Question

suppose that $chi^{2}$ follows a chi - square distribution with 22 degrees of freedom. use the aleks calculator to answer the following. (a) compute $p(chi^{2}leq22)$. round your answer to at least three decimal places. $p(chi^{2}leq22)=square$ (b) find $k$ such that $p(chi^{2}>k)=0.05$. round your answer to at least two decimal places. $k = square$

Explanation:

Step1: Use chi - square distribution table or calculator

For part (a), when using a calculator (like ALEKS), we input the degrees of freedom $df = 22$ and the value $x = 22$ to find the cumulative - distribution function value. The cumulative - distribution function of a chi - square distribution $\chi^{2}(df)$ at a value $x$ gives $P(\chi^{2}\leq x)$.

Step2: Find the critical value for part (b)

For part (b), we know that $P(\chi^{2}>k)=0.05$. This means $P(\chi^{2}\leq k)=1 - 0.05=0.95$. We use the inverse - cumulative distribution function of the chi - square distribution with $df = 22$ to find the value of $k$.

Answer:

(a) Using a chi - square calculator with $df = 22$ and $x = 22$, we get $P(\chi^{2}\leq22)\approx0.500$.
(b) Using the inverse - cumulative distribution function of the chi - square distribution with $df = 22$ and a right - tail probability of $0.05$ (or left - tail probability of $0.95$), we get $k\approx33.92$.