Question

Part A

Suppose you work for a political pollster during an election year. You are tasked with determining the projected winner of the November election. That is, you wish to determine if the number of votes for Candidate 1 is greater than the votes for Candidate 2. What are the hypotheses for this test?

	1) HO: μ1 < μ2 HA: μ1 ≥ μ2
	2) HO: μ1 ≥ μ2 HA: μ1 < μ2
	3) HO: μ1 ≤ μ2 HA: μ1 > μ2
	4) HO: μ1 = μ2 HA: μ1 ≠ μ2
	5) HO: μ1 > μ2 HA: μ1 ≤ μ2

Part B

Does the amount of hazardous material absorbed by the bodies of hazardous waste workers depend on gender? You want to test the hypotheses that the amount absorbed by men (group 1) is different from the amount absorbed by women (group 2). A random sample of 213 male workers and 166 female workers showed an average lead absorption in the blood of 7.85 (SD = 1.322) and 7.74 (SD = 0.999), respectively (measured in micrograms/deciliter). Assuming that the population standard deviations are the same, perform a two independent samples t-test on the hypotheses Null Hypothesis: μ₁ = μ₂, Alternative Hypothesis: μ₁ ≠ μ₂. What is the test statistic and p-value of this test?

	1) Test Statistic: 0.892, P-Value: 0.373
	2) Test Statistic: -0.892, P-Value: 0.373
	3) Test Statistic: 0.892, P-Value: 0.1865
	4) Test Statistic: 0.892, P-Value: 1.8135
	5) Test Statistic: 0.892, P-Value: 0.8135

Part C

You are interested in whether the average lifetime of Duracell AAA batteries is greater than the average lifetime of Energizer AAA batteries. You lay out your hypotheses as follows: Null Hypothesis: μ₁ ≤ μ₂, Alternative Hypothesis: μ₁ > μ₂. After running a two independent samples t-test, you see a p-value of 0.0442. What is the appropriate conclusion?

	1) We did not find enough evidence to say the average lifetime of Duracell AAA batteries is greater than the average lifetime of Energizer AAA batteries.
	2) The average lifetime of Duracell AAA batteries is less than or equal to the average lifetime of Energizer AAA batteries.
	3) The average lifetime of Duracell AAA batteries is significantly less than the average lifetime of Energizer AAA batteries.
	4) The average lifetime of Duracell AAA batteries is significantly greater than the average lifetime of Energizer AAA batteries.
	5) The average lifetime of Duracell AAA batteries is significantly different from the average lifetime of Energizer AAA batteries.

Part D

A new gasoline additive is supposed to make gas burn more cleanly and increase gas mileage in the process. Consumer Protection Anonymous conducted a mileage test to confirm this. They took a sample of their cars, filled it with regular gas, and drove it on I-94 until it was empty. They repeated the process using the same cars, but using the gas additive. Using the data they found, they performed a paired t-test with data calculated as (with additive - without additive) with the following hypotheses: Null Hypothesis: μ_D ≤ 0, Alternative Hypothesis: μ_D > 0. If they calculate a p-value of 0.0169 in the paired samples t-test, what is the appropriate conclusion?

	1) The average difference in gas mileage is significantly less than 0. The average gas mileage was higher without the additive.
	2) The average difference in gas mileage is less than or equal to 0.
	3) We did not find enough evidence to say there was a significantly positive average difference in gas mileage. The additive does not appear to be effective.
	4) The average difference in gas mileage is significantly larger than 0. The average gas mileage was higher with the additive.
	5) The average difference in gas mileage is significantly different from 0. There is a significant difference in gas mileage with and without the additive.

Answer 1

Solution: Part-A: We wish to determine if the number of votes for Candidate 1 is greater than the votes for Candidate 2. The hypotheses are: H:Hi <H2 HA: M1 > uz Hence, Option-(3) is the correct answer. Part-B: Men: ni = 213 X1 = 7.85 Si = 1.322 Women: n2 = 166 x2 = 7.74 S2 = 0.999 Ho:Hi = H2 Ha: M1 # Uz Degrees of Freedom = df = ny + n2 - 2 = 213 + 166 - 2 = 377 Pooled standard deviation = sp = [(n1 – 1)s? + (n2 – 1)s2]/(nı + n2 - 2) Sp = V[(213 – 1) 1.3222 + (166 - 1) 0.9992]/(213 + 166 - 2) Sp = 1.191

Test Statistic: (x1 - x2) t= 1 1 vn N (7.85 - 7.74) =t=- X + (1.191 x 17 V 21 =t= 0.892 From t-distribution table; area under the t-distribution to the right of (t = 0.892) and for (df = 377), is given by: 0.1866 The test is a two-tailed test. Hence, p-value = 0.18662 x 2 = 0.373 Hence, Option-(1) is the correct answer. Part-C: Ho: Mi suz HA:Hi > H2 P-value = 0.0442 Level of significance = a = 0.05 Since p-value is less than the significance level, reject the null hypothesis. Conclusion: The average lifetime of Duracell AAA batteries is significantly greater than the average lifetime of Energizer AAA batteries. Hence, Option-(4) is the correct answer. Part-D: Houp < 0 HA:MD > 0 P-value = 0.0169 Level of significance = a = 0.05 Since p-value is less than the significance level, reject the null hypothesis.

Conclusion: The average difference in gas mileage is significantly larger than 0. The average gas mileage was higher with the additive. Hence, Option-(4) is the correct answer.