Let's examine each of the questions in detail.
Question 2: What is the distribution of the random variable \( Q \)?
Given the formula for \( Q \):
[tex]\[ Q = \sum_{i=1}^k \frac{\left(n_i - E\left(n_i\right)\right)^2}{E\left(n_i\right)} \][/tex]
This formula matches the form of the test statistic used in the chi-square goodness-of-fit test. This test is used to assess whether observed frequencies \(n_i\) differ significantly from expected frequencies \(E(n_i)\) under a specified distribution.
The correct answer is:
A. Chi-square
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Question 3: Which of the following usually describes the value \( E(n_i) \)?
The term \( E(n_i) \) stands for the expected frequency for the ith class/category under the null hypothesis. In the context of a goodness-of-fit test, this is the number of observations expected in a category if the data follows the specified distribution. This is not the mean value of the data, initial class frequency, or expected total sample size.
The correct answer is:
D. Expected class frequency.
In summary:
- The variable \( Q \) follows a Chi-square distribution.
- The value [tex]\( E(n_i) \)[/tex] typically refers to the expected class frequency in the context of a goodness-of-fit test.
