Tutorial Question: Age and Political Preference
It is generally assumed that older individuals are more likely to vote Conservative compared to younger individuals. A survey was conducted to test this assumption:
- Among 400 people over 40, 160 said they would vote Conservative.
- Among 400 people under 40, 120 said they would vote Conservative.
Tasks
- Test the hypothesis that the proportion of Conservative voters is greater among people over 40 than under 40, using a 5% significance level.
- Calculate a 95% confidence interval for the difference in proportions between the two age groups.
Solution
1. Hypothesis Test
Let: - \(p_1\): proportion of over-40s voting Conservative = \(\frac{160}{400} = 0.40\) - \(p_2\): proportion of under-40s voting Conservative = \(\frac{120}{400} = 0.30\)
Hypotheses
- Null Hypothesis \(H_0\): \(p_1 = p_2\) (no difference)
- Alternative Hypothesis \(H_1\): \(p_1 > p_2\) (older people more likely to vote Conservative)
Pooled Proportion
\[ \hat{p} = \frac{160 + 120}{400 + 400} = \frac{280}{800} = 0.35 \]
Standard Error
\[ SE = \sqrt{ \hat{p} (1 - \hat{p}) \left( \frac{1}{n_1} + \frac{1}{n_2} \right) } = \sqrt{ 0.35 \cdot 0.65 \left( \frac{1}{400} + \frac{1}{400} \right) } \approx 0.0341 \]
Test Statistic
\[ z = \frac{p_1 - p_2}{SE} = \frac{0.40 - 0.30}{0.0341} \approx 2.93 \]
Conclusion
- Critical value for one-tailed test at 5%: \(z = 1.645\)
- Since \(z = 2.93 > 1.645\), we reject the null hypothesis.
Interpretation: There is statistically significant evidence that older people are more likely to vote Conservative.
2. Confidence Interval for \(p_1 - p_2\)
We use the formula for the confidence interval for a difference in proportions:
\[ (p_1 - p_2) \pm z \cdot SE \]
Where: - \(z = 1.96\) for 95% confidence - \(SE = \sqrt{ \frac{p_1(1 - p_1)}{n_1} + \frac{p_2(1 - p_2)}{n_2} }\) \[ SE = \sqrt{ \frac{0.40 \cdot 0.60}{400} + \frac{0.30 \cdot 0.70}{400} } \approx 0.0333 \]
Interval Calculation
\[ (0.40 - 0.30) \pm 1.96 \cdot 0.0333 = 0.10 \pm 0.0653 \] \[ \text{95% CI} = (0.0347,\ 0.1653) \]
Interpretation: We are 95% confident that the proportion of older Conservative voters exceeds that of younger voters by between 3.5% and 16.5%.