Worked solution P71-Easy-1. For the homogeneity test, ${\chi }^2=5.043$, df=2, and p=0.0804; expected counts model the same categorical distribution across groups. $$E_{ij}=\frac{({\text{group total}}_i)({\text{response total}}_j)}{120},{\chi }^2=5.043,df=2.$$ Interpretation: A small p-value gives evidence that the categorical response distribution is not homogeneous across the sampled populations or assigned treatments. Validity: Groups arise from independent random samples or randomized treatments, responses are counted once, and the expected-count approximation is adequate. Error to reject: The homogeneity question compares response distributions across groups; it is not framed as two variables measured on one random sample.

Worked solution P71-Easy-2. For the homogeneity test, ${\chi }^2=5.782$, df=2, and p=0.0555; expected counts model the same categorical distribution across groups. $$E_{ij}=\frac{({\text{group total}}_i)({\text{response total}}_j)}{122},{\chi }^2=5.782,df=2.$$ Interpretation: A small p-value gives evidence that the categorical response distribution is not homogeneous across the sampled populations or assigned treatments. Validity: Groups arise from independent random samples or randomized treatments, responses are counted once, and the expected-count approximation is adequate. Error to reject: The homogeneity question compares response distributions across groups; it is not framed as two variables measured on one random sample.

Worked solution P71-Easy-3. For the homogeneity test, ${\chi }^2=6.573$, df=2, and p=0.0374; expected counts model the same categorical distribution across groups. $$E_{ij}=\frac{({\text{group total}}_i)({\text{response total}}_j)}{124},{\chi }^2=6.573,df=2.$$ Interpretation: A small p-value gives evidence that the categorical response distribution is not homogeneous across the sampled populations or assigned treatments. Validity: Groups arise from independent random samples or randomized treatments, responses are counted once, and the expected-count approximation is adequate. Error to reject: The homogeneity question compares response distributions across groups; it is not framed as two variables measured on one random sample.

Worked solution P71-Easy-4. For the homogeneity test, ${\chi }^2=7.412$, df=2, and p=0.0246; expected counts model the same categorical distribution across groups. $$E_{ij}=\frac{({\text{group total}}_i)({\text{response total}}_j)}{126},{\chi }^2=7.412,df=2.$$ Interpretation: A small p-value gives evidence that the categorical response distribution is not homogeneous across the sampled populations or assigned treatments. Validity: Groups arise from independent random samples or randomized treatments, responses are counted once, and the expected-count approximation is adequate. Error to reject: The homogeneity question compares response distributions across groups; it is not framed as two variables measured on one random sample.

Worked solution P71-Easy-5. For the homogeneity test, ${\chi }^2=6.844$, df=2, and p=0.0327; expected counts model the same categorical distribution across groups. $$E_{ij}=\frac{({\text{group total}}_i)({\text{response total}}_j)}{124},{\chi }^2=6.844,df=2.$$ Interpretation: A small p-value gives evidence that the categorical response distribution is not homogeneous across the sampled populations or assigned treatments. Validity: Groups arise from independent random samples or randomized treatments, responses are counted once, and the expected-count approximation is adequate. Error to reject: The homogeneity question compares response distributions across groups; it is not framed as two variables measured on one random sample.

Worked solution P71-Easy-6. For the homogeneity test, ${\chi }^2=5.325$, df=2, and p=0.0698; expected counts model the same categorical distribution across groups. $$E_{ij}=\frac{({\text{group total}}_i)({\text{response total}}_j)}{121},{\chi }^2=5.325,df=2.$$ Interpretation: A small p-value gives evidence that the categorical response distribution is not homogeneous across the sampled populations or assigned treatments. Validity: Groups arise from independent random samples or randomized treatments, responses are counted once, and the expected-count approximation is adequate. Error to reject: The homogeneity question compares response distributions across groups; it is not framed as two variables measured on one random sample.

