Worked solution P69-Easy-1. For the independence test, ${\chi }^2=5.043$, df=2, and p=0.0804; expected counts model no association within the population. $$E_{ij}=\frac{({\text{row total}}_i)({\text{column total}}_j)}{120},{\chi }^2=5.043,df=2.$$ Interpretation: A small p-value gives evidence that the two categorical variables are associated in the population represented by the one-sample design. Validity: One probability sample supplies independent observational units; every unit contributes to exactly one cell, and expected counts support the approximation. Error to reject: The independence conclusion concerns association between two variables in one population, not equality of several separately sampled populations.

Worked solution P69-Easy-2. For the independence test, ${\chi }^2=5.782$, df=2, and p=0.0555; expected counts model no association within the population. $$E_{ij}=\frac{({\text{row total}}_i)({\text{column total}}_j)}{122},{\chi }^2=5.782,df=2.$$ Interpretation: A small p-value gives evidence that the two categorical variables are associated in the population represented by the one-sample design. Validity: One probability sample supplies independent observational units; every unit contributes to exactly one cell, and expected counts support the approximation. Error to reject: The independence conclusion concerns association between two variables in one population, not equality of several separately sampled populations.

Worked solution P69-Easy-3. For the independence test, ${\chi }^2=6.573$, df=2, and p=0.0374; expected counts model no association within the population. $$E_{ij}=\frac{({\text{row total}}_i)({\text{column total}}_j)}{124},{\chi }^2=6.573,df=2.$$ Interpretation: A small p-value gives evidence that the two categorical variables are associated in the population represented by the one-sample design. Validity: One probability sample supplies independent observational units; every unit contributes to exactly one cell, and expected counts support the approximation. Error to reject: The independence conclusion concerns association between two variables in one population, not equality of several separately sampled populations.

Worked solution P69-Easy-4. For the independence test, ${\chi }^2=7.412$, df=2, and p=0.0246; expected counts model no association within the population. $$E_{ij}=\frac{({\text{row total}}_i)({\text{column total}}_j)}{126},{\chi }^2=7.412,df=2.$$ Interpretation: A small p-value gives evidence that the two categorical variables are associated in the population represented by the one-sample design. Validity: One probability sample supplies independent observational units; every unit contributes to exactly one cell, and expected counts support the approximation. Error to reject: The independence conclusion concerns association between two variables in one population, not equality of several separately sampled populations.

Worked solution P69-Easy-5. For the independence test, ${\chi }^2=6.844$, df=2, and p=0.0327; expected counts model no association within the population. $$E_{ij}=\frac{({\text{row total}}_i)({\text{column total}}_j)}{124},{\chi }^2=6.844,df=2.$$ Interpretation: A small p-value gives evidence that the two categorical variables are associated in the population represented by the one-sample design. Validity: One probability sample supplies independent observational units; every unit contributes to exactly one cell, and expected counts support the approximation. Error to reject: The independence conclusion concerns association between two variables in one population, not equality of several separately sampled populations.

Worked solution P69-Easy-6. For the independence test, ${\chi }^2=5.325$, df=2, and p=0.0698; expected counts model no association within the population. $$E_{ij}=\frac{({\text{row total}}_i)({\text{column total}}_j)}{121},{\chi }^2=5.325,df=2.$$ Interpretation: A small p-value gives evidence that the two categorical variables are associated in the population represented by the one-sample design. Validity: One probability sample supplies independent observational units; every unit contributes to exactly one cell, and expected counts support the approximation. Error to reject: The independence conclusion concerns association between two variables in one population, not equality of several separately sampled populations.

Worked solution P69-Easy-7. For the independence test, ${\chi }^2=6.094$, df=2, and p=0.0475; expected counts model no association within the population. $$E_{ij}=\frac{({\text{row total}}_i)({\text{column total}}_j)}{123},{\chi }^2=6.094,df=2.$$ Interpretation: A small p-value gives evidence that the two categorical variables are associated in the population represented by the one-sample design. Validity: One probability sample supplies independent observational units; every unit contributes to exactly one cell, and expected counts support the approximation. Error to reject: The independence conclusion concerns association between two variables in one population, not equality of several separately sampled populations.

