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 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 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 Which is better A low level of confidence or A high level of confidence Which is better 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
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