Question

2. Consider a large population with mean μ and known standard deviation σ = 5. There are two independent simple random samples of this population, one with n 150, and the other with n2 = 400, Denote the two sample means by , and X2, respectively. Let Cli and C12 be the usual 95% confidence intervals, constructed from each of the two samples. What is the probability that at the same time, X E CI2 and X2 E CI?

Answer 1

The margin of error for both the samples are respectivley 0.8 and 0.49. The lowest among the two is 0.49. The difference between $\bar{X_1}$ and $\bar{X_2}$ ie
X1 X2
must be less than 0.49 for $\bar{X_1}\epsilon CI_2$ and $\bar{X_2}\epsilon CI_1$ at the same time. ie to find the required probability all we need to find is
P(|X1-Xy < 0.49) = Pl(리 < 0.49/V50) = 0.055

{since
X1 X2
has normal distribution with mean zero and variance =
52 +52- 50
}