Worked solution P68-Easy-1. The statistic is 5.043, df=2, and p=0.0804; expected counts are computed from row and column totals. $$E_{ij}=\frac{(\text{row total})(\text{column total})}{120},{\chi }^2=5.043,df=(2-1)(3-1)=2.$$ Interpretation: A small p-value indicates evidence against the null distributional claim, but the design controls whether the conclusion concerns association, homogeneity, or treatment effects. Validity: Use random sampling or random assignment as appropriate, independent observations, and sufficiently large expected counts. Error to reject: Check expected counts, not observed counts; a large cell contribution identifies where observed and expected differ most.

Worked solution P68-Easy-2. The statistic is 5.782, df=2, and p=0.0555; expected counts are computed from row and column totals. $$E_{ij}=\frac{(\text{row total})(\text{column total})}{122},{\chi }^2=5.782,df=(2-1)(3-1)=2.$$ Interpretation: A small p-value indicates evidence against the null distributional claim, but the design controls whether the conclusion concerns association, homogeneity, or treatment effects. Validity: Use random sampling or random assignment as appropriate, independent observations, and sufficiently large expected counts. Error to reject: Check expected counts, not observed counts; a large cell contribution identifies where observed and expected differ most.

Worked solution P68-Easy-3. The statistic is 6.573, df=2, and p=0.0374; expected counts are computed from row and column totals. $$E_{ij}=\frac{(\text{row total})(\text{column total})}{124},{\chi }^2=6.573,df=(2-1)(3-1)=2.$$ Interpretation: A small p-value indicates evidence against the null distributional claim, but the design controls whether the conclusion concerns association, homogeneity, or treatment effects. Validity: Use random sampling or random assignment as appropriate, independent observations, and sufficiently large expected counts. Error to reject: Check expected counts, not observed counts; a large cell contribution identifies where observed and expected differ most.

Worked solution P68-Easy-4. The statistic is 7.412, df=2, and p=0.0246; expected counts are computed from row and column totals. $$E_{ij}=\frac{(\text{row total})(\text{column total})}{126},{\chi }^2=7.412,df=(2-1)(3-1)=2.$$ Interpretation: A small p-value indicates evidence against the null distributional claim, but the design controls whether the conclusion concerns association, homogeneity, or treatment effects. Validity: Use random sampling or random assignment as appropriate, independent observations, and sufficiently large expected counts. Error to reject: Check expected counts, not observed counts; a large cell contribution identifies where observed and expected differ most.

Worked solution P68-Easy-5. The statistic is 6.844, df=2, and p=0.0327; expected counts are computed from row and column totals. $$E_{ij}=\frac{(\text{row total})(\text{column total})}{124},{\chi }^2=6.844,df=(2-1)(3-1)=2.$$ Interpretation: A small p-value indicates evidence against the null distributional claim, but the design controls whether the conclusion concerns association, homogeneity, or treatment effects. Validity: Use random sampling or random assignment as appropriate, independent observations, and sufficiently large expected counts. Error to reject: Check expected counts, not observed counts; a large cell contribution identifies where observed and expected differ most.

Worked solution P68-Easy-6. The statistic is 5.325, df=2, and p=0.0698; expected counts are computed from row and column totals. $$E_{ij}=\frac{(\text{row total})(\text{column total})}{121},{\chi }^2=5.325,df=(2-1)(3-1)=2.$$ Interpretation: A small p-value indicates evidence against the null distributional claim, but the design controls whether the conclusion concerns association, homogeneity, or treatment effects. Validity: Use random sampling or random assignment as appropriate, independent observations, and sufficiently large expected counts. Error to reject: Check expected counts, not observed counts; a large cell contribution identifies where observed and expected differ most.

