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?
To determine if the proportion of smokers is lower in Alabama than in Missouri, we can use a hypothesis test for comparing two proportions.
The null and alternative hypothesis for this problem are as follows:
Null Hypothesis (H0): The proportion of smokers in Alabama is not lower than in Missouri. P1 ≥ P2
Alternative Hypothesis (H1): The proportion of smokers in Alabama is lower than in Missouri. P1 < P2
To determine if we can support the claim that the proportion of smokers is lower in Alabama, we need to calculate the test statistic and compare it to the critical value.
The test statistic for comparing two proportions is given by:
z = (p1 - p2) / sqrt(p * (1 - p) * (1/n1 + 1/n2))
Where:
p1 = proportion of smokers in Alabama = 0.247
p2 = proportion of smokers in Missouri = 0.287
n1 = sample size for Alabama = 2000
n2 = sample size for Missouri = 2000
p = pooled proportion
The pooled proportion is calculated as follows:
p = (x1 + x2) / (n1 + n2)
Where:
x1 = number of smokers in Alabama = p1 * n1
x2 = number of smokers in Missouri = p2 * n2
Let's calculate the test statistic:
x1 = 0.247 * 2000 = 494
x2 = 0.287 * 2000 = 574
p = (494 + 574) / (2000 + 2000) = 0.281
z = (0.247 - 0.287) / sqrt(0.281 * (1 - 0.281) * (1/2000 + 1/2000)) ≈ -2.388
To determine if we can support the claim that the proportion of smokers is lower in Alabama, we compare the test statistic (-2.388) to the critical value at α = 0.01.
Since α = 0.01, it is a one-tailed test.
Using a standard normal distribution table or software, we find that the critical value for a one-tailed test at α = 0.01 is approximately -2.326.
Since -2.388 < -2.326, the test statistic falls in the critical region.
Therefore, we can reject the null hypothesis and support the claim that the proportion of smokers is lower in Alabama than in Missouri.