Confidence Interval Estimation for a Population Proportion Lecture 33 Section 9.4 Tue, Mar 28, 2006

Point Estimates 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. 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. An interval estimate is more informative than a point estimate.

Interval Estimates 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.

Approximate 95% Confidence Intervals How do we find a 95% confidence interval for 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

The Target Analogy 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.

The Target Analogy

The Target Analogy

The Target Analogy

The Target Analogy

The Target Analogy

The Target Analogy

The Target Analogy 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?

The Target Analogy

The Target Analogy

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.

The Confidence Interval p

The Confidence Interval 1.96 p^ p

The Confidence Interval 1.96 p^ p

The Confidence Interval 1.96 p^ p

The Confidence Interval 1.96 p^ p

The Confidence Interval 1.96 p^ p

The Confidence Interval 1.96 p^ 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.

The Confidence Interval

The Confidence Interval

The Confidence Interval p^ Where is p?

The Confidence Interval 1.96 p^ p^

The Confidence Interval 1.96 p^ p^ 95% chance that p is within this interval

Approximate 95% Confidence Intervals Thus, the confidence interval is 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^.

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

Example 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. 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.) 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 where z Is given by the previous chart, or Is found in the normal table, or Is obtained using the invNorm function on the TI-83.

Confidence Level Rework Example 9.6, p. 585, by computing a 90% confidence interval. 99% confidence interval. Which one is widest? 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. That is,  is the probability that 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. Confidence Level  invNorm(/2) 90% 0.10 -1.645 95% 0.05 -1.960 99% 0.01 -2.576 99.9% 0.001 -3.291

Think About It Think About It, p. 586. Computing a confidence interval is a procedure that contains one step whose outcome is left to chance. (Which step?) Thus, the confidence interval itself is a random variable.

Interpretation See p. 587. “If we repeated this procedure over and over, yielding many 95% confidence intervals for p, we would expect that approximately 95% of these intervals would contain p and approximately 5% would not.”

Interpretation Compare this to shooting at the target, where the probability of hitting it is 95%. “If we shoot at the target over and over, yielding many bullet holes, we would expect that approximately 95% of these bullet holes would be in the target and approximately 5% would not.”

Interpretation On the other hand, if we see that a particular shot hit the target, then what are the chances that it hit the target? On the other hand, if we see that a particular shot missed the target, then what are the chances that it hit the target? So, for a shot that has already been fired, what is the probability that it hit the target?

Interpretation Therefore, if we are given a particular confidence interval, it either does or does not contain p. That is, the probability is either 0% or 100%, but we do not know which. Therefore, we should not talk about the probability that it contains p. It is the procedure that has a 95% of producing a confidence interval that contains p.

Which Confidence Interval is Best? Which is better? A wider confidence interval, or A narrower confidence interval. A low level of confidence, or A high level of confidence. A smaller sample, or A larger sample.

Which Confidence Interval is Best? What do we mean by “better”? 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. Press STAT. Select TESTS. Select 1-PropZInt.

TI-83 – Confidence Intervals 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.

TI-83 – Confidence Intervals 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?

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