Worked solution P68-Easy-7. The statistic is 6.094, df=2, and p=0.0475; expected counts are computed from row and column totals. $$E_{ij}=\frac{(\text{row total})(\text{column total})}{123},{\chi }^2=6.094,df=(2-1)(3-1)=2.$$ Interpretation: A small p-value indicates evidence against the null distributional claim, but the design controls whether the conclusion concerns association, homogeneity, or treatment effects. Validity: Use random sampling or random assignment as appropriate, independent observations, and sufficiently large expected counts. Error to reject: Check expected counts, not observed counts; a large cell contribution identifies where observed and expected differ most.

Worked solution P68-Easy-8. The statistic is 6.912, df=2, and p=0.0315; expected counts are computed from row and column totals. $$E_{ij}=\frac{(\text{row total})(\text{column total})}{125},{\chi }^2=6.912,df=(2-1)(3-1)=2.$$ Interpretation: A small p-value indicates evidence against the null distributional claim, but the design controls whether the conclusion concerns association, homogeneity, or treatment effects. Validity: Use random sampling or random assignment as appropriate, independent observations, and sufficiently large expected counts. Error to reject: Check expected counts, not observed counts; a large cell contribution identifies where observed and expected differ most.

Worked solution P68-Easy-9. The statistic is 6.385, df=2, and p=0.0411; expected counts are computed from row and column totals. $$E_{ij}=\frac{(\text{row total})(\text{column total})}{123},{\chi }^2=6.385,df=(2-1)(3-1)=2.$$ Interpretation: A small p-value indicates evidence against the null distributional claim, but the design controls whether the conclusion concerns association, homogeneity, or treatment effects. Validity: Use random sampling or random assignment as appropriate, independent observations, and sufficiently large expected counts. Error to reject: Check expected counts, not observed counts; a large cell contribution identifies where observed and expected differ most.

Worked solution P68-Easy-10. The statistic is 7.189, df=2, and p=0.0275; expected counts are computed from row and column totals. $$E_{ij}=\frac{(\text{row total})(\text{column total})}{125},{\chi }^2=7.189,df=(2-1)(3-1)=2.$$ Interpretation: A small p-value indicates evidence against the null distributional claim, but the design controls whether the conclusion concerns association, homogeneity, or treatment effects. Validity: Use random sampling or random assignment as appropriate, independent observations, and sufficiently large expected counts. Error to reject: Check expected counts, not observed counts; a large cell contribution identifies where observed and expected differ most.

Worked solution P68-Easy-11. The statistic is 5.622, df=2, and p=0.0602; expected counts are computed from row and column totals. $$E_{ij}=\frac{(\text{row total})(\text{column total})}{122},{\chi }^2=5.622,df=(2-1)(3-1)=2.$$ Interpretation: A small p-value indicates evidence against the null distributional claim, but the design controls whether the conclusion concerns association, homogeneity, or treatment effects. Validity: Use random sampling or random assignment as appropriate, independent observations, and sufficiently large expected counts. Error to reject: Check expected counts, not observed counts; a large cell contribution identifies where observed and expected differ most.

Worked solution P68-Easy-12. The statistic is 6.418, df=2, and p=0.0404; expected counts are computed from row and column totals. $$E_{ij}=\frac{(\text{row total})(\text{column total})}{124},{\chi }^2=6.418,df=(2-1)(3-1)=2.$$ Interpretation: A small p-value indicates evidence against the null distributional claim, but the design controls whether the conclusion concerns association, homogeneity, or treatment effects. Validity: Use random sampling or random assignment as appropriate, independent observations, and sufficiently large expected counts. Error to reject: Check expected counts, not observed counts; a large cell contribution identifies where observed and expected differ most.

Worked solution P68-Easy-13. The statistic is 5.931, df=2, and p=0.0515; expected counts are computed from row and column totals. $$E_{ij}=\frac{(\text{row total})(\text{column total})}{122},{\chi }^2=5.931,df=(2-1)(3-1)=2.$$ Interpretation: A small p-value indicates evidence against the null distributional claim, but the design controls whether the conclusion concerns association, homogeneity, or treatment effects. Validity: Use random sampling or random assignment as appropriate, independent observations, and sufficiently large expected counts. Error to reject: Check expected counts, not observed counts; a large cell contribution identifies where observed and expected differ most.