Worked solution P69-Easy-8. For the independence test, ${\chi }^2=6.912$, df=2, and p=0.0315; expected counts model no association within the population. $$E_{ij}=\frac{({\text{row total}}_i)({\text{column total}}_j)}{125},{\chi }^2=6.912,df=2.$$ Interpretation: A small p-value gives evidence that the two categorical variables are associated in the population represented by the one-sample design. Validity: One probability sample supplies independent observational units; every unit contributes to exactly one cell, and expected counts support the approximation. Error to reject: The independence conclusion concerns association between two variables in one population, not equality of several separately sampled populations.

Worked solution P69-Easy-9. For the independence test, ${\chi }^2=6.385$, df=2, and p=0.0411; expected counts model no association within the population. $$E_{ij}=\frac{({\text{row total}}_i)({\text{column total}}_j)}{123},{\chi }^2=6.385,df=2.$$ Interpretation: A small p-value gives evidence that the two categorical variables are associated in the population represented by the one-sample design. Validity: One probability sample supplies independent observational units; every unit contributes to exactly one cell, and expected counts support the approximation. Error to reject: The independence conclusion concerns association between two variables in one population, not equality of several separately sampled populations.

Worked solution P69-Easy-10. For the independence test, ${\chi }^2=7.189$, df=2, and p=0.0275; expected counts model no association within the population. $$E_{ij}=\frac{({\text{row total}}_i)({\text{column total}}_j)}{125},{\chi }^2=7.189,df=2.$$ Interpretation: A small p-value gives evidence that the two categorical variables are associated in the population represented by the one-sample design. Validity: One probability sample supplies independent observational units; every unit contributes to exactly one cell, and expected counts support the approximation. Error to reject: The independence conclusion concerns association between two variables in one population, not equality of several separately sampled populations.

Worked solution P69-Easy-11. For the independence test, ${\chi }^2=5.622$, df=2, and p=0.0602; expected counts model no association within the population. $$E_{ij}=\frac{({\text{row total}}_i)({\text{column total}}_j)}{122},{\chi }^2=5.622,df=2.$$ Interpretation: A small p-value gives evidence that the two categorical variables are associated in the population represented by the one-sample design. Validity: One probability sample supplies independent observational units; every unit contributes to exactly one cell, and expected counts support the approximation. Error to reject: The independence conclusion concerns association between two variables in one population, not equality of several separately sampled populations.

Worked solution P69-Easy-12. For the independence test, ${\chi }^2=6.418$, df=2, and p=0.0404; expected counts model no association within the population. $$E_{ij}=\frac{({\text{row total}}_i)({\text{column total}}_j)}{124},{\chi }^2=6.418,df=2.$$ Interpretation: A small p-value gives evidence that the two categorical variables are associated in the population represented by the one-sample design. Validity: One probability sample supplies independent observational units; every unit contributes to exactly one cell, and expected counts support the approximation. Error to reject: The independence conclusion concerns association between two variables in one population, not equality of several separately sampled populations.

Worked solution P69-Easy-13. For the independence test, ${\chi }^2=5.931$, df=2, and p=0.0515; expected counts model no association within the population. $$E_{ij}=\frac{({\text{row total}}_i)({\text{column total}}_j)}{122},{\chi }^2=5.931,df=2.$$ Interpretation: A small p-value gives evidence that the two categorical variables are associated in the population represented by the one-sample design. Validity: One probability sample supplies independent observational units; every unit contributes to exactly one cell, and expected counts support the approximation. Error to reject: The independence conclusion concerns association between two variables in one population, not equality of several separately sampled populations.

Worked solution P69-Easy-14. For the independence test, ${\chi }^2=6.715$, df=2, and p=0.0348; expected counts model no association within the population. $$E_{ij}=\frac{({\text{row total}}_i)({\text{column total}}_j)}{124},{\chi }^2=6.715,df=2.$$ Interpretation: A small p-value gives evidence that the two categorical variables are associated in the population represented by the one-sample design. Validity: One probability sample supplies independent observational units; every unit contributes to exactly one cell, and expected counts support the approximation. Error to reject: The independence conclusion concerns association between two variables in one population, not equality of several separately sampled populations.

Worked solution P69-Easy-15. For the independence test, ${\chi }^2=7.546$, df=2, and p=0.0230; expected counts model no association within the population. $$E_{ij}=\frac{({\text{row total}}_i)({\text{column total}}_j)}{126},{\chi }^2=7.546,df=2.$$ Interpretation: A small p-value gives evidence that the two categorical variables are associated in the population represented by the one-sample design. Validity: One probability sample supplies independent observational units; every unit contributes to exactly one cell, and expected counts support the approximation. Error to reject: The independence conclusion concerns association between two variables in one population, not equality of several separately sampled populations.

