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 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 p 1 p and standard deviation p n 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 p 1 96 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 Approximate 95 Confidence Intervals That is p 1 p p n This is called the standard error of p and is denoted SE p Now the 95 confidence interval is p 1 96 SE p 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 95 99 99 9 1 645 1 960 2 576 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 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 Confiden ce Level 90 95 99 99 9 0 10 0 05 0 01 0 00 1 invNorm 2 1 645 1 960 2 576 3 291 Which Confidence Interval is Best Which is better A large margin of error wide interval or A small margin of error narrow interval Which is better A low level of confidence or 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 Hey why not a margin of error of 0 and a confidence level of 100 Which Confidence Interval is Best Which is better A smaller sample size or 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 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 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
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