Solve the Problem $\chi ^ { 2 }$ Values Can Be Approximated as Follows

Solve the problem. For large numbers of degrees of freedom, the critical $\chi ^ { 2 }$ values can be approximated as follows: $x ^ { 2 } = \frac { 1 } { 2 } ( z + \sqrt { 2 k - 1 } ) ^ { 2 }$ , where $\mathrm { k }$ is the number of degrees of freedom and $z$ is the critical value. To find the lower critical value, the negative $z$ -value is used, to find the upper critical value, the positive $z$ -value is used. Use this approximation to estimate the critical value of $\chi ^ { 2 }$ in a right-tailed hypothesis test with $n = 143$ and $\alpha = 0.01$ .

A) $X ^ { 2 } \approx 189.286$
B) $\chi ^ { 2 } \approx 188.134$
C) $x ^ { 2 } \approx 183.411$
D) $\chi ^ { 2 } \approx 184.549$