Worked solution P69-Tough-1. For the independence test, ${\chi }^2=5.043$, df=2, and p=0.0804; expected counts model no association within the population. $$E_{ij}=\frac{({\text{row total}}_i)({\text{column total}}_j)}{120},{\chi }^2=5.043,df=2.$$ Interpretation: A small p-value gives evidence that the two categorical variables are associated in the population represented by the one-sample design. Validity: One probability sample supplies independent observational units; every unit contributes to exactly one cell, and expected counts support the approximation. Error to reject: The independence conclusion concerns association between two variables in one population, not equality of several separately sampled populations.

Worked solution P69-Tough-2. For the independence test, ${\chi }^2=5.782$, df=2, and p=0.0555; expected counts model no association within the population. $$E_{ij}=\frac{({\text{row total}}_i)({\text{column total}}_j)}{122},{\chi }^2=5.782,df=2.$$ Interpretation: A small p-value gives evidence that the two categorical variables are associated in the population represented by the one-sample design. Validity: One probability sample supplies independent observational units; every unit contributes to exactly one cell, and expected counts support the approximation. Error to reject: The independence conclusion concerns association between two variables in one population, not equality of several separately sampled populations.

Worked solution P69-Tough-3. For the independence test, ${\chi }^2=6.573$, df=2, and p=0.0374; expected counts model no association within the population. $$E_{ij}=\frac{({\text{row total}}_i)({\text{column total}}_j)}{124},{\chi }^2=6.573,df=2.$$ Interpretation: A small p-value gives evidence that the two categorical variables are associated in the population represented by the one-sample design. Validity: One probability sample supplies independent observational units; every unit contributes to exactly one cell, and expected counts support the approximation. Error to reject: The independence conclusion concerns association between two variables in one population, not equality of several separately sampled populations.

Worked solution P69-Tough-4. For the independence test, ${\chi }^2=7.412$, df=2, and p=0.0246; expected counts model no association within the population. $$E_{ij}=\frac{({\text{row total}}_i)({\text{column total}}_j)}{126},{\chi }^2=7.412,df=2.$$ Interpretation: A small p-value gives evidence that the two categorical variables are associated in the population represented by the one-sample design. Validity: One probability sample supplies independent observational units; every unit contributes to exactly one cell, and expected counts support the approximation. Error to reject: The independence conclusion concerns association between two variables in one population, not equality of several separately sampled populations.

Worked solution P69-Tough-5. For the independence test, ${\chi }^2=6.844$, df=2, and p=0.0327; expected counts model no association within the population. $$E_{ij}=\frac{({\text{row total}}_i)({\text{column total}}_j)}{124},{\chi }^2=6.844,df=2.$$ Interpretation: A small p-value gives evidence that the two categorical variables are associated in the population represented by the one-sample design. Validity: One probability sample supplies independent observational units; every unit contributes to exactly one cell, and expected counts support the approximation. Error to reject: The independence conclusion concerns association between two variables in one population, not equality of several separately sampled populations.

Worked solution P69-Tough-6. For the independence test, ${\chi }^2=5.325$, df=2, and p=0.0698; expected counts model no association within the population. $$E_{ij}=\frac{({\text{row total}}_i)({\text{column total}}_j)}{121},{\chi }^2=5.325,df=2.$$ Interpretation: A small p-value gives evidence that the two categorical variables are associated in the population represented by the one-sample design. Validity: One probability sample supplies independent observational units; every unit contributes to exactly one cell, and expected counts support the approximation. Error to reject: The independence conclusion concerns association between two variables in one population, not equality of several separately sampled populations.

Worked solution P69-Tough-7. For the independence test, ${\chi }^2=6.094$, df=2, and p=0.0475; expected counts model no association within the population. $$E_{ij}=\frac{({\text{row total}}_i)({\text{column total}}_j)}{123},{\chi }^2=6.094,df=2.$$ Interpretation: A small p-value gives evidence that the two categorical variables are associated in the population represented by the one-sample design. Validity: One probability sample supplies independent observational units; every unit contributes to exactly one cell, and expected counts support the approximation. Error to reject: The independence conclusion concerns association between two variables in one population, not equality of several separately sampled populations.

