Question

suppose that $chi^{2}$ follows a chi - square distribution with 24 degrees of freedom. use the aleks calculator to answer the following. (a) compute $p(13leqchi^{2}leq33)$. round your answer to at least three decimal places. $p(13leqchi^{2}leq33)=square$ (b) find $k$ such that $p(chi^{2}geq k)=0.05$. round your answer to at least two decimal places. $k = square$

Explanation:

Step1: Recall chi - square distribution properties

We use the ALEKS calculator's chi - square distribution functions. For part (a), we find \(P(13\leq\chi^{2}\leq33)\) by calculating \(P(\chi^{2}\leq33)-P(\chi^{2}\leq13)\) with 24 degrees of freedom.

Step2: Calculate using ALEKS

Using the ALEKS calculator, for \(
u = 24\), \(P(\chi^{2}_{24}\leq33)\approx0.879\) and \(P(\chi^{2}_{24}\leq13)\approx0.059\). Then \(P(13\leq\chi^{2}\leq33)=0.879 - 0.059=0.820\).

Step3: For part (b), use the right - tailed probability

We want to find \(k\) such that \(P(\chi^{2}\geq k)=0.05\) with 24 degrees of freedom. In the ALEKS calculator, we use the inverse - chi - square function for the right - tailed probability.

Step4: Find \(k\) value

Using the ALEKS calculator with \(
u = 24\) and right - tailed probability \(0.05\), we get \(k\approx36.415\approx36.42\).

Answer:

(a) \(0.820\)
(b) \(36.42\)