Based on the z-test and P-value, we fail to reject the null hypothesis. There is no statistically significant difference in average self-image scores between Emerson County recipients and the expected population at the 0.02 level of significance.
Question 1: Null and Alternative Hypotheses
Null Hypothesis (H0):The average self-image score for Emerson County public-assistance recipients is 65 (same as the expected population mean).
Alternative Hypothesis (H1): The average self-image score for Emerson County public-assistance recipients is different from 65 (either higher or lower).
Question 2: Test Statistic and P-value
1. Test Statistic: We use a one-sample z-test since the sample size is large enough (>30), the population standard deviation is unknown (estimated by the sample standard deviation), and we are comparing the sample mean to a known population mean.
2. Calculating the z-score:z = (62.1 - 65) / 5.83 ≈ -0.49
3. P-value:Using a z-table or statistical software, the P-value for a two-tailed test with z = -0.49 is approximately 0.624.
Question 3: Decision and Interpretation
1. Decision Rule: Since the P-value (0.624) is greater than the significance level (0.02), we fail to reject the null hypothesis.
2. Interpretation: At the 0.02 level of significance, there is not sufficient evidence to conclude that the average self-image score for Emerson County public-assistance recipients is statistically different from the expected population mean of 65. The observed difference in means (-0.9) could be due to chance or sampling variation