Question

Gallup recently conducted a survey about the proportion of Americans who use e-cigarettes (vape). The survey data was collected via random-digit-dial telephone interviews conducted July 1-12, 2019 with a random sample of 1525 adults living in all 50 U.S. states with a minimum quota of 70% cellphone respondents and 30% landline respondents, with additional minimum quotas by time zone within region. The following table summarizes their results for men and women:

	Men	Women	Total
Vape	92	31	123
Do Not Vape	924	478	1402
Total	1016	509	1525

Problem 2. Does the data suggest that a greater proportion of men vape compared to women? Conduct a hypothesis test to at the 5% significance level by completing the following steps:

1. Using either the p-value or critical value method, make a decision about the null hypothesis. Be sure to show how your decision was made.
2. Based on the results of the hypothesis test, does it appear that “use of e-cigarettes” and “gender” are independent variables? In other words, is it equally likely for a man to vape as it is for a woman to vape? Explain.
3. Would the results of the hypothesis test change at the 1% significance level? Explain.

Answer 1

Solution Step 1: Frame hypothesis H: There is no significant difference in proportion of men vape compared to women. H: There is significant difference in proportion of men vape compared to women. Step 2: level of significance a = 0.05 (given) also for (0.01) step 3: Formula (critical value approach) chi square test (Independence of Attribute) x² = 310-) ~ xr1104,df 0 - observed frequency E-expected frequency r- no of rows C- no of columns df - degrees of freedom step 4: forming contegency table E
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123x1016- 81.9 Given table: Men Women Total Vape 92 31 123 Do not 924 478 1402 Vape Total 1016 509 1525 To form contegency table We use the formula to find the entries Expected frequency = row total xcolumn total grand total 1st row 1st column = 123 1525 2nd row 2nd column = 1402 1525 Contegency table: Men Women Total Vape 82 41 123 Do not 934 468 1402 Vape Total 1016 509 1525 Write the total value as it as from the given table, write the calculated value and fond other value by subtracting the total value by calculated value (i.e 123 - 82 = 41 and 1402 - 468 = 934) 1402509 467.9
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E 0-E (O-E) (O-E)? 92 924 82 31 478 41 468 61 446 41 466 3721 3721 120 198,916 416 1681 41 217,156 464 [ = 1041 934 x==105%* ~ xm-1961, df ~ x*2-12-1) df ~ xdf at (0.05) ~3.841 x = 1041 ~ 3.841 Step 5: Inference 1041 > 3.841 Calculated value of x² = 1041 is greater than table value ~ 3.841. hence we reject H.
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Step 6: Conclusion Therefore we accept H,. There is significant difference in proportion of men vape compared to women. Answers to the questions followed: Based on the results of the hypothesis test, we say that use of cigarettes and gender are independent variables No it is not equally likely as there is significant difference in proportion of men compared to women. The results of hypothesis does not change at 1% level of significance ~ xudf at (0.01) is 6.635 which is also less than 1041 calculated x value i.e 1041 > 6.635
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