We are given an estimator $T(X)$ for a parameter $\mu$ and two probability statements:

$P(3T(X) + \mu > 8) = 0.025$

$P(-3T(X) - \mu < -2) = 0.975$

We need to determine the 95% confidence interval (CI) for $\mu$ given that $t(x) = 2$.

Step 1: Express in Terms of Probability Quantiles

From the given probability statements:

1. The first equation can be rewritten as:

   $P\left( \mu > 8 - 3T(X) \right) = 0.025$

   This suggests that the upper 2.5% quantile corresponds to $8 - 3T(X)$.

2. The second equation can be rewritten as:

   $P\left( \mu < -2 + 3T(X) \right) = 0.975$

   This suggests that the lower 2.5% quantile corresponds to $-2 + 3T(X)$.

Thus, the 95% confidence interval is given by:

$-2 + 3T(X) \leq \mu \leq 8 - 3T(X)$

Step 2: Substitute $t(x) = 2$

Since we observe $t(x) = 2$, we substitute:

$-2 + 3(2) \leq \mu \leq 8 - 3(2)$

$-2 + 6 \leq \mu \leq 8 - 6$

$4 \leq \mu \leq 2$

Step 3: Interpret and Correct the Interval

Since the usual notation for confidence intervals has the lower bound on the left, we rewrite:

$2 \leq \mu \leq 4$

Thus, the 95% confidence interval for $\mu$ is:

$(2,4)$

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Would you like a detailed explanation of any step?

Related Questions:

1. How do confidence intervals relate to hypothesis testing?
2. What is the role of quantiles in confidence interval construction?
3. How can we interpret confidence intervals in real-world scenarios?
4. What are the assumptions behind this confidence interval calculation?
5. How would this change if $T(X)$ had a known variance?

Tip: Confidence intervals do not guarantee that the true parameter lies within them for every sample—rather, 95% of similarly constructed intervals will contain the true parameter.