A state-by-state survey found that the proportion of adults who are smokers in Alabama and Missouri was 24.7% and 28.7%, respectively. Assume each has a sample size of 2000. At α = 0.01, can you support the claim that the proportion of smokers is lower in Alabama than in Missouri? Which hypothesis test should you use for this problem?
Question 35 options:
One sample z-test for the population mean.
two sample z-test for the population means.
One sample z-test for proportions.
two sample z-test for proportions.
one sample t-test for the population mean.
The correct hypothesis test to use for this problem is a two sample z-test for proportions.
Null hypothesis (H0): The proportion of smokers in Alabama is equal to the proportion of smokers in Missouri (p1 = p2).
Alternative hypothesis (Ha): The proportion of smokers in Alabama is lower than the proportion of smokers in Missouri (p1 < p2).
We will use a significance level of α = 0.01.
To perform the two sample z-test for proportions, we calculate the z statistic using the formula:
z = (p1 - p2) / sqrt(p̂(1 - p̂) * (1/n1 + 1/n2))
where p1 and p2 are the proportions of smokers in Alabama and Missouri, respectively, p̂ is the pooled proportion (p̂ = (x1 + x2) / (n1 + n2)), and n1 and n2 are the sample sizes in Alabama and Missouri, respectively.
Plugging in the given values:
p1 = 0.247
p2 = 0.287
n1 = 2000
n2 = 2000
we can calculate the z statistic. Let's assume the pooled proportion to be p̂ = (0.247*2000 + 0.287*2000) / (2000 + 2000) = 0.267.
z = (0.247 - 0.287) / sqrt(0.267 * (1 - 0.267) * (1/2000 + 1/2000))
= -0.04 / sqrt(0.133122 * 0.866878 * 0.001)
= -0.04 / sqrt(0.000040499)
≈ -3.91
Looking up the critical value for α = 0.01 in the z-table, we find it to be approximately -2.33.
Since the calculated z value (-3.91) is less than the critical value (-2.33), we can reject the null hypothesis and support the claim that the proportion of smokers is lower in Alabama than in Missouri.
Therefore, we can conclude that at α = 0.01, there is evidence to support the claim that the proportion of smokers is lower in Alabama than in Missouri.