Question

Across the states, 12.2% of all families (urban, rural, and suburban) send their children to private schools. A researcher believes that the percentage of suburban families who send their children to private schools is higher than the average for all families.

a) Formulate the hypotheses and rejection rule for this problem.
b) A nationwide sample of 1,000 suburban families showed that 162 send their children to private schools. Obtain the p-value.
c) At α = .05, would you conclude that the percentage of suburban families sending their children to private schools is higher than the percentage of all families? Explain.

Answer 1

a) The hypotheses are H0: The percentage of suburban families sending their children to private schools is not higher than the percentage of all families, and Ha: The percentage of suburban families sending their children to private schools is higher than the percentage of all families.

b) The p-value is approximately 0.0001.

c) Yes, at α = .05, we would conclude that the percentage of suburban families sending their children to private schools is higher than the percentage of all families.

Step-by-step explanation:

a) H0: The percentage of suburban families sending their children to private schools is not higher than the percentage of all families.

Ha: The percentage of suburban families sending their children to private schools is higher than the percentage of all families.

Rejection rule: Reject H0 if the test statistic is greater than the critical value from the chi-square distribution at the given significance level.

b) To find the p-value, we need to perform a hypothesis test using the given sample data. The test statistic is calculated as:

X^2 = (n * (P - P0)^2) / P0

Where n is the sample size, P is the observed proportion, and P0 is the hypothesized proportion.

In this case, n = 1000, P = 162/1000 = 0.162, and P0 = 0.122.

Substituting the values, we get:

X^2 = (1000 * (0.162 - 0.122)^2) / 0.122 ≈ 14.52

The degrees of freedom for this test is 1 (since we are comparing one proportion to a known proportion), and the critical value at α = 0.05 is approximately 3.841.

The p-value is the probability of obtaining a test statistic as extreme as the one calculated or more extreme, assuming H0 is true. Using a chi-square distribution with 1 degree of freedom, we can look up the p-value associated with X^2 = 14.52, which is approximately 0.0001.

c) Since the p-value (0.0001) is less than the significance level α (0.05), we reject the null hypothesis. This means that there is evidence to suggest that the percentage of suburban families sending their children to private schools is higher than the percentage of all families.