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Chapter 7: Problem 12

Exercises 7.9 to 7.12 give a null hypothesis for a goodness-of-fit test and a frequency table from a sample. For each table, find: (a) The expected count for the category labeled B. (b) The contribution to the sum of the chi-square statistic for the category labeled \(\mathrm{B}\). (c) The degrees of freedom for the chi-square distribution for that table. $$ \begin{aligned} &H_{0}: p_{a}=0.2, p_{b}=0.80\\\ &H_{a}: \text { Some } p_{i} \text { is wrong }\\\ &\begin{array}{ll} \mathrm{A} & \mathrm{B} \end{array}\\\ &132 \quad 468 \end{aligned} $$

Short Answer

Expert verified
The expected count for category B is 480, the contribution to the chi-square statistic for category B is 0.3, and the degrees of freedom for the chi-square distribution for this table is 1.

Step by step solution

01

Finding the Expected Count for Category B

The expected count is the total count times the probability under the null hypothesis. The total count here is the sum of the counts for categories A and B which is \( 132 + 468 = 600 \). The given null hypothesis probability for category B is \(0.80\). Therefore, the expected count for category B would be \(0.80 * 600 = 480\)
02

Contribution to the chi-square statistic for category B

The formula for the chi-square statistic is \((observed-expected)^2/expected\). Here, the observed count for category B is 468, so the contribution to the chi-square statistic from category B would be \((468 - 480)^2 / 480 = 0.3\)
03

Degrees of freedom for the chi-square distribution table

The degrees of freedom are typically calculated as the number of categories minus 1. In this case, there are 2 categories (A and B), thus the degrees of freedom would be \(2 - 1 = 1\)

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Most popular questions from this chapter

Give a two-way table and specify a particular cell for that table. In each case find the expected count for that cell and the contribution to the chi- square statistic for that cell. (Group 3. Yes) cell $$ \begin{array}{l|rr|r} \hline & \text { Yes } & \text { No } & \text { Total } \\ \hline \text { Group 1 } & 56 & 44 & 100 \\ \text { Group 2 } & 132 & 68 & 200 \\ \text { Group 3 } & 72 & 28 & 100 \\ \hline \text { Total } & 260 & 140 & 400 \\ \hline \end{array} $$

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