Chi-square Examples

Chi-square Examples 1. Goodness of Fit Pennsylvania Lottery (Data from Utts/Heckard) The PA Daily Number is a state lottery where a 3 digit number is constructed by drawing a digit between 0 and 9 at random from each of 3 different containers. Assume a local official is concerned about recent draws because 4 draws in the past week started with the number 9. Focusing on the first container, he collects 500 draws. (Data below are the starting values for the actual lottery numbers drawn from July 19, 1999 to November 29, 2000.) If the draws from the container are random, what proportion of draws should be 9s? What hypotheses would you test to see if the container was drawing numbers between 0 and 9 at random? The following table summarizes the 500 draws obtained by the official. Number 0 1 2 3 4 5 6 7 8 9 Total Observed 47 50 55 46 53 39 55 55 44 56 500 Expected 50 50 50 50 50 50 50 50 500 (O-E)^2 25 16 9 121 25 25 16 36 - (O-E)^2/E 25/50 16/50 9/50 121/50 25/50 25/50 16/50 36/50 - What are the values of the missing expected counts? Are the assumptions for the test met? Compute the chi-square test statistic (part of this has been done for you). 2 =  (O − E ) 2 1 16 9 121 1 1 36 36 = ____ +____ + + + + + + + + = 6.04 E 2 50 50 50 2 2 50 50 Determine the df for the test statistic and then the p-value. What is your conclusion? 2. Homogeneity A survey in Newsweek in 1999 asked 747 randomly selected women “how satisfied are you with your overall appearance?” 4 possible responses were given. The women were also asked to provide their age. Results are shown in the table, with some expected counts in parentheses. Age/Satisfaction Very Somewhat ) 82 ( ) Not Too Not at all Total 10 (18.50) 4 (4.15) 141 Under 30 45 ( 30-49 73 (88.16) 168 (158.61) 47 (38.57) 6 (8.66) 294 Over 50 106 (93.56) 153 (168.32) 41 (40.93) 12 (9.19) 312 Total 224 403 98 22 747 What proportion of women surveyed were “somewhat” satisfied with their overall appearance? What proportion of women surveyed were over 50 years of age? If you wanted to know if the distribution of satisfaction was the same for all the age groups, this is treating age as a variable which denotes ________________________________. What hypotheses should you test? Check your assumptions. Fill in the missing parts of the test statistic computation. 2 =  (O − E ) 2 (10 − 18.50) 2 (12 − 9.19) 2 = ___________ +____________ + + ... + = 14.2782 E 18.50 9.19 What distribution will you use to determine the p-value? Determine the p-value. What is your conclusion at a .05 significance level? 3. Independence In 2002, the General Social Survey conducted by UChicago asked participants the following questions: 1. Do you favor or oppose the death penalty for persons convicted of murder? 2. Do you think the use of marijuana should be made legal or not? The results are displayed in the table, with most expected counts in parentheses Death Pen/Marijuana Legal Not Legal Total Favor 191 ( Oppose 104 (100.53) Total 295 ) 341 (337.53) 532 171 (174.47) 275 512 807 If you want to know if there is a relationship between death penalty and marijuana beliefs, what hypotheses should you test? What are the assumptions for your test? Do they check out? Finish the test statistic computation. 2 =  (O − E ) 2 (104 − 100.53) 2 (171 − 174.47) 2 = _____________ +______________ + + = .2864 E 100.53 174.47 Determine the appropriate df and p-value. Sketch the distribution you used to obtain the p-value. Interpret your p-value in context. What is your decision and conclusion at a .05 significance level? Differentiating between the Three Chi-square Tests For each scenario below, see if you can identify the appropriate chi-square test procedure. To help you, focus on determining what the variables are, the number of categorical variables, the number of populations or samples, and key words of interest. 1. A reader for the AP statistics examination wants to compare the distribution of AP statistics scores to the distribution of scores on the BC Calculus examination. The reader obtains scores from students who only took one of the two exams, not both. 2. The same reader wants to know if there is an association between gender and AP statistics scores. 3. A parking lot attendant wants to know if all 5 levels of the parking garage are used equally on a rainy day (the top level is not covered). You can assume all levels have the same number of parking spaces. 4. A study wants to investigate relationships between anger levels (high, med, and low) and whether or not a person suffers from chronic heart disease. They survey over 8000 people with normal blood pressure and administer an anger test. 5. Nationally, the top 10 causes of death for adults are known and reported each year by various bureaus. A bureau in the Northeast wants to compare its distribution of top causes with the given national distribution. 6. An elementary school cafeteria collects data on ice cream served one week at lunch for first, third, and fifth graders. The three ice cream flavors served were vanilla, chocolate, and strawberry. The number of each flavor served to each grade was recorded. The intent was just to get the students to look at their own data, but administrators realize they can also check to see if the three grades consumed ice cream flavors in the same proportions.