Question

a chi-square distribution with 4 degrees of freedom is graphed below. the region under the curve to the right of 6 is shaded. find the area of the shaded region. round your answer to three decimal places.

Explanation:

Step1: Recall chi - square distribution

We know that for a chi - square distribution with \(k\) degrees of freedom, the cumulative distribution function \(P(\chi^{2}\leq x)\) gives the area to the left of \(x\). We want the area to the right of \(x = 6\) for a chi - square distribution with \(k=4\) degrees of freedom, which is \(P(\chi^{2}_{4}>6)=1 - P(\chi^{2}_{4}\leq6)\)

Step2: Use chi - square table or calculator

Using a chi - square calculator or statistical software (or a chi - square table), we find the cumulative probability for \(\chi^{2}\) with \(df = 4\) and \(x = 6\).

Using a chi - square calculator: The cumulative distribution function for a chi - square random variable with \(df = 4\) at \(x = 6\) can be calculated. The formula for the chi - square cumulative distribution is \(P(\chi^{2}\leq x)=\int_{0}^{x}\frac{1}{2^{\frac{k}{2}}\Gamma(\frac{k}{2})}t^{\frac{k}{2}-1}e^{-\frac{t}{2}}dt\), where \(k = 4\), \(\Gamma(\frac{4}{2})=\Gamma(2)=(2 - 1)!=1\), \(2^{\frac{4}{2}}=4\). So the pdf is \(f(t)=\frac{1}{4\times1}t^{2 - 1}e^{-\frac{t}{2}}=\frac{1}{4}te^{-\frac{t}{2}}\) for \(t\geq0\).

But it's easier to use a calculator. Using a chi - square calculator, we find that \(P(\chi^{2}_{4}\leq6)\approx0.715\)

Step3: Calculate the area to the right

Then \(P(\chi^{2}_{4}>6)=1 - P(\chi^{2}_{4}\leq6)=1 - 0.715 = 0.285\) (rounded to three decimal places)

Answer:

\(0.285\)