Worked solution P68-Easy-14. The statistic is 6.715, df=2, and p=0.0348; expected counts are computed from row and column totals. $$E_{ij}=\frac{(\text{row total})(\text{column total})}{124},{\chi }^2=6.715,df=(2-1)(3-1)=2.$$ Interpretation: A small p-value indicates evidence against the null distributional claim, but the design controls whether the conclusion concerns association, homogeneity, or treatment effects. Validity: Use random sampling or random assignment as appropriate, independent observations, and sufficiently large expected counts. Error to reject: Check expected counts, not observed counts; a large cell contribution identifies where observed and expected differ most.

Worked solution P68-Easy-15. The statistic is 7.546, df=2, and p=0.0230; expected counts are computed from row and column totals. $$E_{ij}=\frac{(\text{row total})(\text{column total})}{126},{\chi }^2=7.546,df=(2-1)(3-1)=2.$$ Interpretation: A small p-value indicates evidence against the null distributional claim, but the design controls whether the conclusion concerns association, homogeneity, or treatment effects. Validity: Use random sampling or random assignment as appropriate, independent observations, and sufficiently large expected counts. Error to reject: Check expected counts, not observed counts; a large cell contribution identifies where observed and expected differ most.

Worked solution P68-Tough-1. The statistic is 5.043, df=2, and p=0.0804; expected counts are computed from row and column totals. $$E_{ij}=\frac{(\text{row total})(\text{column total})}{120},{\chi }^2=5.043,df=(2-1)(3-1)=2.$$ Interpretation: A small p-value indicates evidence against the null distributional claim, but the design controls whether the conclusion concerns association, homogeneity, or treatment effects. Validity: Use random sampling or random assignment as appropriate, independent observations, and sufficiently large expected counts. Error to reject: Check expected counts, not observed counts; a large cell contribution identifies where observed and expected differ most.

Worked solution P68-Tough-2. The statistic is 5.782, df=2, and p=0.0555; expected counts are computed from row and column totals. $$E_{ij}=\frac{(\text{row total})(\text{column total})}{122},{\chi }^2=5.782,df=(2-1)(3-1)=2.$$ Interpretation: A small p-value indicates evidence against the null distributional claim, but the design controls whether the conclusion concerns association, homogeneity, or treatment effects. Validity: Use random sampling or random assignment as appropriate, independent observations, and sufficiently large expected counts. Error to reject: Check expected counts, not observed counts; a large cell contribution identifies where observed and expected differ most.

Worked solution P68-Tough-3. The statistic is 6.573, df=2, and p=0.0374; expected counts are computed from row and column totals. $$E_{ij}=\frac{(\text{row total})(\text{column total})}{124},{\chi }^2=6.573,df=(2-1)(3-1)=2.$$ Interpretation: A small p-value indicates evidence against the null distributional claim, but the design controls whether the conclusion concerns association, homogeneity, or treatment effects. Validity: Use random sampling or random assignment as appropriate, independent observations, and sufficiently large expected counts. Error to reject: Check expected counts, not observed counts; a large cell contribution identifies where observed and expected differ most.

Worked solution P68-Tough-4. The statistic is 7.412, df=2, and p=0.0246; expected counts are computed from row and column totals. $$E_{ij}=\frac{(\text{row total})(\text{column total})}{126},{\chi }^2=7.412,df=(2-1)(3-1)=2.$$ Interpretation: A small p-value indicates evidence against the null distributional claim, but the design controls whether the conclusion concerns association, homogeneity, or treatment effects. Validity: Use random sampling or random assignment as appropriate, independent observations, and sufficiently large expected counts. Error to reject: Check expected counts, not observed counts; a large cell contribution identifies where observed and expected differ most.

