Confidence interval of a constant value

Question

If W₁,W₂,... are i.i.d with N(0,σ²), where σ² is unknown. Let X_i = m + W_i, where m is some constant. Suppose M₁₀ = 8.5 and S₁₀ = 2. Find the 95% confidence interval for m.

What I've tried:

Since the sample mean of W_i is going to be 0, and since the sample mean of X_i is 8.5, then the confidence interval is going to be 8.5 $\pm$ (1.96$\frac 2 {\sqrt{10}})$. This isn't the correct answer, and I'm not sure what I'm doing wrong here. Can someone help me out here?

Thanks

Answer 1

$$ \frac{M_{10} - m}{\sigma/\sqrt{10}} \sim N(0,1) $$ Therefore $$ \Pr\left( -1.96 < \frac{M_{10}-m}{\sigma/\sqrt{10}} < 1.96 \right) = 0.95, $$ and so $$ \Pr\left(M_{10} - 1.96\cdot \frac \sigma{\sqrt{10}} < m < M_{10} + 1.96 \cdot\frac\sigma {\sqrt{10}} \right) = 0.95. $$ But $\sigma$ is unknown. $$ \frac{M_{10} - m}{S_{10}/\sqrt{10}} \sim t_9 $$ (Student's t-distribution with $9$ degrees of freedom.) $$ \Pr\left( -2.262 < \frac{M_{10}-m}{S_{10}/\sqrt{10}} < 2.262 \right) = 0.95 $$ $$ \Pr\left( M_{10} - 2.262\cdot \frac {S_{10}} {\sqrt{10}} < m < M_{10} + 2.262\cdot\frac {S_{10}} {\sqrt{10}} \right) = 0.95 $$