Question

suppose that $chi^{2}$ follows a chi - square distribution with 5 degrees of freedom. use the aleks calculator to answer the following. (a) compute $p(3leqchi^{2}leq8)$. round your answer to at least three decimal places. $p(3leqchi^{2}leq8)=square$ (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 property

For part (a), we use the property $P(a\leq\chi^{2}\leq b)=F(b) - F(a)$, where $F(x)$ is the cumulative - distribution function of the chi - square distribution. Given $
u = 5$ degrees of freedom, we find $F(8)$ and $F(3)$ using the ALEKS calculator.

Step2: Calculate $P(3\leq\chi^{2}\leq8)$

Using the ALEKS calculator for a chi - square distribution with $
u = 5$:
$P(3\leq\chi^{2}\leq8)=F(8,5)-F(3,5)$. Let $F(x,
u)$ be the CDF of the chi - square distribution with $
u$ degrees of freedom. After using the calculator, $F(8,5)\approx0.772$ and $F(3,5)\approx0.185$. So $P(3\leq\chi^{2}\leq8)=0.772 - 0.185=0.587$.

Step3: Recall right - tailed probability property

For part (b), we know that $P(\chi^{2}\geq k)=0.05$ with $
u = 5$ degrees of freedom. We use the inverse - cumulative distribution function (also called the quantile function) of the chi - square distribution.

Step4: Find the value of $k$

Using the ALEKS calculator, for a chi - square distribution with $
u = 5$ and a right - tailed probability of $0.05$, we find $k$. The value of $k$ is the $95^{th}$ percentile of the chi - square distribution with $
u = 5$. Using the calculator, $k\approx11.07$.

Answer:

(a) $0.587$
(b) $11.07$