Let's break down the tasks from the image, which deals with confidence intervals and hypothesis testing:

Part a:

i. Construct a 95% confidence interval for the mean agreement on "Abortion is justifiable":

• The sample size $n = 825$.
• Sample mean $\bar{x} = 7$.
• Sample variance $s^2 = 5.5$.

To construct the confidence interval, we'll use the formula for the confidence interval for the mean: $CI = \bar{x} \pm Z_{\alpha/2} \times \left( \frac{s}{\sqrt{n}} \right)$ Where:

• $Z_{\alpha/2}$ is the critical value for a 95% confidence level.
• $s$ is the sample standard deviation (which is the square root of the sample variance).

Since it's a large sample, we'll assume a normal distribution for the critical value.

ii. Hypothesis test:

• Null hypothesis $H_0: \mu = 4$.
• The z-obtained value is 2.50.

You'll need to interpret the z-value and compare it with the critical value at $\alpha = 0.05$.

Part b:

i. Confidence interval for proportions:

• Sample size $n = 180$.
• Proportion $p = \frac{75}{180}$.

The formula for the confidence interval for a proportion is: $CI = \hat{p} \pm Z_{\alpha/2} \times \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$ Where $\hat{p}$ is the sample proportion.

ii. Hypothesis test for proportions:

• Null hypothesis $H_0: p = 0.61$.
• The z-obtained value is -1.82.

We can use this to draw conclusions about whether to reject the null hypothesis based on the critical z-value.

Would you like me to walk through each calculation step-by-step?