At the 5% significance level, we do not have sufficient evidence to conclude that the percentage of adults who approve of labor unions has decreased since 1936.
To determine if there is sufficient evidence to conclude that the proportion of adults who approve of labor unions has decreased, we can perform a two-tailed z-test.
Step 1: State the Hypotheses
Null Hypothesis (H₀): The proportion of adults who approve of labor unions is the same in 2023 as it was in 1936.
Alternative Hypothesis (H₁): The proportion of adults who approve of labor unions has decreased since 1936.
Step 2: Calculate the Sample Proportions
Proportion in 2023 (p₀) = 63% = 0.63
Proportion in 1936 (p₁) = 66% = 0.66
Step 3: Determine the Pooled Proportion
Pooled Proportion (P) = (p₀ + p₁) / 2 = (0.63 + 0.66) / 2 = 0.645
Step 4: Calculate the Standard Error
Standard Error (SE) = √[P(1 - P) / n] = √[0.645(1 - 0.645) / 1008] ≈ 0.018
Step 5: Calculate the Test Statistic (z)
z = (p₀ - p₁) / SE = (0.63 - 0.66) / 0.018 ≈ -1.67
Step 6: Determine the Critical Values
For a two-tailed z-test at the 5% significance level (α = 0.05), the critical values are -1.96 and 1.96.
Step 7: Make a Decision
Since the test statistic (z = -1.67) falls within the non-rejection region (-1.96 < z < 1.96), we fail to reject the null hypothesis.