Final answer:
To test the claim that the proportion of Americans who believe in space aliens has changed, we can use a one-proportion hypothesis test. The null hypothesis assumes that the proportion is still 24%, and the alternative hypothesis assumes that the proportion has changed. Using a significance level of 0.02, we calculate the test statistic and p-value to determine if we can reject the null hypothesis.
Step-by-step explanation:
To test the claim that the proportion of Americans who believe in space aliens has changed, we can use a one-proportion hypothesis test. The null hypothesis (H0) assumes that the proportion is still 24%, and the alternative hypothesis (Ha) assumes that the proportion has changed. We will use a significance level of 0.02.
The test statistic for a one-proportion hypothesis test is calculated using the formula: z = (p - P) / sqrt((P * (1 - P)) / n), where p is the sample proportion, P is the hypothesized proportion, and n is the sample size.
In this case, the sample proportion is 16%, the hypothesized proportion is 24%, and the sample size is 200. Using these values, we can calculate the test statistic:
z = (0.16 - 0.24) / sqrt((0.24 * (1 - 0.24)) / 200) = -3.86
Next, we need to find the p-value associated with this test statistic. The p-value represents the probability of getting a test statistic as extreme as the one we observed, assuming the null hypothesis is true.
Using a standard normal distribution table or a calculator, we can find that the p-value for a test statistic of -3.86 is approximately 0.0001.
Since the p-value (0.0001) is less than the significance level (0.02), we can reject the null hypothesis. This provides evidence to support the claim that the proportion of Americans who believe in space aliens has changed.
Keywords: one-proportion hypothesis test, significance level, null hypothesis, alternative hypothesis, sample proportion, hypothesized proportion, sample size, test statistic, p-value