Worked solution P68-Tough-5. The statistic is 6.844, df=2, and p=0.0327; expected counts are computed from row and column totals. $$E_{ij}=\frac{(\text{row total})(\text{column total})}{124},{\chi }^2=6.844,df=(2-1)(3-1)=2.$$ Interpretation: A small p-value indicates evidence against the null distributional claim, but the design controls whether the conclusion concerns association, homogeneity, or treatment effects. Validity: Use random sampling or random assignment as appropriate, independent observations, and sufficiently large expected counts. Error to reject: Check expected counts, not observed counts; a large cell contribution identifies where observed and expected differ most.

Worked solution P68-Tough-6. The statistic is 5.325, df=2, and p=0.0698; expected counts are computed from row and column totals. $$E_{ij}=\frac{(\text{row total})(\text{column total})}{121},{\chi }^2=5.325,df=(2-1)(3-1)=2.$$ Interpretation: A small p-value indicates evidence against the null distributional claim, but the design controls whether the conclusion concerns association, homogeneity, or treatment effects. Validity: Use random sampling or random assignment as appropriate, independent observations, and sufficiently large expected counts. Error to reject: Check expected counts, not observed counts; a large cell contribution identifies where observed and expected differ most.

Worked solution P68-Tough-7. The statistic is 6.094, df=2, and p=0.0475; expected counts are computed from row and column totals. $$E_{ij}=\frac{(\text{row total})(\text{column total})}{123},{\chi }^2=6.094,df=(2-1)(3-1)=2.$$ Interpretation: A small p-value indicates evidence against the null distributional claim, but the design controls whether the conclusion concerns association, homogeneity, or treatment effects. Validity: Use random sampling or random assignment as appropriate, independent observations, and sufficiently large expected counts. Error to reject: Check expected counts, not observed counts; a large cell contribution identifies where observed and expected differ most.

Worked solution P68-Tough-8. The statistic is 6.912, df=2, and p=0.0315; expected counts are computed from row and column totals. $$E_{ij}=\frac{(\text{row total})(\text{column total})}{125},{\chi }^2=6.912,df=(2-1)(3-1)=2.$$ Interpretation: A small p-value indicates evidence against the null distributional claim, but the design controls whether the conclusion concerns association, homogeneity, or treatment effects. Validity: Use random sampling or random assignment as appropriate, independent observations, and sufficiently large expected counts. Error to reject: Check expected counts, not observed counts; a large cell contribution identifies where observed and expected differ most.

Worked solution P68-Tough-9. The statistic is 6.385, df=2, and p=0.0411; expected counts are computed from row and column totals. $$E_{ij}=\frac{(\text{row total})(\text{column total})}{123},{\chi }^2=6.385,df=(2-1)(3-1)=2.$$ Interpretation: A small p-value indicates evidence against the null distributional claim, but the design controls whether the conclusion concerns association, homogeneity, or treatment effects. Validity: Use random sampling or random assignment as appropriate, independent observations, and sufficiently large expected counts. Error to reject: Check expected counts, not observed counts; a large cell contribution identifies where observed and expected differ most.

Worked solution P68-Tough-10. The statistic is 7.189, df=2, and p=0.0275; expected counts are computed from row and column totals. $$E_{ij}=\frac{(\text{row total})(\text{column total})}{125},{\chi }^2=7.189,df=(2-1)(3-1)=2.$$ Interpretation: A small p-value indicates evidence against the null distributional claim, but the design controls whether the conclusion concerns association, homogeneity, or treatment effects. Validity: Use random sampling or random assignment as appropriate, independent observations, and sufficiently large expected counts. Error to reject: Check expected counts, not observed counts; a large cell contribution identifies where observed and expected differ most.

Worked solution P68-Tough-11. The statistic is 5.622, df=2, and p=0.0602; expected counts are computed from row and column totals. $$E_{ij}=\frac{(\text{row total})(\text{column total})}{122},{\chi }^2=5.622,df=(2-1)(3-1)=2.$$ Interpretation: A small p-value indicates evidence against the null distributional claim, but the design controls whether the conclusion concerns association, homogeneity, or treatment effects. Validity: Use random sampling or random assignment as appropriate, independent observations, and sufficiently large expected counts. Error to reject: Check expected counts, not observed counts; a large cell contribution identifies where observed and expected differ most.

