Confidence Interval Estimation for a Population Proportion Lecture 33 Section 9.4 Mon, Nov 6, 2006

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. That is, we have no idea of how large the error may be. That is, we have no idea of how large the error may be.

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.

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

The Target Analogy Suppose a shooter hits within 4 rings (4 inches) of the bull’s eye 95% of the time. Suppose a shooter hits within 4 rings (4 inches) of the bull’s eye 95% of the time. Then each individual shot has a 95% chance of hitting within 4 inches. Then each individual shot has a 95% chance of hitting within 4 inches.

The Target Analogy

Now suppose we are shown where the shot hit, but we are not shown where the bull’s eye is. Now suppose we are shown where the shot hit, but we are not shown where the bull’s eye is. What is the probability that the bull’s eye is within 4 inches of that shot? What is the probability that the bull’s eye is within 4 inches of that shot?

The Target Analogy

Where is the bull’s eye?

The Target Analogy 4 inches

The Target Analogy 4 inches 95% chance that the bull’s eye is within this circle.

The Confidence Interval In a similar way, 95% of the sample proportions p ^ should lie within 1.96 standard deviations (  p^ ) of the parameter p. In a similar way, 95% of the sample proportions p ^ should lie within 1.96 standard deviations (  p^ ) of the parameter p.

The Confidence Interval p

p 1.96  p^

The Confidence Interval p 1.96  p^

The Confidence Interval p 1.96  p^

The Confidence Interval p 1.96  p^

The Confidence Interval p 1.96  p^

The Confidence Interval p 1.96  p^

The Confidence Interval Therefore, if we compute a single p ^, then we expect that there is a 95% chance that it lies within a distance 1.96  p^ of p. Therefore, if we compute a single p ^, then we expect that there is a 95% chance that it lies within a distance 1.96  p^ of p.

The Confidence Interval

Where is p? p^p^

The Confidence Interval 1.96  p^ p^p^

The Confidence Interval 1.96  p^ 95% chance that p is within this interval p^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. 585 – Study: Chronic Fatigue Common. Example 9.6, p. 585 – Study: Chronic Fatigue Common. Rework the problem supposing that 350 out of 3066 people reported that they suffer from chronic fatigue syndrome. Rework the problem supposing that 350 out of 3066 people reported that they suffer from chronic fatigue syndrome. How should we interpret the confidence interval? How should we interpret the confidence interval?

Standard Confidence Levels The standard confidence levels are 90%, 95%, 99%, and 99.9%. (See p. 588 and Table III, p. A-6.) The standard confidence levels are 90%, 95%, 99%, and 99.9%. (See p. 588 and Table III, p. A-6.) Confidence Levelz 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 given by the previous chart, or Is found in the normal table, or Is found in the normal table, or Is obtained using the invNorm function on the TI-83. Is obtained using the invNorm function on the TI-83.

Confidence Level Rework Example 9.6, p. 585, by computing a Rework Example 9.6, p. 585, by computing a 90% confidence interval. 90% confidence interval. 99% confidence interval. 99% confidence interval. Which one is widest? Which one is widest? In which one do we have the most confidence? In which one do we have the most confidence?

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. ConfidenceLevel  invNorm(  /2) 90% % % %

Which Confidence Interval is Best? Which is better? Which is better? A large margin of error (wide interval), or A large margin of error (wide interval), or A small margin of error (narrow interval). A small margin of error (narrow 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.

Which Confidence Interval is Best? Why not get a confidence interval that has a small margin of error and has a high level of confidence associated with it? Why not get a confidence interval that has a small margin of error and has a high level of confidence associated with it? Hey, why not a margin of error of 0 and a confidence level of 100%? Hey, why not a margin of error of 0 and a confidence level of 100%?

Which Confidence Interval is Best? Which is better? Which is better? A smaller sample size, or A smaller sample size, or A larger sample size. A larger sample size.

Which Confidence Interval is Best? A larger sample size is better only up to the point where its cost is not worth its benefit. A larger sample size is better only up to the point where its cost is not worth its benefit. That is why we settle for a certain margin of error and a confidence level of less than 100%. That is why we settle for a certain margin of error and a confidence level of less than 100%.

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 Rework Example 9.6, p. 585, using the TI-83. Rework Example 9.6, p. 585, using the TI-83.