A chi-square distribution with 4 degrees of freedom is graphed below. The region under the curve to the right of $\chi_{0.045}^{2}$ is shaded. The area of this region is 0.045 .
Find the value of $\chi_{0.045}^{2}$. Round your answer to three decimal places.
\[
x_{0.045}^{2}=
\]
\[
\times \quad 5
\]

Step 1 ：The problem is asking for the chi-square value corresponding to the right tail probability of 0.045 with 4 degrees of freedom. This is a standard problem of finding the quantile of a chi-square distribution. We can use the inverse of the cumulative distribution function (CDF) to find this value. The inverse of the CDF, also known as the quantile function or the percent-point function, gives the value below which a given percentage of the data falls.

Step 2 ：Given that the degrees of freedom (df) is 4 and the probability (p) is 0.045, we can calculate the chi-square value.

Step 3 ：After calculation, we find that the chi-square value is 9.742.

Step 4 ：So, the value of \(\chi_{0.045}^{2}\) is \(\boxed{9.742}\).