Find the critical value or values of $\chi ^ { 2 }$ based on the given information.
-For large numbers of degrees of freedom, the critical $x ^ { 2 }$ values can be approximated as follows: $\chi ^ { 2 } = \frac { 1 } { 2 } ( z + \sqrt { 2 k - 1 } ) ^ { 2 }$ where $\mathrm { k }$ is the number of degrees of freedom and $\mathrm { z }$ 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) $\chi ^ { 2 } \approx 189.286$
B) $\chi ^ { 2 } \approx 188.134$
C) $X ^ { 2 } \approx 184.549$
D) $\chi ^ { 2 } \approx 183.411$

Question

Quizplus · Accepted Answer

Using the formula given, we have: $\chi ^ { 2 } = \frac { 1 } { 2 } ( z + \sqrt { 2 k - 1 } ) ^ { 2 }$. 
We know that the level of significance is $\alpha = 0.01$, so the area in the right tail is $0.01$. This means that the area in the left tail is $0.99$. From a chi-square distribution table with $k = 143$ degrees of freedom and an area of $0.99$ in the left tail, we find the critical value of $\chi^{2}$ to be approximately $94.326$. 
Next, we solve for $z$ in the equation: $\chi ^ { 2 } = \frac { 1 } { 2 } ( z + \sqrt { 2 k - 1 } ) ^ { 2 }$ using the known values of $\chi^{2}$ and $k$. Substituting in the values of $\chi^{2}$ and $k$, we get: 
$$94.326 = \frac { 1 } { 2 } ( z + \sqrt { 2(143) - 1 } ) ^ { 2 }$$
Solving for $z$, we get: 
$$z \approx 5.5$$
Therefore, the critical value of $\chi^{2}$ for a right-tailed hypothesis test with $k = 143$ and $\alpha = 0.01$ is approximately:
$$\chi ^ { 2 } = \frac { 1 } { 2 } ( 5.5 + \sqrt { 2 ( 143 ) - 1 } ) ^ { 2 } \approx 183.411$$ 
Thus, the correct choice is D.

Find the Critical Value or Values Of χ2\chi ^ { 2 }χ2

Find the Critical Value or Values Of $\chi ^ { 2 }$