Worked solution P68-Tough-12. The statistic is 6.418, df=2, and p=0.0404; expected counts are computed from row and column totals. $$E_{ij}=\frac{(\text{row total})(\text{column total})}{124},{\chi }^2=6.418,df=(2-1)(3-1)=2.$$ Interpretation: A small p-value indicates evidence against the null distributional claim, but the design controls whether the conclusion concerns association, homogeneity, or treatment effects. Validity: Use random sampling or random assignment as appropriate, independent observations, and sufficiently large expected counts. Error to reject: Check expected counts, not observed counts; a large cell contribution identifies where observed and expected differ most.

Worked solution P68-Tough-13. The statistic is 5.931, df=2, and p=0.0515; expected counts are computed from row and column totals. $$E_{ij}=\frac{(\text{row total})(\text{column total})}{122},{\chi }^2=5.931,df=(2-1)(3-1)=2.$$ Interpretation: A small p-value indicates evidence against the null distributional claim, but the design controls whether the conclusion concerns association, homogeneity, or treatment effects. Validity: Use random sampling or random assignment as appropriate, independent observations, and sufficiently large expected counts. Error to reject: Check expected counts, not observed counts; a large cell contribution identifies where observed and expected differ most.

Worked solution P68-Tough-14. The statistic is 6.715, df=2, and p=0.0348; expected counts are computed from row and column totals. $$E_{ij}=\frac{(\text{row total})(\text{column total})}{124},{\chi }^2=6.715,df=(2-1)(3-1)=2.$$ Interpretation: A small p-value indicates evidence against the null distributional claim, but the design controls whether the conclusion concerns association, homogeneity, or treatment effects. Validity: Use random sampling or random assignment as appropriate, independent observations, and sufficiently large expected counts. Error to reject: Check expected counts, not observed counts; a large cell contribution identifies where observed and expected differ most.

Worked solution P68-Tough-15. The statistic is 7.546, df=2, and p=0.0230; expected counts are computed from row and column totals. $$E_{ij}=\frac{(\text{row total})(\text{column total})}{126},{\chi }^2=7.546,df=(2-1)(3-1)=2.$$ Interpretation: A small p-value indicates evidence against the null distributional claim, but the design controls whether the conclusion concerns association, homogeneity, or treatment effects. Validity: Use random sampling or random assignment as appropriate, independent observations, and sufficiently large expected counts. Error to reject: Check expected counts, not observed counts; a large cell contribution identifies where observed and expected differ most.

Worked solution P68-Toughest-1. The statistic is 5.043, df=2, and p=0.0804; expected counts are computed from row and column totals. $$E_{ij}=\frac{(\text{row total})(\text{column total})}{120},{\chi }^2=5.043,df=(2-1)(3-1)=2.$$ Interpretation: A small p-value indicates evidence against the null distributional claim, but the design controls whether the conclusion concerns association, homogeneity, or treatment effects. Validity: Use random sampling or random assignment as appropriate, independent observations, and sufficiently large expected counts. Error to reject: Check expected counts, not observed counts; a large cell contribution identifies where observed and expected differ most.

Worked solution P68-Toughest-2. The statistic is 5.782, df=2, and p=0.0555; expected counts are computed from row and column totals. $$E_{ij}=\frac{(\text{row total})(\text{column total})}{122},{\chi }^2=5.782,df=(2-1)(3-1)=2.$$ Interpretation: A small p-value indicates evidence against the null distributional claim, but the design controls whether the conclusion concerns association, homogeneity, or treatment effects. Validity: Use random sampling or random assignment as appropriate, independent observations, and sufficiently large expected counts. Error to reject: Check expected counts, not observed counts; a large cell contribution identifies where observed and expected differ most.

