Confidence Interval Estimation for a Population Proportion Lecture 31 Section 9.4 Wed, Nov 17, 2004

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

Interval Estimates Interval estimate – an interval of numbers that has a stated probability (often 95%) of containing the parameter. Interval estimate – an interval of numbers that has a stated probability (often 95%) of containing the parameter. An interval estimate is more informative than a point estimate. An interval estimate is more informative than a point estimate. Consider again the example of the 100,000 Iraqi civilian “deaths,” plus or minus 94,000. Consider again the example of the 100,000 Iraqi civilian “deaths,” plus or minus 94,000.

Interval Estimates Confidence level – The probability that is associated with the interval. Confidence level – The probability that is associated with the interval. If the confidence level is 95%, then the interval is called a 95% confidence interval. 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? How do we find a 95% confidence interval for p? Begin with the sample size n and the sampling distribution of p^. Begin with the sample size n and the sampling distribution of p^. We know that the sampling distribution is normal with mean p and standard deviation We know that the sampling distribution is normal with mean p and standard deviation

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

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

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

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

Let’s Do It! Let’s do it! 9.5, p. 540 – When Do You Turn Off Your Cell Phone? 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 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. 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%. A 95% confidence interval cuts off the upper 2.5% and the lower 2.5%. What values of z do that? What values of z do that?

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

The Confidence Interval The confidence interval is given by the formula 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 Rework Let’s Do It! 9.5, p. 540, by computing a Rework Let’s Do It! 9.5, p. 540, by computing a 90% confidence interval. 90% confidence interval. 99% confidence interval. 99% confidence interval. 80% confidence interval. 80% confidence interval.

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

Probability of Error Thus, the area in each tail is  /2. Thus, the area in each tail is  /2. The value of z can be found by using the invNorm function on the TI-83. The value of z can be found by using the invNorm function on the TI-83. For example, For example, 90% CI:  =0.10; invNorm(0.05) = – % CI:  =0.10; invNorm(0.05) = – % CI:  =0.05; invNorm(0.025) = – % CI:  =0.05; invNorm(0.025) = – % CI:  =0.01; invNorm(0.005) = – % CI:  =0.01; invNorm(0.005) = – % CI:  =0.001; invNorm(0.0005) = – % CI:  =0.001; invNorm(0.0005) = –3.291.

Values of z Confidence Level  invNorm(  /2) 90% % % %

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

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

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

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

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

TI-83 – Confidence Intervals Work Let’s Do It! 9.5, p. 540, using the TI-83. Work Let’s Do It! 9.5, p. 540, using the TI-83.