91Ó°ÊÓ

True or False: The expected frequencies in a chi-square test for independence are found using the formula Expected frequency \(=\frac{(\text { row total })(\text { column total })}{\text { table total }}\)

The following table contains observed values and expected values in parentheses for two categorical variables, \(X\) and \(Y\), where variable \(X\) has three categories and variable \(Y\) has two categories: $$ \begin{array}{cccc} & \boldsymbol{X}_{\mathbf{1}} & \boldsymbol{X}_{\mathbf{2}} & \boldsymbol{X}_{3} \\ \hline \boldsymbol{Y}_{\mathbf{1}} & 34(36.26) & 43(44.63) & 52(48.11) \\ \hline \boldsymbol{Y}_{\mathbf{2}} & 18(15.74) & 21(19.37) & 17(20.89) \\ \hline \end{array} $$ (a) Compute the value of the chi-square test statistic. (b) Test the hypothesis that \(X\) and \(Y\) are independent at the \(\alpha=0.05\) level of significance.

The following table contains the number of successes and failures for three categories of a variable. $$ \begin{array}{lccc} & \text { Category } 1 & \text { Category } 2 & \text { Category } 3 \\ \hline \text { Success } & 76 & 84 & 69 \\ \hline \text { Failure } & 44 & 41 & 49 \\ \hline \end{array} $$ Test whether the proportions are equal for each category at the \(\alpha=0.01\) level of significance.

Determine the expected counts for each outcome. $$ \begin{array}{lllll} \hline \boldsymbol{n}=\mathbf{5 0 0} & & & & \\ \hline p_{i} & 0.2 & 0.1 & 0.45 & 0.25 \\ \hline \text { Expected counts } & & & & \\ \hline \end{array} $$

Determine the expected counts for each outcome. $$ \begin{array}{lllll} \hline \boldsymbol{n}=\mathbf{7 0 0} & & & & \\ \hline p_{i} & 0.15 & 0.3 & 0.35 & 0.20 \\ \hline{\text {Expected counts }} & & & & \\ \hline \end{array} $$

Determine \((a)\) the \(\chi^{2}\) test statistic, \((b)\) the degrees of freedom, (c) the critical value using \(\alpha=0.05,\) and (d) test the hypothesis at the \(\alpha=0.05\) level of significance. \(H_{0}: p_{\mathrm{A}}=p_{\mathrm{B}}=p_{\mathrm{C}}=p_{\mathrm{D}}=\frac{1}{4}\) \(H_{1}\) : At least one of the proportions is different from the others. $$ \begin{array}{lcccc} \hline\text { Outcome } & \mathbf{A} & \mathbf{B} & \mathbf{C} & \mathbf{D} \\\ \hline \text { Observed } & 30 & 20 & 28 & 22 \\ \hline \text { Expected } & 25 & 25 & 25 & 25 \\ \hline \end{array} $$

Family Structure and Sexual Activity A sociologist wants to discover whether the sexual activity of females between the ages of 15 and 19 years and family structure are associated. She randomly selects 380 females between the ages of 15 and 19 years and asks each to disclose her family structure at age 14 and whether she has had sexual intercourse. The results are shown in the table. Data are based on information obtained from the National Center for Health Statistics. $$ \begin{array}{lcccc} &&{\text { Family Structure }} \\ \hline & \text { Both Biological } & & & \\ \text { Had Sexual } & \text { or Adoptive } & \text { Single } & \text { Parent and } & \text { Nonparental } \\ \text { Intercourse } & \text { Parents } & \text { Parent } & \text { Stepparent } & \text { Guardian } \\ \hline \text { Yes } & 64 & 59 & 44 & 32 \\ \hline \text { No } & 86 & 41 & 36 & 18 \\ \hline \end{array} $$ (a) Compute the expected values of each cell under the assumption of independence. (b) Verify that the requirements for performing a chi-square test of independence are satisfied. (c) Compute the chi-square test statistic. (d) Test whether family structure and sexual activity of 15 - to 19-year-old females are independent at the \(\alpha=0.05\) level of significance. (e) Compare the observed frequencies with the expected frequencies. Which cell contributed most to the test statistic? Was the expected frequency greater than or less than the observed frequency? What does this information tell you? (f) Construct a conditional distribution by family structure and draw a bar graph. Does this evidence support your conclusion in part (d)?