Worked solution P68-Toughest-3. The statistic is 6.573, df=2, and p=0.0374; expected counts are computed from row and column totals. $$E_{ij}=\frac{(\text{row total})(\text{column total})}{124},{\chi }^2=6.573,df=(2-1)(3-1)=2.$$ Interpretation: A small p-value indicates evidence against the null distributional claim, but the design controls whether the conclusion concerns association, homogeneity, or treatment effects. Validity: Use random sampling or random assignment as appropriate, independent observations, and sufficiently large expected counts. Error to reject: Check expected counts, not observed counts; a large cell contribution identifies where observed and expected differ most.

Worked solution P68-Toughest-4. The statistic is 7.412, df=2, and p=0.0246; expected counts are computed from row and column totals. $$E_{ij}=\frac{(\text{row total})(\text{column total})}{126},{\chi }^2=7.412,df=(2-1)(3-1)=2.$$ Interpretation: A small p-value indicates evidence against the null distributional claim, but the design controls whether the conclusion concerns association, homogeneity, or treatment effects. Validity: Use random sampling or random assignment as appropriate, independent observations, and sufficiently large expected counts. Error to reject: Check expected counts, not observed counts; a large cell contribution identifies where observed and expected differ most.

Worked solution P68-Toughest-5. The statistic is 6.844, df=2, and p=0.0327; expected counts are computed from row and column totals. $$E_{ij}=\frac{(\text{row total})(\text{column total})}{124},{\chi }^2=6.844,df=(2-1)(3-1)=2.$$ Interpretation: A small p-value indicates evidence against the null distributional claim, but the design controls whether the conclusion concerns association, homogeneity, or treatment effects. Validity: Use random sampling or random assignment as appropriate, independent observations, and sufficiently large expected counts. Error to reject: Check expected counts, not observed counts; a large cell contribution identifies where observed and expected differ most.

Worked solution P68-Toughest-6. The statistic is 5.325, df=2, and p=0.0698; expected counts are computed from row and column totals. $$E_{ij}=\frac{(\text{row total})(\text{column total})}{121},{\chi }^2=5.325,df=(2-1)(3-1)=2.$$ Interpretation: A small p-value indicates evidence against the null distributional claim, but the design controls whether the conclusion concerns association, homogeneity, or treatment effects. Validity: Use random sampling or random assignment as appropriate, independent observations, and sufficiently large expected counts. Error to reject: Check expected counts, not observed counts; a large cell contribution identifies where observed and expected differ most.

Worked solution P68-Toughest-7. The statistic is 6.094, df=2, and p=0.0475; expected counts are computed from row and column totals. $$E_{ij}=\frac{(\text{row total})(\text{column total})}{123},{\chi }^2=6.094,df=(2-1)(3-1)=2.$$ Interpretation: A small p-value indicates evidence against the null distributional claim, but the design controls whether the conclusion concerns association, homogeneity, or treatment effects. Validity: Use random sampling or random assignment as appropriate, independent observations, and sufficiently large expected counts. Error to reject: Check expected counts, not observed counts; a large cell contribution identifies where observed and expected differ most.

Worked solution P68-Toughest-8. The statistic is 6.912, df=2, and p=0.0315; expected counts are computed from row and column totals. $$E_{ij}=\frac{(\text{row total})(\text{column total})}{125},{\chi }^2=6.912,df=(2-1)(3-1)=2.$$ Interpretation: A small p-value indicates evidence against the null distributional claim, but the design controls whether the conclusion concerns association, homogeneity, or treatment effects. Validity: Use random sampling or random assignment as appropriate, independent observations, and sufficiently large expected counts. Error to reject: Check expected counts, not observed counts; a large cell contribution identifies where observed and expected differ most.

Worked solution P68-Toughest-9. The statistic is 6.385, df=2, and p=0.0411; expected counts are computed from row and column totals. $$E_{ij}=\frac{(\text{row total})(\text{column total})}{123},{\chi }^2=6.385,df=(2-1)(3-1)=2.$$ Interpretation: A small p-value indicates evidence against the null distributional claim, but the design controls whether the conclusion concerns association, homogeneity, or treatment effects. Validity: Use random sampling or random assignment as appropriate, independent observations, and sufficiently large expected counts. Error to reject: Check expected counts, not observed counts; a large cell contribution identifies where observed and expected differ most.

