1. **Problem statement:** We have a sample of 60 grade 9 students with a sample mean age of 15.3 years. The population mean is given as 16 years. We need to find: a. The point estimate for the mean. b. The 95% confidence interval for the mean. c. The 99% confidence interval for the mean. 2. **Point estimate:** The point estimate for the population mean is the sample mean itself. $$\text{Point estimate} = \bar{x} = 15.3$$ 3. **Confidence interval formula:** For a large sample (n=60), we use the z-distribution to calculate confidence intervals: $$\text{CI} = \bar{x} \pm z_{\alpha/2} \times \frac{s}{\sqrt{n}}$$ where: - $\bar{x}$ is the sample mean - $z_{\alpha/2}$ is the z-score for the desired confidence level - $s$ is the sample standard deviation (not given, so we assume population standard deviation or estimate) - $n$ is the sample size 4. **Important note:** The problem does not provide the sample standard deviation $s$ or population standard deviation. Without $s$, we cannot compute the confidence intervals exactly. Assuming the population standard deviation $\sigma$ is known or approximated, but since it is not given, we cannot calculate the intervals numerically. 5. **If we assume the population standard deviation $\sigma$ is known, say $\sigma = 1$ (for example), then:** - For 95% confidence, $z_{0.025} = 1.96$ - For 99% confidence, $z_{0.005} = 2.576$ 6. **Calculate margin of error (ME):** $$\text{ME} = z_{\alpha/2} \times \frac{\sigma}{\sqrt{n}}$$ For 95%: $$\text{ME}_{95} = 1.96 \times \frac{1}{\sqrt{60}} = 1.96 \times 0.1291 = 0.253$$ For 99%: $$\text{ME}_{99} = 2.576 \times \frac{1}{\sqrt{60}} = 2.576 \times 0.1291 = 0.333$$ 7. **Confidence intervals:** - 95% CI: $$15.3 \pm 0.253 = (15.047, 15.553)$$ - 99% CI: $$15.3 \pm 0.333 = (14.967, 15.633)$$ **Final answers:** - a. Point estimate for mean: $15.3$ - b. 95% confidence interval: approximately $(15.05, 15.55)$ - c. 99% confidence interval: approximately $(14.97, 15.63)$ *Note: Actual intervals depend on the true or sample standard deviation, which is not provided.*