Worked solution P71-Easy-7. For the homogeneity test, ${\chi }^2=6.094$, df=2, and p=0.0475; expected counts model the same categorical distribution across groups. $$E_{ij}=\frac{({\text{group total}}_i)({\text{response total}}_j)}{123},{\chi }^2=6.094,df=2.$$ Interpretation: A small p-value gives evidence that the categorical response distribution is not homogeneous across the sampled populations or assigned treatments. Validity: Groups arise from independent random samples or randomized treatments, responses are counted once, and the expected-count approximation is adequate. Error to reject: The homogeneity question compares response distributions across groups; it is not framed as two variables measured on one random sample.

Worked solution P71-Easy-8. For the homogeneity test, ${\chi }^2=6.912$, df=2, and p=0.0315; expected counts model the same categorical distribution across groups. $$E_{ij}=\frac{({\text{group total}}_i)({\text{response total}}_j)}{125},{\chi }^2=6.912,df=2.$$ Interpretation: A small p-value gives evidence that the categorical response distribution is not homogeneous across the sampled populations or assigned treatments. Validity: Groups arise from independent random samples or randomized treatments, responses are counted once, and the expected-count approximation is adequate. Error to reject: The homogeneity question compares response distributions across groups; it is not framed as two variables measured on one random sample.

Worked solution P71-Easy-9. For the homogeneity test, ${\chi }^2=6.385$, df=2, and p=0.0411; expected counts model the same categorical distribution across groups. $$E_{ij}=\frac{({\text{group total}}_i)({\text{response total}}_j)}{123},{\chi }^2=6.385,df=2.$$ Interpretation: A small p-value gives evidence that the categorical response distribution is not homogeneous across the sampled populations or assigned treatments. Validity: Groups arise from independent random samples or randomized treatments, responses are counted once, and the expected-count approximation is adequate. Error to reject: The homogeneity question compares response distributions across groups; it is not framed as two variables measured on one random sample.

Worked solution P71-Easy-10. For the homogeneity test, ${\chi }^2=7.189$, df=2, and p=0.0275; expected counts model the same categorical distribution across groups. $$E_{ij}=\frac{({\text{group total}}_i)({\text{response total}}_j)}{125},{\chi }^2=7.189,df=2.$$ Interpretation: A small p-value gives evidence that the categorical response distribution is not homogeneous across the sampled populations or assigned treatments. Validity: Groups arise from independent random samples or randomized treatments, responses are counted once, and the expected-count approximation is adequate. Error to reject: The homogeneity question compares response distributions across groups; it is not framed as two variables measured on one random sample.

Worked solution P71-Easy-11. For the homogeneity test, ${\chi }^2=5.622$, df=2, and p=0.0602; expected counts model the same categorical distribution across groups. $$E_{ij}=\frac{({\text{group total}}_i)({\text{response total}}_j)}{122},{\chi }^2=5.622,df=2.$$ Interpretation: A small p-value gives evidence that the categorical response distribution is not homogeneous across the sampled populations or assigned treatments. Validity: Groups arise from independent random samples or randomized treatments, responses are counted once, and the expected-count approximation is adequate. Error to reject: The homogeneity question compares response distributions across groups; it is not framed as two variables measured on one random sample.

Worked solution P71-Easy-12. For the homogeneity test, ${\chi }^2=6.418$, df=2, and p=0.0404; expected counts model the same categorical distribution across groups. $$E_{ij}=\frac{({\text{group total}}_i)({\text{response total}}_j)}{124},{\chi }^2=6.418,df=2.$$ Interpretation: A small p-value gives evidence that the categorical response distribution is not homogeneous across the sampled populations or assigned treatments. Validity: Groups arise from independent random samples or randomized treatments, responses are counted once, and the expected-count approximation is adequate. Error to reject: The homogeneity question compares response distributions across groups; it is not framed as two variables measured on one random sample.

Worked solution P71-Easy-13. For the homogeneity test, ${\chi }^2=5.931$, df=2, and p=0.0515; expected counts model the same categorical distribution across groups. $$E_{ij}=\frac{({\text{group total}}_i)({\text{response total}}_j)}{122},{\chi }^2=5.931,df=2.$$ Interpretation: A small p-value gives evidence that the categorical response distribution is not homogeneous across the sampled populations or assigned treatments. Validity: Groups arise from independent random samples or randomized treatments, responses are counted once, and the expected-count approximation is adequate. Error to reject: The homogeneity question compares response distributions across groups; it is not framed as two variables measured on one random sample.

