Confidence Intervals for Proportions Robb T Koether Homework Review Other Confidence Levels Confidence Intervals in the TI 83 Which Confidence Interval is Best Assignment Confidence Intervals for Proportions Lecture 32 Sections 9 4 Robb T Koether Hampden Sydney College Tue Oct 27 2009 Outline Confidence Intervals for Proportions Robb T Koether Homework Review 1 Homework Review 2 Other Confidence Levels 3 Confidence Intervals in the TI 83 4 Which Confidence Interval is Best 5 Assignment Other Confidence Levels Confidence Intervals in the TI 83 Which Confidence Interval is Best Assignment Outline Confidence Intervals for Proportions Robb T Koether Homework Review 1 Homework Review 2 Other Confidence Levels 3 Confidence Intervals in the TI 83 4 Which Confidence Interval is Best 5 Assignment Other Confidence Levels Confidence Intervals in the TI 83 Which Confidence Interval is Best Assignment Homework Review Confidence Intervals for Proportions Robb T Koether Homework Review Other Confidence Levels Confidence Intervals in the TI 83 Which Confidence Interval is Best Assignment Exercise 9 20 page 595 A simple random sample of 400 voters is taken after they have just voted yes or no on a certain proposal in a small city Of the 400 polled 36 voted yes and 64 voted no The total number of voters in the election is 40 000 a Give a 95 confidence interval for the proportion of all voters in the election who voted yes b Interpret your interval and explain what the 95 confidence level means c How would your interval change if the total number of voters in the election were 400 000 instead of 40 000 d How would your interval change if 4000 voters were polled instead of 400 and 36 voted yes Homework Review Confidence Intervals for Proportions Robb T Koether Homework Review Other Confidence Levels Confidence Intervals in the TI 83 Which Confidence Interval is Best Assignment Solution a We are told that n 400 and p 0 36 For a 95 confidence interval we should use the coefficient 1 96 The confidence interval is r r p 1 p 0 36 0 64 0 36 1 96 p 1 96 n 400 0 36 1 96 0 024 0 36 0 047 Homework Review Confidence Intervals for Proportions Robb T Koether Homework Review Other Confidence Levels Confidence Intervals in the TI 83 Which Confidence Interval is Best Assignment Solution b For the standard interpretation see page 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 More briefly 95 of all such intervals contain p and 5 do not Homework Review Confidence Intervals for Proportions Robb T Koether Homework Review Other Confidence Levels Confidence Intervals in the TI 83 Which Confidence Interval is Best Assignment Solution c If the population size were N 400 000 instead of N 40 000 it would not change the confidence interval at all The parameter N does not appear anywhere in the formula d If the sample size were n 4000 instead of n 400 and p were still 0 36 the confidence interval would be narrower That is because n appears in the denominator A larger n gives a smaller margin of error Outline Confidence Intervals for Proportions Robb T Koether Homework Review 1 Homework Review 2 Other Confidence Levels 3 Confidence Intervals in the TI 83 4 Which Confidence Interval is Best 5 Assignment Other Confidence Levels Confidence Intervals in the TI 83 Which Confidence Interval is Best Assignment Standard Confidence Levels Confidence Intervals for Proportions Robb T Koether Homework Review Other Confidence Levels Confidence Intervals in the TI 83 Which Confidence Interval is Best Assignment The most common confidence levels are 90 95 99 and 99 9 Confidence Level z 90 1 645 95 1 960 99 2 576 99 9 3 291 The Confidence Interval Confidence Intervals for Proportions Robb T Koether Homework Review Other Confidence Levels Confidence Intervals in the TI 83 Which Confidence Interval is Best Assignment The confidence interval is given by the formula r p 1 p p z n where z is obtained using the invNorm function on the TI 83 and depends on the confidence level Probability of Error Confidence Intervals for Proportions Robb T Koether Homework Review Other Confidence Levels Confidence Intervals in the TI 83 Which Confidence Interval is Best Assignment 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 The area in each tail is 2 Probability of Error Confidence Intervals for Proportions Robb T Koether Let zA denote the value of z that cuts off a lower tail of area A Homework Review For example z0 05 is the 5th percentile of Z Other Confidence Levels The book uses z but it is often better to specify the value Confidence Intervals in the TI 83 The confidence interval requires z 2 Which Confidence Interval is Best The confidence interval is Assignment To find z 2 we should compute invNorm 2 r p z 2 p 1 p n Confidence Level Confidence Intervals for Proportions Robb T Koether Homework Review Other Confidence Levels Confidence Intervals in the TI 83 Which Confidence Interval is Best Assignment Compute new confidence intervals for Exercise 9 20 using these confidence levels 90 confidence interval 99 confidence interval 99 9 confidence interval Outline Confidence Intervals for Proportions Robb T Koether Homework Review 1 Homework Review 2 Other Confidence Levels 3 Confidence Intervals in the TI 83 4 Which Confidence Interval is Best 5 Assignment Other Confidence Levels Confidence Intervals in the TI 83 Which Confidence Interval is Best Assignment TI 83 Confidence Intervals Confidence Intervals for Proportions Robb T Koether Homework Review Other Confidence Levels Confidence Intervals in the TI 83 TI 83 Confidence intervals for p Press STAT Select TESTS Select 1 PropZInt A display appears requesting information Enter x the numerator of the sample proportion Which Confidence Interval is Best Enter n the sample size Assignment Select Calculate and press ENTER Enter the confidence level as a decimal TI 83 Confidence Intervals Confidence Intervals for Proportions Robb T Koether Homework Review Other Confidence Levels Confidence Intervals in the TI 83 Which Confidence Interval is Best Assignment TI 83 Confidence intervals for p A display appears with several items The title 1 PropZInt The confidence interval in interval notation The sample proportion p The sample size
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