Question

suppose that $chi^{2}$ follows a chi - square distribution with 27 degrees of freedom. use the aleks calculator to answer the following. (a) compute $p(18leqchi^{2}leq35)$. round your answer to at least three decimal places. $p(18leqchi^{2}leq35)=square$ (b) find $k$ such that $p(chi^{2}>k)=0.025$. round your answer to at least two decimal places. $k = square$

Explanation:

Step1: Recall chi - square distribution properties

We use the cumulative - distribution function of the chi - square distribution. Let $\chi^2_{27}$ be a chi - square random variable with 27 degrees of freedom. We know that $P(18\leq\chi^2\leq35)=F(35)-F(18)$, where $F(x)$ is the cumulative distribution function of $\chi^2_{27}$.

Step2: Use ALEKS calculator

On the ALEKS calculator, we input the degrees of freedom $df = 27$, and find $F(35)$ and $F(18)$. Then $P(18\leq\chi^2\leq35)=\text{chi - square cdf}(35,27)-\text{chi - square cdf}(18,27)$. After using the calculator, we get $P(18\leq\chi^2\leq35)\approx0.729$.

Step3: For part (b)

We know that if $P(\chi^2>k) = 0.025$, then $P(\chi^2\leq k)=1 - 0.025=0.975$.

Step4: Use ALEKS calculator for inverse - chi - square

On the ALEKS calculator, we use the inverse - cumulative distribution function (inverse chi - square) with degrees of freedom $df = 27$ and cumulative probability $0.975$. We find that $k=\text{chi - square invcdf}(0.975,27)\approx43.19$.

Answer:

(a) $0.729$
(b) $43.19$