Question

suppose that $\chi^{2}$ follows a chi - square distribution with 20 degrees of freedom. use the aleks calculator to answer the following. (a) compute $p(\chi^{2}\leq16)$. round your answer to at least three decimal places. $p(\chi^{2}\leq16)=\square$ (b) find $k$ such that $p(\chi^{2}\geq k)=0.025$. round your answer to at least two decimal places. $k = \square$

Explanation:

Part (a)

Step1: Use the chi - square cumulative distribution function

For a chi - square random variable \(\chi^{2}\) with \(n = 20\) degrees of freedom, we want to find \(P(\chi^{2}\leq16)\). Using a chi - square calculator (such as ALEKS calculator), we input the value \(x = 16\) and degrees of freedom \(df=20\) into the cumulative distribution function \(P(\chi^{2}\leq x)\).

Part (b)

Step1: Use the chi - square inverse cumulative distribution function

We know that \(P(\chi^{2}\geq k)=0.025\). By the property of the chi - square distribution \(P(\chi^{2}\geq k)=1 - P(\chi^{2}\lt k)\), so \(P(\chi^{2}\lt k)=1 - 0.025=0.975\). Using the inverse cumulative distribution function (also called the quantile function) of the chi - square distribution with \(df = 20\), we input the probability \(p = 0.975\) and degrees of freedom \(df = 20\).

Answer:

a. \(P(\chi^{2}\leq16)\approx0.304\)
b. \(k\approx34.17\)