Worked solution P69-Tough-8. For the independence test, ${\chi }^2=6.912$, df=2, and p=0.0315; expected counts model no association within the population. $$E_{ij}=\frac{({\text{row total}}_i)({\text{column total}}_j)}{125},{\chi }^2=6.912,df=2.$$ Interpretation: A small p-value gives evidence that the two categorical variables are associated in the population represented by the one-sample design. Validity: One probability sample supplies independent observational units; every unit contributes to exactly one cell, and expected counts support the approximation. Error to reject: The independence conclusion concerns association between two variables in one population, not equality of several separately sampled populations.

Worked solution P69-Tough-9. For the independence test, ${\chi }^2=6.385$, df=2, and p=0.0411; expected counts model no association within the population. $$E_{ij}=\frac{({\text{row total}}_i)({\text{column total}}_j)}{123},{\chi }^2=6.385,df=2.$$ Interpretation: A small p-value gives evidence that the two categorical variables are associated in the population represented by the one-sample design. Validity: One probability sample supplies independent observational units; every unit contributes to exactly one cell, and expected counts support the approximation. Error to reject: The independence conclusion concerns association between two variables in one population, not equality of several separately sampled populations.

Worked solution P69-Tough-10. For the independence test, ${\chi }^2=7.189$, df=2, and p=0.0275; expected counts model no association within the population. $$E_{ij}=\frac{({\text{row total}}_i)({\text{column total}}_j)}{125},{\chi }^2=7.189,df=2.$$ Interpretation: A small p-value gives evidence that the two categorical variables are associated in the population represented by the one-sample design. Validity: One probability sample supplies independent observational units; every unit contributes to exactly one cell, and expected counts support the approximation. Error to reject: The independence conclusion concerns association between two variables in one population, not equality of several separately sampled populations.

Worked solution P69-Tough-11. For the independence test, ${\chi }^2=5.622$, df=2, and p=0.0602; expected counts model no association within the population. $$E_{ij}=\frac{({\text{row total}}_i)({\text{column total}}_j)}{122},{\chi }^2=5.622,df=2.$$ Interpretation: A small p-value gives evidence that the two categorical variables are associated in the population represented by the one-sample design. Validity: One probability sample supplies independent observational units; every unit contributes to exactly one cell, and expected counts support the approximation. Error to reject: The independence conclusion concerns association between two variables in one population, not equality of several separately sampled populations.

Worked solution P69-Tough-12. For the independence test, ${\chi }^2=6.418$, df=2, and p=0.0404; expected counts model no association within the population. $$E_{ij}=\frac{({\text{row total}}_i)({\text{column total}}_j)}{124},{\chi }^2=6.418,df=2.$$ Interpretation: A small p-value gives evidence that the two categorical variables are associated in the population represented by the one-sample design. Validity: One probability sample supplies independent observational units; every unit contributes to exactly one cell, and expected counts support the approximation. Error to reject: The independence conclusion concerns association between two variables in one population, not equality of several separately sampled populations.

Worked solution P69-Tough-13. For the independence test, ${\chi }^2=5.931$, df=2, and p=0.0515; expected counts model no association within the population. $$E_{ij}=\frac{({\text{row total}}_i)({\text{column total}}_j)}{122},{\chi }^2=5.931,df=2.$$ Interpretation: A small p-value gives evidence that the two categorical variables are associated in the population represented by the one-sample design. Validity: One probability sample supplies independent observational units; every unit contributes to exactly one cell, and expected counts support the approximation. Error to reject: The independence conclusion concerns association between two variables in one population, not equality of several separately sampled populations.

Worked solution P69-Tough-14. For the independence test, ${\chi }^2=6.715$, df=2, and p=0.0348; expected counts model no association within the population. $$E_{ij}=\frac{({\text{row total}}_i)({\text{column total}}_j)}{124},{\chi }^2=6.715,df=2.$$ Interpretation: A small p-value gives evidence that the two categorical variables are associated in the population represented by the one-sample design. Validity: One probability sample supplies independent observational units; every unit contributes to exactly one cell, and expected counts support the approximation. Error to reject: The independence conclusion concerns association between two variables in one population, not equality of several separately sampled populations.

Worked solution P69-Tough-15. For the independence test, ${\chi }^2=7.546$, df=2, and p=0.0230; expected counts model no association within the population. $$E_{ij}=\frac{({\text{row total}}_i)({\text{column total}}_j)}{126},{\chi }^2=7.546,df=2.$$ Interpretation: A small p-value gives evidence that the two categorical variables are associated in the population represented by the one-sample design. Validity: One probability sample supplies independent observational units; every unit contributes to exactly one cell, and expected counts support the approximation. Error to reject: The independence conclusion concerns association between two variables in one population, not equality of several separately sampled populations.