Worked solution P71-Easy-14. For the homogeneity test, ${\chi }^2=6.715$, df=2, and p=0.0348; expected counts model the same categorical distribution across groups. $$E_{ij}=\frac{({\text{group total}}_i)({\text{response total}}_j)}{124},{\chi }^2=6.715,df=2.$$ Interpretation: A small p-value gives evidence that the categorical response distribution is not homogeneous across the sampled populations or assigned treatments. Validity: Groups arise from independent random samples or randomized treatments, responses are counted once, and the expected-count approximation is adequate. Error to reject: The homogeneity question compares response distributions across groups; it is not framed as two variables measured on one random sample.

Worked solution P71-Easy-15. For the homogeneity test, ${\chi }^2=7.546$, df=2, and p=0.0230; expected counts model the same categorical distribution across groups. $$E_{ij}=\frac{({\text{group total}}_i)({\text{response total}}_j)}{126},{\chi }^2=7.546,df=2.$$ Interpretation: A small p-value gives evidence that the categorical response distribution is not homogeneous across the sampled populations or assigned treatments. Validity: Groups arise from independent random samples or randomized treatments, responses are counted once, and the expected-count approximation is adequate. Error to reject: The homogeneity question compares response distributions across groups; it is not framed as two variables measured on one random sample.

Worked solution P71-Tough-1. For the homogeneity test, ${\chi }^2=5.043$, df=2, and p=0.0804; expected counts model the same categorical distribution across groups. $$E_{ij}=\frac{({\text{group total}}_i)({\text{response total}}_j)}{120},{\chi }^2=5.043,df=2.$$ Interpretation: A small p-value gives evidence that the categorical response distribution is not homogeneous across the sampled populations or assigned treatments. Validity: Groups arise from independent random samples or randomized treatments, responses are counted once, and the expected-count approximation is adequate. Error to reject: The homogeneity question compares response distributions across groups; it is not framed as two variables measured on one random sample.

Worked solution P71-Tough-2. For the homogeneity test, ${\chi }^2=5.782$, df=2, and p=0.0555; expected counts model the same categorical distribution across groups. $$E_{ij}=\frac{({\text{group total}}_i)({\text{response total}}_j)}{122},{\chi }^2=5.782,df=2.$$ Interpretation: A small p-value gives evidence that the categorical response distribution is not homogeneous across the sampled populations or assigned treatments. Validity: Groups arise from independent random samples or randomized treatments, responses are counted once, and the expected-count approximation is adequate. Error to reject: The homogeneity question compares response distributions across groups; it is not framed as two variables measured on one random sample.

Worked solution P71-Tough-3. For the homogeneity test, ${\chi }^2=6.573$, df=2, and p=0.0374; expected counts model the same categorical distribution across groups. $$E_{ij}=\frac{({\text{group total}}_i)({\text{response total}}_j)}{124},{\chi }^2=6.573,df=2.$$ Interpretation: A small p-value gives evidence that the categorical response distribution is not homogeneous across the sampled populations or assigned treatments. Validity: Groups arise from independent random samples or randomized treatments, responses are counted once, and the expected-count approximation is adequate. Error to reject: The homogeneity question compares response distributions across groups; it is not framed as two variables measured on one random sample.

Worked solution P71-Tough-4. For the homogeneity test, ${\chi }^2=7.412$, df=2, and p=0.0246; expected counts model the same categorical distribution across groups. $$E_{ij}=\frac{({\text{group total}}_i)({\text{response total}}_j)}{126},{\chi }^2=7.412,df=2.$$ Interpretation: A small p-value gives evidence that the categorical response distribution is not homogeneous across the sampled populations or assigned treatments. Validity: Groups arise from independent random samples or randomized treatments, responses are counted once, and the expected-count approximation is adequate. Error to reject: The homogeneity question compares response distributions across groups; it is not framed as two variables measured on one random sample.

