definition: An ideal point estimator will have no bias and low variability. Since variability is almost always present when calculating statistics from different samples, we must extend our thinking about estimating parameters to include an acknowledgement that repeated sampling could yield different results.

In a review of ALL criminal cases heard by the Supreme Courts of 11 states from 2000 to 2004, 391 of the 1488 cases were decided in favor of the defendant. Let p be the proportion of all cases reviewed that decided in favor of the defendant.

Estimating a Population Proportion Margin of Error

The margin of error of a statistic is the

If a variable has a standard normal distribution, about 95% of the time the value of variable will be between and Central Area = 0.95 Upper tail area =.025Lower tail area =.025 0

Central Area = 0.95 Upper tail area =.025Lower tail area =.025 p

Margin of Error for Estimating a Population Proportion p Appropriate when the following conditions are met

Margin of Error for Estimating a Population Proportion p Continued... When these conditions are met Interpretation of margin of error It would be unusual for the sample proportion to differ from the actual value of the population proportion by more than the margin of error. For 95% of all random samples, the estimation error will be less than the margin of error.

Check conditions:

Compute margin of error Interpretation

A Large Sample Confidence Interval for a Population Proportion Confidence Interval Confidence Level

The Idea of a Confidence Interval estimate ± margin of error Definition: We usually choose a confidence level of 90% or higher because we want to be quite sure of our conclusions. The most common confidence level is 95%.

Developing a Confidence Interval p

Confidence Intervals A confidence interval (CI)

Confidence level If this method was used to generate an interval estimate over and over again from different random samples, in the long run 95% of the resulting intervals would include the actual value of the characteristic being estimated.

The diagram to the right is % confidence intervals for p computed from 100 different random samples. Note that the ones with asterisks do not capture p. If we were to compute 100 more confidence intervals for p from 100 different random samples, would we get the same results?

Other Confidence Levels Suppose we wanted to create confidence intervals with a 90% confidence level... Suppose we wanted to create confidence intervals with a 99% confidence level...

The Large-Sample Confidence Interval for p Appropriate when the following conditions are met:

The Large-Sample Confidence Interval for p Continued... When these conditions are met, a confidence interval for the population proportion is

The Large-Sample Confidence Interval for p Continued... Interpretation of Confidence Interval Interpretation of Confidence Level

Interpreting Confidence Levels and Confidence Intervals Interpreting Confidence Level and Confidence Intervals

Interpreting Confidence Levels and Confidence Intervals The confidence level tells us how likely it is that the method we are using will produce an interval that captures the population parameter if we use it many times. Instead, the confidence interval gives us a set of plausible values for the parameter. We interpret confidence levels and confidence intervals in much the same way whether we are estimating a population mean, proportion, or some other parameter. The confidence level does not tell us the chance that a particular confidence interval captures the population parameter.

Recall from Chapter 7... Four Key Questions: Q Estimate or hypothesis testing? SSample data or experimental data? TOne variable or two? Categorical or numerical? NHow many samples or treatments? E (Estimate) – Explain what population characteristic you plan to estimate. M (Method) – Select a method using QSTN C (Check) – Verify that the conditions are met C (Calculate) – Perform the necessary calculations C (Communicate) – Interpret the confidence interval 5 Steps:

Of 1100 drivers surveyed, 990 admitted to careless or aggressive driving during the previous 6 months. Assuming that it is reasonable to regard this sample of 1100 as representative of the population of drivers, compute a 90% confidence interval to estimate p, the proportion of all drivers who have engaged in careless or aggressive driving in the last 6 months. Step 1 (E): The proportion of drivers who have engaged in careless or aggressive driving during the last 6 months, p, will be estimated. Step 2 (M): Because the answers to the four key questions are Q: estimation, S: sample data, T: one categorical variable, N: one sample, a confidence interval for a population proportion will be considered.

Careless or Aggressive Driving Continued... Step 3 (C): There are two conditions that need to be met for the confidence interval of this section to be appropriate. Step 4 (C): Calculate the interval 1. You do not know how the sample was selected. In order to proceed, you MUST assume that the sample was representative of the population. 2. Sample size is large enough because

Careless or Aggressive Driving Continued... Step 5 (C): Communicate results Interpret Confidence Interval: Interpret Confidence level:

Three Things that Affect the Width of a Confidence Interval

An Alternative to the Large- Sample z Interval Even when the sample size conditions are met, sometimes the actual confidence level associated with the method may be noticeably different from the reported confidence level.

Choosing a Sample Size to Achieve a Desired Margin of Error

Choosing a Sample Size

Why is the conservative estimate for p = 0.5? 0.1(0.9) = (0.8) = (0.7) = (0.6) = (0.5) = 0.25 By using 0.5 for p, we are using the largest possible value for p(1 – p) in our calculations.

Researchers have found biochemical markers of cancers in the exhaled breath of cancer patients, but chemical analysis of breath specimens has not yet proven effective in diagnosing cancer. How many different breath specimens should be used if you want to estimate the long-run proportion of correct identifications for this dog with a margin of error of 0.10? A study is to be performed to investigate whether a dog can be trained to identify the presence or absence of cancer by sniffing breath specimens.

Avoid These Common Mistakes

If a 90% confidence interval for p, the proportion of students at a particular college who own a computer, is (0.56, 0.78), you might say “You can be 90% confident that between 56% and 78% of the students at this college own a computer.” Interpretation of interval You have used a method to produce this estimate that is successful in capturing the actual population proportion about 90% of the time. Interpretation of confidence level Don’t get these two statements confused!

Avoid These Common Mistakes 1.

Avoid These Common Mistakes 2.

Avoid These Common Mistakes 3.

Avoid These Common Mistakes 4.

Avoid These Common Mistakes 5.