To solve these questions, we need to work with the chi-square distribution for a given degree of freedom. Let's tackle each part:

Part (a): Compute $P(\chi^2 \leq 42)$ for 28 degrees of freedom

This requires finding the cumulative probability for the chi-square distribution with 28 degrees of freedom at the point $\chi^2 = 42$.

1. Approach: Use a chi-square cumulative distribution function (CDF) table or an online chi-square calculator to find $P(\chi^2 \leq 42)$.
2. Result: The cumulative probability is approximately 0.923. This means the probability that a chi-square value is less than or equal to 42 with 28 degrees of freedom is 0.923.

Part (b): Find $k$ such that $P(\chi^2 > k) = 0.05$

Here, we want to find the critical value $k$ for which the probability of observing a chi-square value greater than $k$ is 0.05. This corresponds to the 95th percentile of the chi-square distribution with 28 degrees of freedom.

1. Approach: Use a chi-square inverse CDF or a calculator to find the value of $k$ such that $P(\chi^2 \leq k) = 0.95$.
2. Result: The value of $k$ is approximately 41.34. This means that 5% of the distribution is to the right of this value.

Let me know if you'd like further details or have any questions about these solutions.

Related Questions

1. What is the 50th percentile (median) for a chi-square distribution with 28 degrees of freedom?
2. How does the shape of the chi-square distribution change with increasing degrees of freedom?
3. What is the mean and variance of a chi-square distribution with 28 degrees of freedom?
4. How would the probability change in part (a) if the degrees of freedom were 40 instead of 28?
5. What is the 90th percentile for a chi-square distribution with 28 degrees of freedom?

Tip

Remember that the mean of a chi-square distribution is equal to its degrees of freedom, and the variance is twice the degrees of freedom.