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 $k$ 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 two-tailed hypothesis test with $n = 104$ and $\alpha = 0.10$.
A) $\chi ^ { 2 } \approx 80.300$ and $\chi ^ { 2 } \approx 127.406$
B) $\chi ^ { 2 } \approx 84.992$ and $\chi ^ { 2 } \approx 121.646$
C) $\chi ^ { 2 } \approx 85.903$ and $\chi ^ { 2 } \approx 122.735$
D) $\chi ^ { 2 } \approx 81.186$ and $\chi ^ { 2 } \approx 128.520$

Question

Quizplus · Accepted Answer

The critical values of $\chi ^ { 2 }$ can be approximated as follows: $$x ^ { 2 } = \frac { 1 } { 2 } ( z + \sqrt { 2 k - 1 } ) ^ { 2 }$$where $k$ 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.For a two-tailed hypothesis test with $n = 104$ and $\alpha = 0.10$, the degrees of freedom is $k = n - 1 = 103$. Using a standard normal distribution table, we find that the critical values of $z$ are $-1.645$ and $1.645$.Substituting these values into the formula, we get:$$x ^ { 2 } = \frac { 1 } { 2 } ( - 1.645 + \sqrt { 2 ( 103 ) - 1 } ) ^ { 2 } \approx 80.300$$$$x ^ { 2 } = \frac { 1 } { 2 } ( 1.645 + \sqrt { 2 ( 103 ) - 1 } ) ^ { 2 } \approx 127.406$$Therefore, the critical values of $\chi ^ { 2 }$ are approximately $80.300$ and $127.406$.

Solve the Problem χ2\chi ^ { 2 }χ2 Values Can Be Approximated as Follows

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