Worked solution P71-Tough-5. For the homogeneity test, ${\chi }^2=6.844$, df=2, and p=0.0327; expected counts model the same categorical distribution across groups. $$E_{ij}=\frac{({\text{group total}}_i)({\text{response total}}_j)}{124},{\chi }^2=6.844,df=2.$$ Interpretation: A small p-value gives evidence that the categorical response distribution is not homogeneous across the sampled populations or assigned treatments. Validity: Groups arise from independent random samples or randomized treatments, responses are counted once, and the expected-count approximation is adequate. Error to reject: The homogeneity question compares response distributions across groups; it is not framed as two variables measured on one random sample.

Worked solution P71-Tough-6. For the homogeneity test, ${\chi }^2=5.325$, df=2, and p=0.0698; expected counts model the same categorical distribution across groups. $$E_{ij}=\frac{({\text{group total}}_i)({\text{response total}}_j)}{121},{\chi }^2=5.325,df=2.$$ Interpretation: A small p-value gives evidence that the categorical response distribution is not homogeneous across the sampled populations or assigned treatments. Validity: Groups arise from independent random samples or randomized treatments, responses are counted once, and the expected-count approximation is adequate. Error to reject: The homogeneity question compares response distributions across groups; it is not framed as two variables measured on one random sample.

Worked solution P71-Tough-7. For the homogeneity test, ${\chi }^2=6.094$, df=2, and p=0.0475; expected counts model the same categorical distribution across groups. $$E_{ij}=\frac{({\text{group total}}_i)({\text{response total}}_j)}{123},{\chi }^2=6.094,df=2.$$ Interpretation: A small p-value gives evidence that the categorical response distribution is not homogeneous across the sampled populations or assigned treatments. Validity: Groups arise from independent random samples or randomized treatments, responses are counted once, and the expected-count approximation is adequate. Error to reject: The homogeneity question compares response distributions across groups; it is not framed as two variables measured on one random sample.

Worked solution P71-Tough-8. For the homogeneity test, ${\chi }^2=6.912$, df=2, and p=0.0315; expected counts model the same categorical distribution across groups. $$E_{ij}=\frac{({\text{group total}}_i)({\text{response total}}_j)}{125},{\chi }^2=6.912,df=2.$$ Interpretation: A small p-value gives evidence that the categorical response distribution is not homogeneous across the sampled populations or assigned treatments. Validity: Groups arise from independent random samples or randomized treatments, responses are counted once, and the expected-count approximation is adequate. Error to reject: The homogeneity question compares response distributions across groups; it is not framed as two variables measured on one random sample.

Worked solution P71-Tough-9. For the homogeneity test, ${\chi }^2=6.385$, df=2, and p=0.0411; expected counts model the same categorical distribution across groups. $$E_{ij}=\frac{({\text{group total}}_i)({\text{response total}}_j)}{123},{\chi }^2=6.385,df=2.$$ Interpretation: A small p-value gives evidence that the categorical response distribution is not homogeneous across the sampled populations or assigned treatments. Validity: Groups arise from independent random samples or randomized treatments, responses are counted once, and the expected-count approximation is adequate. Error to reject: The homogeneity question compares response distributions across groups; it is not framed as two variables measured on one random sample.

Worked solution P71-Tough-10. For the homogeneity test, ${\chi }^2=7.189$, df=2, and p=0.0275; expected counts model the same categorical distribution across groups. $$E_{ij}=\frac{({\text{group total}}_i)({\text{response total}}_j)}{125},{\chi }^2=7.189,df=2.$$ Interpretation: A small p-value gives evidence that the categorical response distribution is not homogeneous across the sampled populations or assigned treatments. Validity: Groups arise from independent random samples or randomized treatments, responses are counted once, and the expected-count approximation is adequate. Error to reject: The homogeneity question compares response distributions across groups; it is not framed as two variables measured on one random sample.

Worked solution P71-Tough-11. For the homogeneity test, ${\chi }^2=5.622$, df=2, and p=0.0602; expected counts model the same categorical distribution across groups. $$E_{ij}=\frac{({\text{group total}}_i)({\text{response total}}_j)}{122},{\chi }^2=5.622,df=2.$$ Interpretation: A small p-value gives evidence that the categorical response distribution is not homogeneous across the sampled populations or assigned treatments. Validity: Groups arise from independent random samples or randomized treatments, responses are counted once, and the expected-count approximation is adequate. Error to reject: The homogeneity question compares response distributions across groups; it is not framed as two variables measured on one random sample.

