STAT 400 Lecture AL1 Spring 2015 Dalpiaz Examples for 9 1 Pearson s 2 Test for Goodness of Fit Based on Large n A random sample of size n is classified into k categories or cells k Yi n Let Y 1 Y 2 Y k denote the respective cell frequencies i 1 Denote the cell probabilities by p 1 p 2 p k k p i 0 1 H 0 p 1 p 10 p 2 p 20 p k p k 0 i 1 1 2 k Total Observed frequency O Y1 Y2 Yk n Probability under H 0 p 10 p 20 pk0 1 Expected frequency E under H 0 n p 10 n p 20 n pk 0 n Test Statistic Q k 1 2 k Y np i0 i n p i0 i 1 k O E 2 i i i 1 Ei O E 2 cells Rejection Region 2 Reject H 0 if Q k 1 d f k 1 number of cells 1 Pearson s 2 test is an approximate test that is valid only for large samples As a rule of thumb n should be large enough so that expected frequency of each cell is at least 5 E 1 Alex buys a package of Sour Jelly Beans On the package it says that 50 of all jelly beans are lemon 30 are cherry and 20 are lime When Alex opens the package he finds 15 lemon 9 cherry and 12 lime jelly beans Is there enough evidence to conclude that the true proportions are different from the ones listed on the package Use 0 05 a State H 0 and H 1 b Find the expected frequencies under H 0 c Calculate the values of the Q 2 test statistic d 2 Find the critical value e State your decision Reject H 0 or Do NOT Reject H 0 at 0 05 2 The financial manager in charge of accounts receivable department is concerned about the current economic slowdown because customers sometimes wait longer to pay their bills She wishes to check on this year s performance of the department by comparing the current outstanding accounts with records from the past few years Historical records show the following percentages in the respective classifications Age of Accounts Receivable Percent of Total Receivables Less than 30 days Between 30 and 60 days Between 60 and 90 days Over 90 days 50 25 15 10 To avoid the time required for a complete audit of the accounts receivable the financial manager chooses a random sample of 60 accounts and finds 24 18 15 and 3 accounts respectively in the above categories H 0 p 1 0 50 p 2 0 25 p 3 0 15 p 4 0 10 Perform the appropriate test at the 0 05 level of significance 3 The developers of a math proficiency exam to be used at Anytown State University believe that 60 of all incoming freshmen will be able to pass the exam In a random sample of 200 incoming freshmen 105 pass the exam Does this contradict the claim of the developers a Use 0 05 to perform a goodness of fit test H 0 p 1 0 60 p 2 0 40 b H 1 H 0 is not true Use 0 05 to perform a large sample test H 0 p 0 60 c vs vs H 1 p 0 60 Compare the test statistics Q 1 part a and Z part b Compare the critical values 02 05 1 d f part a and z 0 025 part b 4 It is claimed that X has a Poisson distribution and 120 observations of X are given in the frequency distribution x frequency 0 1 2 3 4 5 6 7 8 12 14 22 24 23 17 5 2 1 Use a chi square goodness of fit test to verify this claim
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