Confidence Interval Estimation for a Population Proportion Lecture 43 Section 9.4 Fri, Apr 9, 2004

Point Estimates The statistic p^ gives us a point estimate. A point estimate is a single number that estimates the parameter. The problem with point estimates is that we have no idea how close we can expect it to be to the parameter.

Interval Estimates A more informative estimate is an interval estimate. An interval estimate is an interval of numbers that has a certain probability of containing the parameter. The probability associated with the interval is called the confidence level.

Interval Estimates If the confidence level is 95%, then the interval is called a 95% confidence interval.

Approximate 95% Confidence Intervals How do we find a 95% confidence interval for p? Begin with the sample size and the sampling distribution of p^. We know that the sampling distribution is normal with mean p and standard deviation p^ = (p(1 – p)/n).

Approximate 95% Confidence Intervals Therefore, approximately 95% of the time, p^ is within 2 standard deviations of p. That is, for a single random p^, there is a 95% probability that it is within 2 standard deviations of p. Therefore, there is a 95% chance that p is within 2 standard deviations of p^ (for a single random p^).

Approximate 95% Confidence Intervals Thus, the confidence interval is p^  2p^ . The trouble is, to know p^, we must know p. The best we can do is to use p^ in place of p to estimate p^.

Approximate 95% Confidence Intervals That is, p^  (p^(1 – p^)/n). This is called the standard error of p^ and is denoted SE(p^). Now the 95% confidence interval is p^  2SE(p^).

Example See Example 9.6, p. 539. The answer is (0.178, 0.206). That means that we are 95% confident, or sure, that p is somewhere between 0.178 and 0.206.

Let’s Do It! 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. We can get a more precise answer if we use the normal tables. A 95% confidence interval cuts off the upper 2.5% and the lower 2.5%. What value of z does that?

Standard Confidence Levels The standard confidence levels are 90%, 95%, 99%, and 99.9%. Confidence Level z 90% 1.645 95% 1.960 99% 2.576 99.9% 3.291

The Confidence Interval The confidence interval is given by the formula p^  zSE(p^). where z is given by the previous chart or is found in the normal table.

Probability of Error We use the symbol  to represent the probability that the confidence interval is in error. That is, p is not in the confidence interval. In a 95% confidence interval,  = 0.05.

Probability of Error Thus, the area in each tail is /2. The value of z can be found by using the invNorm function on the TI-83. For example, invNorm(0.05) = –1.645. invNorm(0.025) = –1.960. invNorm(0.005) = –2.576. invNorm(0.0005) = –3.291.

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

Think About It Which is better? A smaller sample. A larger sample. Is it possible to increase the level of confidence and make the confidence narrower at the same time?

Confidence Intervals on the TI-83 The TI-83 will compute a confidence interval for a population proportion. Press STAT. Select TESTS. Select 1-PropZInt.

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

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

Assignment To be posted later today.