Determine \((a)\) the \(\chi^{2}\) test statistic, \((b)\) the degrees of freedom, (c) the critical value using \(\alpha=0.05,\) and (d) test the hypothesis at the \(\alpha=0.05\) level of significance. \(H_{0}:\) The random variable \(X\) is binomial with \(n=4, p=0.3\) \(H_{1}:\) The random variable \(X\) is not binomial with \(n=4\) \(p=0.3\) $$ \begin{array}{llllll} \boldsymbol{X} & \mathbf{0} & \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} \\ \hline \text { Observed } & 260 & 400 & 280 & 50 & 10 \\ \hline \text { Expected } & 240.1 & 411.6 & 264.6 & 75.6 & 8.1 \\ \hline \end{array} $$

According to the manufacturer of M\&Ms, \(13 \%\) of the plain M\&Ms in a bag should be brown, \(14 \%\) yellow, \(13 \%\) red, \(24 \%\) blue \(, 20 \%\) orange, and \(16 \%\) green. A student randomly selected a bag of plain M\&Ms. He counted the number of \(\mathrm{M} \& \mathrm{Ms}\) that were each color and obtained the results shown in the table. Test whether plain M\&Ms follow the distribution stated by M\&M/Mars at the \(\alpha=0.05\) level of significance. $$ \begin{array}{lc} \text { Color } & \text { Frequency } \\ \hline \text { Brown } & 57 \\ \hline \text { Yellow } & 64 \\ \hline \text { Red } & 54 \\ \hline \text { Blue } & 75 \\ \hline \text { Orange } & 86 \\ \hline \text { Green } & 64\\\ \hline \end{array} $$

Does the location of your seat in a classroom play a role in attendance or grade? To answer this question, professors randomly assigned 400 students a general education physics course to one of four groups. The 100 students in group 1 sat 0 to 4 meters from the front of the class, the 100 students in group 2 sat 4 to 6.5 meters from the front, the 100 students in group 3 sat 6.5 to 9 meters from the front, and the 100 students in group 4 sat 9 to 12 meters from the front. (a) For the first half of the semester, the attendance for the whole class averaged \(83 \% .\) So, if there is no effect due to seat location, we would expect \(83 \%\) of students in each group to attend. The data show the attendance history for each group. How many students in each group attended, on average? Is there a significant difference among the groups in attendance patterns? Use the \(\alpha=0.05\) level of significance. (b) For the second half of the semester, the groups were rotated so that group 1 students moved to the back of class and group 4 students moved to the front. The same switch took place between groups 2 and \(3 .\) The attendance for the second half of the semester averaged \(80 \% .\) The data show the attendance records for the original groups (group 1 is now in back, group 2 is 6.5 to 9 meters from the front, and so on ). How many students in each group attended, on average? Is there a significant difference in attendance patterns? Use the \(\alpha=0.05\) level of significance. Do you find anything curious about these data? $$ \begin{array}{lllll} \hline \text { Group } & 1 & 2 & 3 & 4 \\ \hline \text { Attendance } & 0.84 & 0.81 & 0.78 & 0.76\\\ \hline \end{array} $$ (c) At the end of the semester, the proportion of students in the top \(20 \%\) of the class was determined. Of the students in group \(1,25 \%\) were in the top \(20 \%\); of the students in group \(2,21 \%\) were in the top \(20 \%\); of the students in group \(3,15 \%\) were in the top \(20 \%\); of the students in group \(4,19 \%\) were in the top \(20 \% .\) How many students would we expect to be in the top \(20 \%\) of the class if seat location plays no role in grades? Is there a significant difference in the number of students in the top \(20 \%\) of the class by group? (d) In earlier sections, we discussed results that were statistically significant, but did not have any practical significance. Discuss the practical significance of these results. In other words, given the choice, would you prefer sitting in the front or back?