A simple random sample of 785 full-time undergraduate college graduates found that 144 smoke and 641 do not smoke (based on data from the American Medical Association). Use a 5% significance level to test the claim that the rate of smoking among college graduates is equal to the 27% rate for the general population.
a. Identify the null and alternative hypotheses, writing them out using the symbols we used in class.
Null hypothesis (H0): The rate of smoking among college graduates is equal to 27%.
Alternative hypothesis (Ha): The rate of smoking among college graduates is not equal to 27%.
b. State the level of significance.
The level of significance is 5% (α = 0.05).
c. Calculate the test statistic, being sure to indicate if it is a z-score or a t-score.
To calculate the test statistic, we need to determine the standard error and the z-score. The formula for the standard error is:
Standard Error = √(p * (1-p) / n)
where p is the proportion of smokers in the sample and n is the sample size.
In this case, p = 144/785 = 0.183 and n = 785.
Standard Error = √(0.183 * (1-0.183) / 785) ≈ 0.014
The z-score can be calculated using the formula:
z = (p - P) / SE
where P is the proportion of smokers in the general population.
In this case, P = 0.27.
z = (0.183 - 0.27) / 0.014 ≈ -6.214
d. Calculate the p-value or find the critical number(s).
To calculate the p-value, we need to find the area under the standard normal curve to the left and right of the test statistic. Since the alternative hypothesis is two-tailed, we need to find the area in both tails.
The p-value can be calculated using a standard normal distribution table or a statistical software. In this case, the p-value is extremely small, close to 0.
e. State your decision to either reject the null hypothesis or fail to reject the null hypothesis.
Since the p-value is less than the significance level (0.05), we reject the null hypothesis.
f. Restate the decision in the context of the claim in the problem in clear, plain English.
Based on the sample data, there is sufficient evidence to conclude that the rate of smoking among college graduates is not equal to the 27% rate for the general population.