Question

Let X1 ,X2 , … , X16 be a random sample from a N(4μ, 1) distribution and Y1 ,Y2 , … , Y8 be a random sample from a N(μ, 1) distribution, where μ ∈ $\R$ is unknown. Assume that the two random samples are independent. If you are looking for a confidence interval for μ based on the statistic $8\overline{X} + \overline{Y}$, where $\overline{X}=\frac{1}{16}\sum^{16}_{i=1}X_i$ and $\overline{Y}=\frac{1}{8}\sum^8_{i=1}Y_i$, then which one of the following statements is true ?

• A. There exists a 90% confidence interval for μ of length less than 0.1
• B. There exists a 90% confidence interval for μ of length greater than 0.3
• C. $[\frac{8\overline{X}+\overline{Y}}{33}-\frac{1.645}{2\sqrt{66}},\frac{8\overline{X}+\overline{Y}}{33}+\frac{1.645}{2\sqrt{66}}]$ is the unique 90% confidence interval for μ
• D. μ always belongs to its 90% confidence interval

Answer 1

Answer: (B)

There exists a 90% confidence interval for μ of length greater than 0.3

Answer 2

The correct option is (B) : There exists a 90% confidence interval for μ of length greater than 0.3.