Worked solution P68-Toughest-10. The statistic is 7.189, df=2, and p=0.0275; expected counts are computed from row and column totals. $$E_{ij}=\frac{(\text{row total})(\text{column total})}{125},{\chi }^2=7.189,df=(2-1)(3-1)=2.$$ Interpretation: A small p-value indicates evidence against the null distributional claim, but the design controls whether the conclusion concerns association, homogeneity, or treatment effects. Validity: Use random sampling or random assignment as appropriate, independent observations, and sufficiently large expected counts. Error to reject: Check expected counts, not observed counts; a large cell contribution identifies where observed and expected differ most.

Worked solution P68-Toughest-11. The statistic is 5.622, df=2, and p=0.0602; expected counts are computed from row and column totals. $$E_{ij}=\frac{(\text{row total})(\text{column total})}{122},{\chi }^2=5.622,df=(2-1)(3-1)=2.$$ Interpretation: A small p-value indicates evidence against the null distributional claim, but the design controls whether the conclusion concerns association, homogeneity, or treatment effects. Validity: Use random sampling or random assignment as appropriate, independent observations, and sufficiently large expected counts. Error to reject: Check expected counts, not observed counts; a large cell contribution identifies where observed and expected differ most.

Worked solution P68-Toughest-12. The statistic is 6.418, df=2, and p=0.0404; expected counts are computed from row and column totals. $$E_{ij}=\frac{(\text{row total})(\text{column total})}{124},{\chi }^2=6.418,df=(2-1)(3-1)=2.$$ Interpretation: A small p-value indicates evidence against the null distributional claim, but the design controls whether the conclusion concerns association, homogeneity, or treatment effects. Validity: Use random sampling or random assignment as appropriate, independent observations, and sufficiently large expected counts. Error to reject: Check expected counts, not observed counts; a large cell contribution identifies where observed and expected differ most.

Worked solution P68-Toughest-13. The statistic is 5.931, df=2, and p=0.0515; expected counts are computed from row and column totals. $$E_{ij}=\frac{(\text{row total})(\text{column total})}{122},{\chi }^2=5.931,df=(2-1)(3-1)=2.$$ Interpretation: A small p-value indicates evidence against the null distributional claim, but the design controls whether the conclusion concerns association, homogeneity, or treatment effects. Validity: Use random sampling or random assignment as appropriate, independent observations, and sufficiently large expected counts. Error to reject: Check expected counts, not observed counts; a large cell contribution identifies where observed and expected differ most.

Worked solution P68-Toughest-14. The statistic is 6.715, df=2, and p=0.0348; expected counts are computed from row and column totals. $$E_{ij}=\frac{(\text{row total})(\text{column total})}{124},{\chi }^2=6.715,df=(2-1)(3-1)=2.$$ Interpretation: A small p-value indicates evidence against the null distributional claim, but the design controls whether the conclusion concerns association, homogeneity, or treatment effects. Validity: Use random sampling or random assignment as appropriate, independent observations, and sufficiently large expected counts. Error to reject: Check expected counts, not observed counts; a large cell contribution identifies where observed and expected differ most.

Worked solution P68-Toughest-15. The statistic is 7.546, df=2, and p=0.0230; expected counts are computed from row and column totals. $$E_{ij}=\frac{(\text{row total})(\text{column total})}{126},{\chi }^2=7.546,df=(2-1)(3-1)=2.$$ Interpretation: A small p-value indicates evidence against the null distributional claim, but the design controls whether the conclusion concerns association, homogeneity, or treatment effects. Validity: Use random sampling or random assignment as appropriate, independent observations, and sufficiently large expected counts. Error to reject: Check expected counts, not observed counts; a large cell contribution identifies where observed and expected differ most.