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. Determine the p-value or the critical value (depending on the method you prefer). Either way, be sure to show your work by drawing a sketch.
2. 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.
3. 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.
4. Would the results of the hypothesis test change at the 1% significance level? Explain.

Answer 1

The following cross tablulation have been provided. The row and column total have been calculated and they are shown below: Total Men Women Vape 92 31 123 Do not Vape 478 1402 924 Total 1016 509 1525 The expected values are computed in terms of row and column totals. In fact, the formula is E RxCwhere R; corresponds to the total sum of elements in row i, C; corresponds to the total sum T of elements in column 1, and T is the grand total. The table below shows the calculations to obtain the table with expected values: Total Expected Values Men Women 1016x 123 1525 509x123 1525 123 Vape 81.946 41.054 1016x1402 1525 509x1402 1525 Do not Vape 467.946 934.054 1402 Total 1016 509 1525

Based on the observed and expected values, the squared distances can be computed according to the following formula: (E O)2/E The table with squared distances is shown below: Squared Distances Men Women (92-81.946) (31-41.054)2 Vape 1.233 2.462 81.946 41.054 (478-467.946) (924-934.054)2 Do not Vape 0.216 0.108 934.054 467.946 Null and Alternative Hypotheses The following null and alternative hypotheses need to be tested Hn The two variables are independent H The two variables are dependent This corresponds to a Chi-Square test of independence Rejection Region Based on the information provided, the significance level is a 0.05 , the number of degrees of freedom is df 1) 1, so then the rejection region for this test is R {x2 : x2 (2 1) x (2 3.841

Test Statistics The Chi-Squared statistic is computed as follows: (Oij Eij )2 Erj 1.2330108+2.462+0.216 = 4.02 i=1 Decision about the null hypothesis Since it is observed that y2 = 4.02 > x = 3.841, it is then concluded that the null hypothesis is rejected Conclusion It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that the two variables are dependent, at the 0.05 significance level. The corresponding p-value for the test is p Pr(xi 0.045 4.02)

Chi-square 4.02, df 1 = Right-tail p-value is 0.04496 R command: pchisq(4.02, 1, lower.tail FALSE) 1-pchisq(4.02, 1) or + 0 2 6 7 1 4 LO

If α=0.01

Decision about the null hypothesis Since it is observed that y2 = 4.02 < x2 = 6.635, it is then concluded that the null hypothesis is not rejected Conclusion It is concluded that the null hypothesis Ho is not rejected. Therefore, there is NOT enough evidence to claim that the two variables are dependent, at the 0.01 significance level. The corresponding p-value for the test is p Pr(xi > 4.02) = 0.045