Worked solution P71-Tough-12. For the homogeneity test, ${\chi }^2=6.418$, df=2, and p=0.0404; expected counts model the same categorical distribution across groups. $$E_{ij}=\frac{({\text{group total}}_i)({\text{response total}}_j)}{124},{\chi }^2=6.418,df=2.$$ Interpretation: A small p-value gives evidence that the categorical response distribution is not homogeneous across the sampled populations or assigned treatments. Validity: Groups arise from independent random samples or randomized treatments, responses are counted once, and the expected-count approximation is adequate. Error to reject: The homogeneity question compares response distributions across groups; it is not framed as two variables measured on one random sample.

Worked solution P71-Tough-13. For the homogeneity test, ${\chi }^2=5.931$, df=2, and p=0.0515; expected counts model the same categorical distribution across groups. $$E_{ij}=\frac{({\text{group total}}_i)({\text{response total}}_j)}{122},{\chi }^2=5.931,df=2.$$ Interpretation: A small p-value gives evidence that the categorical response distribution is not homogeneous across the sampled populations or assigned treatments. Validity: Groups arise from independent random samples or randomized treatments, responses are counted once, and the expected-count approximation is adequate. Error to reject: The homogeneity question compares response distributions across groups; it is not framed as two variables measured on one random sample.

Worked solution P71-Tough-14. For the homogeneity test, ${\chi }^2=6.715$, df=2, and p=0.0348; expected counts model the same categorical distribution across groups. $$E_{ij}=\frac{({\text{group total}}_i)({\text{response total}}_j)}{124},{\chi }^2=6.715,df=2.$$ Interpretation: A small p-value gives evidence that the categorical response distribution is not homogeneous across the sampled populations or assigned treatments. Validity: Groups arise from independent random samples or randomized treatments, responses are counted once, and the expected-count approximation is adequate. Error to reject: The homogeneity question compares response distributions across groups; it is not framed as two variables measured on one random sample.

Worked solution P71-Tough-15. For the homogeneity test, ${\chi }^2=7.546$, df=2, and p=0.0230; expected counts model the same categorical distribution across groups. $$E_{ij}=\frac{({\text{group total}}_i)({\text{response total}}_j)}{126},{\chi }^2=7.546,df=2.$$ Interpretation: A small p-value gives evidence that the categorical response distribution is not homogeneous across the sampled populations or assigned treatments. Validity: Groups arise from independent random samples or randomized treatments, responses are counted once, and the expected-count approximation is adequate. Error to reject: The homogeneity question compares response distributions across groups; it is not framed as two variables measured on one random sample.

Worked solution P71-Toughest-1. For the homogeneity test, ${\chi }^2=5.043$, df=2, and p=0.0804; expected counts model the same categorical distribution across groups. $$E_{ij}=\frac{({\text{group total}}_i)({\text{response total}}_j)}{120},{\chi }^2=5.043,df=2.$$ Interpretation: A small p-value gives evidence that the categorical response distribution is not homogeneous across the sampled populations or assigned treatments. Validity: Groups arise from independent random samples or randomized treatments, responses are counted once, and the expected-count approximation is adequate. Error to reject: The homogeneity question compares response distributions across groups; it is not framed as two variables measured on one random sample.

Worked solution P71-Toughest-2. For the homogeneity test, ${\chi }^2=5.782$, df=2, and p=0.0555; expected counts model the same categorical distribution across groups. $$E_{ij}=\frac{({\text{group total}}_i)({\text{response total}}_j)}{122},{\chi }^2=5.782,df=2.$$ Interpretation: A small p-value gives evidence that the categorical response distribution is not homogeneous across the sampled populations or assigned treatments. Validity: Groups arise from independent random samples or randomized treatments, responses are counted once, and the expected-count approximation is adequate. Error to reject: The homogeneity question compares response distributions across groups; it is not framed as two variables measured on one random sample.

