Confidence Interval Estimation for a Population Proportion Lecture

Confidence Interval Estimation for a Population Proportion Lecture 33 Section 9. 4 Tue, Mar 22, 2005

Point Estimates Point estimate – A single value of the statistic used to estimate the parameter. n The problem with point estimates is that we have no idea how close we can expect them to be to the parameter. n That is, we have no idea of how large the error may be. n

Interval Estimates Interval estimate – an interval of numbers that has a stated probability (often 95%) of containing the parameter. n An interval estimate is more informative than a point estimate. n

Interval Estimates Confidence level – The probability that is associated with the interval. n If the confidence level is 95%, then the interval is called a 95% confidence interval. n

Approximate 95% Confidence Intervals How do we find a 95% confidence interval for p? n Begin with the sample size n and the sampling distribution of p^. n We know that the sampling distribution is normal with mean p and standard deviation n

Approximate 95% Confidence Intervals n Therefore… n n Approximately 95% of all values of p^ are within 2 standard deviations of p. For a single random p^, there is a 95% chance that it is within 2 standard deviations of p. Therefore… n There is a 95% chance that p is within 2 standard deviations of a single random p^.

Approximate 95% Confidence Intervals n Thus, the confidence interval is The trouble is, to know p^, we must know p. (See the formula for p^. ) n The best we can do is to use p^ in place of p to estimate p^. n

Approximate 95% Confidence Intervals n That is, n This is called the standard error of p^ and is denoted SE(p^). n Now the 95% confidence interval is

Example n Example 9. 6, p. 539. The answer is (0. 178, 0. 206). n That means that we are 95% confident, or sure, that p is somewhere between 0. 178 and 0. 206. n

Let’s Do It! n Let’s do it! 9. 5, p. 540 – When Do You Turn Off Your Cell Phone?

Confidence Intervals We are using the number 2 as a rough approximation for a 95% confidence interval. n We can get a more precise answer if we use the normal tables. n A 95% confidence interval cuts off the upper 2. 5% and the lower 2. 5%. n What values of z do that? n

Standard Confidence Levels n The standard confidence levels are 90%, 95%, 99%, and 99. 9%. (See p. 542. ) Confidence Level z 90% 1. 645 95% 1. 960 99% 2. 576 99. 9% 3. 291

The Confidence Interval n The confidence interval is given by the formula where z is given by the previous chart or is found in the normal table.

Confidence Level n Rework Let’s Do It! 9. 5, p. 540, by computing a 95% confidence interval. n 90% confidence interval. n 99% confidence interval. n Which one is widest? n Which one is best? n

Probability of Error We use the symbol to represent the probability that the confidence interval is in error. n That is, is the probability that p is not in the confidence interval. n In a 95% confidence interval, = 0. 05. n

Probability of Error Thus, the area in each tail is /2. n The value of z can be found by using the inv. Norm function on the TI-83. n For example, n 90% CI: =0. 10; inv. Norm(0. 05) = – 1. 645. n 95% CI: =0. 05; inv. Norm(0. 025) = – 1. 960. n 99% CI: =0. 01; inv. Norm(0. 005) = – 2. 576. n 99. 9% CI: =0. 001; inv. Norm(0. 0005) = – 3. 291. n

Values of z Confidence Level 90% 95% 99. 9% 0. 10 0. 05 0. 01 0. 001 inv. Norm( /2) -1. 645 -1. 960 -2. 576 -3. 291

Think About It Think about it, p. 542. n Which is better? n A wider confidence interval, or n A narrower confidence interval. n n Which is better? A low level of confidence, or n A high level of confidence. n

Think About It n Which is better? A smaller sample, or n A larger sample. n What do we mean by “better”? n Is it possible to increase the level of confidence and make the confidence narrower at the same time? n

TI-83 – Confidence Intervals The TI-83 will compute a confidence interval for a population proportion. n Press STAT. n Select TESTS. n Select 1 -Prop. ZInt. n

TI-83 – Confidence Intervals A display appears requesting information. n Enter x, the numerator of the sample proportion. n Enter n, the sample size. n Enter the confidence level, as a decimal. n Select Calculate and press ENTER. n

TI-83 – Confidence Intervals n A display appears with several items. The title “ 1 -Prop. ZInt. ” n The confidence interval, in interval notation. n The sample proportion p^. n The sample size. n n How would you find the margin of error?

TI-83 – Confidence Intervals n Rework Let’s Do It! 9. 5, p. 540, using the TI 83.