Question5.3.7. Five independent samples, each of size $n$, are to be drawn from a normal distribution where $\sigma$ is known. For each sample, the interval $\left(\bar{y}-0.96 \cdot \frac{d}{\sqrt{n}}, \bar{y}+1.06 \cdot \frac{d}{\sqrt{n}}\right)$ will be constructed. What is the probability that at least four of the intervals will contain the unknown $\mu$ ?

Studdy Solution
Calculate $P(X = 4)$:
$P(X = 4) = 5 \times (0.6869)^4 \times (0.3131) \approx 0.3717$
Calculate $P(X = 5)$:
$P(X = 5) = 1 \times (0.6869)^5 \approx 0.1610$
Thus, the probability that at least four intervals contain $\mu$ is:
$P(X \geq 4) = 0.3717 + 0.1610 = 0.5327$
The probability that at least four of the intervals will contain the unknown $\mu$ is:
$\boxed{0.5327}$