Worked solution P71-Toughest-3. For the homogeneity test, ${\chi }^2=6.573$, df=2, and p=0.0374; expected counts model the same categorical distribution across groups. $$E_{ij}=\frac{({\text{group total}}_i)({\text{response total}}_j)}{124},{\chi }^2=6.573,df=2.$$ Interpretation: A small p-value gives evidence that the categorical response distribution is not homogeneous across the sampled populations or assigned treatments. Validity: Groups arise from independent random samples or randomized treatments, responses are counted once, and the expected-count approximation is adequate. Error to reject: The homogeneity question compares response distributions across groups; it is not framed as two variables measured on one random sample.

Worked solution P71-Toughest-4. For the homogeneity test, ${\chi }^2=7.412$, df=2, and p=0.0246; expected counts model the same categorical distribution across groups. $$E_{ij}=\frac{({\text{group total}}_i)({\text{response total}}_j)}{126},{\chi }^2=7.412,df=2.$$ Interpretation: A small p-value gives evidence that the categorical response distribution is not homogeneous across the sampled populations or assigned treatments. Validity: Groups arise from independent random samples or randomized treatments, responses are counted once, and the expected-count approximation is adequate. Error to reject: The homogeneity question compares response distributions across groups; it is not framed as two variables measured on one random sample.

Worked solution P71-Toughest-5. For the homogeneity test, ${\chi }^2=6.844$, df=2, and p=0.0327; expected counts model the same categorical distribution across groups. $$E_{ij}=\frac{({\text{group total}}_i)({\text{response total}}_j)}{124},{\chi }^2=6.844,df=2.$$ Interpretation: A small p-value gives evidence that the categorical response distribution is not homogeneous across the sampled populations or assigned treatments. Validity: Groups arise from independent random samples or randomized treatments, responses are counted once, and the expected-count approximation is adequate. Error to reject: The homogeneity question compares response distributions across groups; it is not framed as two variables measured on one random sample.

Worked solution P71-Toughest-6. For the homogeneity test, ${\chi }^2=5.325$, df=2, and p=0.0698; expected counts model the same categorical distribution across groups. $$E_{ij}=\frac{({\text{group total}}_i)({\text{response total}}_j)}{121},{\chi }^2=5.325,df=2.$$ Interpretation: A small p-value gives evidence that the categorical response distribution is not homogeneous across the sampled populations or assigned treatments. Validity: Groups arise from independent random samples or randomized treatments, responses are counted once, and the expected-count approximation is adequate. Error to reject: The homogeneity question compares response distributions across groups; it is not framed as two variables measured on one random sample.

Worked solution P71-Toughest-7. For the homogeneity test, ${\chi }^2=6.094$, df=2, and p=0.0475; expected counts model the same categorical distribution across groups. $$E_{ij}=\frac{({\text{group total}}_i)({\text{response total}}_j)}{123},{\chi }^2=6.094,df=2.$$ Interpretation: A small p-value gives evidence that the categorical response distribution is not homogeneous across the sampled populations or assigned treatments. Validity: Groups arise from independent random samples or randomized treatments, responses are counted once, and the expected-count approximation is adequate. Error to reject: The homogeneity question compares response distributions across groups; it is not framed as two variables measured on one random sample.

Worked solution P71-Toughest-8. For the homogeneity test, ${\chi }^2=6.912$, df=2, and p=0.0315; expected counts model the same categorical distribution across groups. $$E_{ij}=\frac{({\text{group total}}_i)({\text{response total}}_j)}{125},{\chi }^2=6.912,df=2.$$ Interpretation: A small p-value gives evidence that the categorical response distribution is not homogeneous across the sampled populations or assigned treatments. Validity: Groups arise from independent random samples or randomized treatments, responses are counted once, and the expected-count approximation is adequate. Error to reject: The homogeneity question compares response distributions across groups; it is not framed as two variables measured on one random sample.

Worked solution P71-Toughest-9. For the homogeneity test, ${\chi }^2=6.385$, df=2, and p=0.0411; expected counts model the same categorical distribution across groups. $$E_{ij}=\frac{({\text{group total}}_i)({\text{response total}}_j)}{123},{\chi }^2=6.385,df=2.$$ Interpretation: A small p-value gives evidence that the categorical response distribution is not homogeneous across the sampled populations or assigned treatments. Validity: Groups arise from independent random samples or randomized treatments, responses are counted once, and the expected-count approximation is adequate. Error to reject: The homogeneity question compares response distributions across groups; it is not framed as two variables measured on one random sample.

