A shoe salesman wants to see if his female customers have a preference in the color of sh purchased. Compute the test value for the following data.
$\begin{array} { l l l l l } \text { Black } & \text { Brown } & \text { Red } & \text { Navy } & \text { White } \ 48 & 32 & 13 & 18 & 23 \end{array}$

Question

Quizplus · Accepted Answer

To compute the test value (Chi-square statistic) for the given data, we first need to calculate the expected frequencies assuming there is no preference among the colors. The total number of shoes purchased is $48 + 32 + 13 + 18 + 23 = 134$. With 5 colors, if there were no preference, we would expect each color to be chosen equally, so the expected frequency for each color is $134 / 5 = 26.8$.The Chi-square statistic is calculated using the formula: $\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}$, where $O_i$ is the observed frequency for each category, and $E_i$ is the expected frequency for each category.Plugging in the numbers:$$\chi^2 = \frac{(48 - 26.8)^2}{26.8} + \frac{(32 - 26.8)^2}{26.8} + \frac{(13 - 26.8)^2}{26.8} + \frac{(18 - 26.8)^2}{26.8} + \frac{(23 - 26.8)^2}{26.8}$$$$\chi^2 = \frac{(21.2)^2}{26.8} + \frac{(5.2)^2}{26.8} + \frac{(-13.8)^2}{26.8} + \frac{(-8.8)^2}{26.8} + \frac{(-3.8)^2}{26.8}$$$$\chi^2 = \frac{449.44}{26.8} + \frac{27.04}{26.8} + \frac{190.44}{26.8} + \frac{77.44}{26.8} + \frac{14.44}{26.8}$$$$\chi^2 = 16.76 + 1.01 + 7.1 + 2.89 + 0.54$$$$\chi^2 = 28.3$$Therefore, the test value is approximately 28.31, which corresponds to choice A.

A Shoe Salesman Wants to See If His Female Customers