Question

A political candidate testing the effectiveness of a new advertisement takes a simple random sample of 300 voters before the ad goes on air and another sample of 250 voters afterwards. 158 of the 300 voters in the "before" sample say they support his candidacy, and 140 voters in the "after" sample say they support him. If p,= proportion approving his candidacy in the after sample and P,= proportion approving his candidacy in the before sample, which of the following is the correct test statistic for H₀ : Pₐ- Pᵦ = 0?

(a) z = 0.56 – 0.527 /√0.56∙0.44/250 + 0.527∙0.473/300
(b) z = 0.56– 0.527 /√0.542∙0.458 1/250 + 1/300
(c) z = 0.56 – 0.527/√0.542∙0.458/550
(d) z = 0.56 – 0.527/√0.56∙0.44/250 + √0.527∙0.473/300

Answer 1

The correct test statistic for the hypothesis that the proportion of voters supporting a candidate before and after an advertisement is the same is found using the z-test for two proportions. The correct calculation involves the proportions before and after the ad, the pooled proportion, and the respective sample sizes. Option (a) z = 0.56 - 0.527 /√(0.56*0.44/250 + 0.527*0.473/300) is the correctly structured answer.

Step-by-step explanation:

The question pertains to finding the correct test statistic for the hypothesis test that the proportion of voters supporting a candidate before and after seeing an advertisement is the same. The correct test statistic can be found using the formula for the z-test of two proportions, which is:

z = (pʂ - pα) / √[p (1-p) (⅑/nα + ⅑/nʂ)]

where:

• pα = Proportion of approval before the advertisement
• pʂ = Proportion of approval after the advertisement
• p = Pooled proportion of approval
• nα = Number of voters before the ad
• nʂ = Number of voters after the ad

Here, pα = 158/300 = 0.527 and pʂ = 140/250 = 0.56. The pooled proportion (p) is calculated using both samples, (158+140)/(300+250) which is not directly used in the answer choices provided. However, the correct answer choice should have the structure of the formula above and not include a square root in the denominator of the fraction after the plus sign, therefore option (a) is the correct one:

z = 0.56 − 0.527 /√(0.56∙0.44/250 + 0.527∙0.473/300)