Worked solution P71-Toughest-10. For the homogeneity test, ${\chi }^2=7.189$, df=2, and p=0.0275; expected counts model the same categorical distribution across groups. $$E_{ij}=\frac{({\text{group total}}_i)({\text{response total}}_j)}{125},{\chi }^2=7.189,df=2.$$ Interpretation: A small p-value gives evidence that the categorical response distribution is not homogeneous across the sampled populations or assigned treatments. Validity: Groups arise from independent random samples or randomized treatments, responses are counted once, and the expected-count approximation is adequate. Error to reject: The homogeneity question compares response distributions across groups; it is not framed as two variables measured on one random sample.

Worked solution P71-Toughest-11. For the homogeneity test, ${\chi }^2=5.622$, df=2, and p=0.0602; expected counts model the same categorical distribution across groups. $$E_{ij}=\frac{({\text{group total}}_i)({\text{response total}}_j)}{122},{\chi }^2=5.622,df=2.$$ Interpretation: A small p-value gives evidence that the categorical response distribution is not homogeneous across the sampled populations or assigned treatments. Validity: Groups arise from independent random samples or randomized treatments, responses are counted once, and the expected-count approximation is adequate. Error to reject: The homogeneity question compares response distributions across groups; it is not framed as two variables measured on one random sample.

Worked solution P71-Toughest-12. For the homogeneity test, ${\chi }^2=6.418$, df=2, and p=0.0404; expected counts model the same categorical distribution across groups. $$E_{ij}=\frac{({\text{group total}}_i)({\text{response total}}_j)}{124},{\chi }^2=6.418,df=2.$$ Interpretation: A small p-value gives evidence that the categorical response distribution is not homogeneous across the sampled populations or assigned treatments. Validity: Groups arise from independent random samples or randomized treatments, responses are counted once, and the expected-count approximation is adequate. Error to reject: The homogeneity question compares response distributions across groups; it is not framed as two variables measured on one random sample.

Worked solution P71-Toughest-13. For the homogeneity test, ${\chi }^2=5.931$, df=2, and p=0.0515; expected counts model the same categorical distribution across groups. $$E_{ij}=\frac{({\text{group total}}_i)({\text{response total}}_j)}{122},{\chi }^2=5.931,df=2.$$ Interpretation: A small p-value gives evidence that the categorical response distribution is not homogeneous across the sampled populations or assigned treatments. Validity: Groups arise from independent random samples or randomized treatments, responses are counted once, and the expected-count approximation is adequate. Error to reject: The homogeneity question compares response distributions across groups; it is not framed as two variables measured on one random sample.

Worked solution P71-Toughest-14. For the homogeneity test, ${\chi }^2=6.715$, df=2, and p=0.0348; expected counts model the same categorical distribution across groups. $$E_{ij}=\frac{({\text{group total}}_i)({\text{response total}}_j)}{124},{\chi }^2=6.715,df=2.$$ Interpretation: A small p-value gives evidence that the categorical response distribution is not homogeneous across the sampled populations or assigned treatments. Validity: Groups arise from independent random samples or randomized treatments, responses are counted once, and the expected-count approximation is adequate. Error to reject: The homogeneity question compares response distributions across groups; it is not framed as two variables measured on one random sample.

Worked solution P71-Toughest-15. For the homogeneity test, ${\chi }^2=7.546$, df=2, and p=0.0230; expected counts model the same categorical distribution across groups. $$E_{ij}=\frac{({\text{group total}}_i)({\text{response total}}_j)}{126},{\chi }^2=7.546,df=2.$$ Interpretation: A small p-value gives evidence that the categorical response distribution is not homogeneous across the sampled populations or assigned treatments. Validity: Groups arise from independent random samples or randomized treatments, responses are counted once, and the expected-count approximation is adequate. Error to reject: The homogeneity question compares response distributions across groups; it is not framed as two variables measured on one random sample.