Question

Let \( X_1, X_2, X_3, X_4 \) be a random sample from an \( N(\mu, \sigma^2) \) distribution. Let \( \bar{X} = \frac{1}{4} \sum_{i=1}^4 X_i \) and \[ \tilde{Y} = \frac{15}{7} \sum_{i=1}^4 X_i. \] If \( \bar{X} \) has a t-distribution, then \( (\nu - k) \) equals

Hint:

For a t-distribution with sample size \( n \), the degrees of freedom are \( n - 1 \).

Answer 1

Correct Answer: 3

Step 1: Understanding the t-distribution.
The statistic \( \bar{X} \) is the sample mean of a random sample from a normal distribution, and it follows a t-distribution with \( n - 1 \) degrees of freedom, where \( n \) is the sample size.
Step 2: Degrees of freedom.
Since the sample size is 4, the degrees of freedom for the t-distribution are \( n - 1 = 4 - 1 = 3 \).
Step 3: Conclusion.
Thus, \( \nu - k = 3 \), where \( \nu \) is the degrees of freedom and \( k \) is the number of parameters estimated.