Question

What is the approximate \(P\) -value for the following values of \(X^{2}\) and df? a. \(X^{2}=14.44, \mathrm{df}=6\) b. \(X^{2}=16.91, \mathrm{df}=9\) c. \(X^{2}=32.32, \mathrm{df}=20\)

Short answer

The exact P-values cannot be calculated without the table or a statistical tool. But the process to calculate the P-value from a chi-square distribution table was highlighted in the step-by-step solution.

Step-by-step solution

- Understand Chi-Square Distribution

The chi-square (X^2) distribution is a family of distributions that take only positive values and are skewed to the right (a long tail on the right). The degrees of freedom (df) define the specific member of this family.

- Use of a Chi-Square Distribution Table

A chi-square distribution table is used to find the P-value. The table shows the boundaries for areas under the curve for different degrees of freedom. Find the row corresponding to the given df and move along the row to find the value that is closest to the given X^2 value.

- Approximate P-value for a. \(X^{2}=14.44, \mathrm{df}=6\)

Refer to the chi-square distribution table. For df=6, look for 14.44 under the table. If this exact value isn't available, find the value that's closest. The P-value is the area to the right of this value.

- Approximate P-value for b. \(X^{2}=16.91, \mathrm{df}=9\)

For df=9, do the same as in the previous case, but now find 16.91 (or the closest value to it) under the table. The P-value is the area to the right of this value.

- Approximate P-value for c. \(X^{2}=32.32, \mathrm{df}=20\)

For df=20, find 32.32 in the chi square distribution table and take the area to the right of this value. If 32.32 is not available, use the closest value.