Part II. Confidence Intervals for Proportions The Central Limit Theorem also applies for binomial distributions, so we can make confidence intervals for proportions. We'll simulate 10,000 draws from a binomial with n = 47 and p = .5, e.g., counting the number of heads on 47 flips of a fair coin. Then we'll compute a confidence interval for the population proportion. Since we simulated the data, we can see how often the confidence intervals contain the true value. Open a New Data Table from the JMP Starter window under File and double-click on the header for Column 1 (double-click in the space above where it says Column 1). A dialog box will appear. Near the bottom, change the Number of rows from 0 to 10000 and hit OK to get 10000 rows of missing values. Right-click on the column header and choose Formula.... From Functions, scroll down and choose Random and Random Binomial. For n enter in 47, and for p enter in 0.5, and click on "OK". You should get 10000 draws of your binomial (with all values from 0 to 47) Here is what JMP looks like for n=20: Column 1 Filter X 1 Columns + -x x 749GX 52 Row Column 1 Numeric Transcendental Trigonometric Character Comparison Conditional Probability Discrete Probability Statistical Random Random Binomial (20 , 0.5 ) Date Time Row State Assignment Parametric Model Finance Constants C U. L N - Help Apply Cancel OK untitled 3 untitled 3 V Column 1 7 2 11 11 A 13 (1/1) 5 10untitled 3 untitled 3 Column 1 7 11 11 13 Columns (1/1) 10 D OO V O U A W N Column 1 + 10 10 11 12 10 JC 13 7 Rows All rows 10,000 14 11 Selected 15 10 O Let's compute the estimated probability, p, and put that in the next column. First create a new column by double-clicking the header area for the new column. Label this new column p- hat. Then right-click the header and bring up the Formula... dialog box. From the upper left Table Columns menu, click on Column 1, then click on ":" and type in 47 (which is the size of each binomial sample) and hit OK. Here is what JMP looks like for n=20: untitled 3 untitled 3 Column 1 p-hat 7 11 A W N - 11 13 p-hat Filter X 2 Columns + - X x 7 95X 5 Column 1 Row p-hat Numeric Transcendental Trigonometric Character Comparison Conditional Probability Discrete Probability Statistical Column 1 Random 20 Date Time Row State Assignment Parametric Model Finance Constants 1 - ONext we'll compute the margin of error and put it in the third column. Again Create a new column by double-clicking the header area for the new column. Label this new column E (for margin of error). . O . untitled 3 iuntitied 3 b 4 [3 El Column 1 p-hat E 1 7 0.35 2 11 0.55 3 11 0.55 4 13 0.65 ' 5 10 0.5 i] Columns (3/1) 6 8 0.4 0 A Column1 + 7 11 0.55 1 F"halt 1' a a 0.4 ' E 9 7 0.35 10 10 0.5 11 8 0.4 12 10 0.5 13 7 0.35 14 11 0.55 15 10 0.5 :l Raw: 16 9 0.45 Then right-click the header and bring up the Formula dialog box. The margin of error for a binomial is 13(1 13) n . E: 1.96 x So type in 1.96 (which should appear in the box in the formula area) and hit \"X\" then hit \"ny\" and then click on p- hat in the upper left to put it in the box inside the square root Sign. Then click on \"X\" and type in l and hit \"-\" and click on p-hat again. You should now have p-hat * (l - p-hat) inside of the square root sign. Now click on the grey box that goes around that whole expression (everything inside of the square root sign) and then click on - and type in 47 into the denominator. Click on OK. Again, here is what IMP looks like for n=20: .0. E Filler X 83mlumns ['3 'l' - x + '5' 'i/E % t= '9 94 '3' \\'I X '3 a AColumnt > Row ' p-hat b Numeric A E b Transcendental P Trigonometric D Character > Comparison > Conditional D Probability > Discrete Probability D Statistical > Random > Date Time h Row State D Assignment b Parametric Model > Finance Constants Finally, let's create a column of "1"'s and "0''s to see which 95% confidence intervals contain the true population proportion of .5. Create a new column and go to its formula dialog box. From Functions (grouped), choose Comparison and a