Question

Gallup recently conducted a survey about the proportion of Americans who use e-cigarettes (vape). The following example is based on hypothetical data that is meant to be similar to the data collected by Gallup.

The following table considers survey data about annual household income and whether or not a person vapes.

Problem 2. We now wish to decide if there “use of e-cigarettes” and “income category” are dependent. To assist with this process, the table from before has been augmented with most of the expected frequencies (listed in parentheses): <$35,000 $35,000-$99,999 $100,000+ Total Vape 47 (34) 57 (58) 19 (31) 123 Do Not Vape 381 (394) 659 (???) 362 (???) 1402 Total 428 716 381 1525 Find the missing two expected frequencies, labeled as (???). For credit, be sure to show your work. From a brief analysis of the frequencies, do you believe that there is convincing evidence of a dependence relationship between income category and whether or not a person vapes? Explain. Based on the given information, the test statistic computes to be 10.47. Based on this, what is the conclusion of the hypothesis test at the 0.05 significance level? Be sure to justify your answer by using either the p-value or critical value method.

Answer 1

2.

Given table data is as below MATRIX col1 col2 col3 TOTALS row 1 34 58 31 123 row 2 394 658 350 1402 TOTALS 428 716 381 N = 1525 ------------------------------------------------------------------ calculation formula for E table matrix E-TABLE col1 col2 col3 row 1 row1*col1/N row1*col2/N row1*col3/N row 2 row2*col1/N row2*col2/N row2*col3/N ------------------------------------------------------------------ expected frequencies calculated by applying E - table matrix formula E-TABLE col1 col2 col3 row 1 34.521 57.75 30.73 row 2 393.479 658.25 350.27 ------------------------------------------------------------------ calculate chisquare test statistic using given observed frequencies, calculated expected frequencies from above Oi Ei Oi-Ei (Oi-Ei)^2 (Oi-Ei)^2/Ei 34 34.521 -0.521 0.271 0.008 58 57.75 0.25 0.063 0.001 31 30.73 0.27 0.073 0.002 394 393.479 0.521 0.271 0.001 658 658.25 -0.25 0.063 0 350 350.27 -0.27 0.073 0 ᴪ^2 o = 0.012 ------------------------------------------------------------------ set up null vs alternative as null, Ho: no relation b/w X and Y OR X and Y are independent alternative, H1: exists a relation b/w X and Y OR X and Y are dependent level of significance, α = 0.05 from standard normal table, chi square value at right tailed, ᴪ^2 α/2 =5.991 since our test is right tailed,reject Ho when ᴪ^2 o > 5.991 we use test statistic ᴪ^2 o = Σ(Oi-Ei)^2/Ei from the table , ᴪ^2 o = 0.012 critical value the value of |ᴪ^2 α| at los 0.05 with d.f (r-1)(c-1)= ( 2 -1 ) * ( 3 - 1 ) = 1 * 2 = 2 is 5.991 we got | ᴪ^2| =0.012 & | ᴪ^2 α | =5.991 make decision hence value of | ᴪ^2 o | < | ᴪ^2 α | and here we do not reject Ho ᴪ^2 p_value =0.994 ANSWERS --------------- a. null, Ho: no relation b/w X and Y OR X and Y are independent alternative, H1: exists a relation b/w X and Y OR X and Y are dependent c. test statistic: 0.012 critical value: 5.991 p-value:0.994 decision: do not reject Ho

b.

we do not have enough evidence to support the claim that there is convincing evidence of a dependence relationship between income category.