Worked solution P69-Toughest-1. For the independence test, ${\chi }^2=5.043$, df=2, and p=0.0804; expected counts model no association within the population. $$E_{ij}=\frac{({\text{row total}}_i)({\text{column total}}_j)}{120},{\chi }^2=5.043,df=2.$$ Interpretation: A small p-value gives evidence that the two categorical variables are associated in the population represented by the one-sample design. Validity: One probability sample supplies independent observational units; every unit contributes to exactly one cell, and expected counts support the approximation. Error to reject: The independence conclusion concerns association between two variables in one population, not equality of several separately sampled populations.

Worked solution P69-Toughest-2. For the independence test, ${\chi }^2=5.782$, df=2, and p=0.0555; expected counts model no association within the population. $$E_{ij}=\frac{({\text{row total}}_i)({\text{column total}}_j)}{122},{\chi }^2=5.782,df=2.$$ Interpretation: A small p-value gives evidence that the two categorical variables are associated in the population represented by the one-sample design. Validity: One probability sample supplies independent observational units; every unit contributes to exactly one cell, and expected counts support the approximation. Error to reject: The independence conclusion concerns association between two variables in one population, not equality of several separately sampled populations.

Worked solution P69-Toughest-3. For the independence test, ${\chi }^2=6.573$, df=2, and p=0.0374; expected counts model no association within the population. $$E_{ij}=\frac{({\text{row total}}_i)({\text{column total}}_j)}{124},{\chi }^2=6.573,df=2.$$ Interpretation: A small p-value gives evidence that the two categorical variables are associated in the population represented by the one-sample design. Validity: One probability sample supplies independent observational units; every unit contributes to exactly one cell, and expected counts support the approximation. Error to reject: The independence conclusion concerns association between two variables in one population, not equality of several separately sampled populations.

Worked solution P69-Toughest-4. For the independence test, ${\chi }^2=7.412$, df=2, and p=0.0246; expected counts model no association within the population. $$E_{ij}=\frac{({\text{row total}}_i)({\text{column total}}_j)}{126},{\chi }^2=7.412,df=2.$$ Interpretation: A small p-value gives evidence that the two categorical variables are associated in the population represented by the one-sample design. Validity: One probability sample supplies independent observational units; every unit contributes to exactly one cell, and expected counts support the approximation. Error to reject: The independence conclusion concerns association between two variables in one population, not equality of several separately sampled populations.

Worked solution P69-Toughest-5. For the independence test, ${\chi }^2=6.844$, df=2, and p=0.0327; expected counts model no association within the population. $$E_{ij}=\frac{({\text{row total}}_i)({\text{column total}}_j)}{124},{\chi }^2=6.844,df=2.$$ Interpretation: A small p-value gives evidence that the two categorical variables are associated in the population represented by the one-sample design. Validity: One probability sample supplies independent observational units; every unit contributes to exactly one cell, and expected counts support the approximation. Error to reject: The independence conclusion concerns association between two variables in one population, not equality of several separately sampled populations.

Worked solution P69-Toughest-6. For the independence test, ${\chi }^2=5.325$, df=2, and p=0.0698; expected counts model no association within the population. $$E_{ij}=\frac{({\text{row total}}_i)({\text{column total}}_j)}{121},{\chi }^2=5.325,df=2.$$ Interpretation: A small p-value gives evidence that the two categorical variables are associated in the population represented by the one-sample design. Validity: One probability sample supplies independent observational units; every unit contributes to exactly one cell, and expected counts support the approximation. Error to reject: The independence conclusion concerns association between two variables in one population, not equality of several separately sampled populations.

Worked solution P69-Toughest-7. For the independence test, ${\chi }^2=6.094$, df=2, and p=0.0475; expected counts model no association within the population. $$E_{ij}=\frac{({\text{row total}}_i)({\text{column total}}_j)}{123},{\chi }^2=6.094,df=2.$$ Interpretation: A small p-value gives evidence that the two categorical variables are associated in the population represented by the one-sample design. Validity: One probability sample supplies independent observational units; every unit contributes to exactly one cell, and expected counts support the approximation. Error to reject: The independence conclusion concerns association between two variables in one population, not equality of several separately sampled populations.

Worked solution P69-Toughest-8. For the independence test, ${\chi }^2=6.912$, df=2, and p=0.0315; expected counts model no association within the population. $$E_{ij}=\frac{({\text{row total}}_i)({\text{column total}}_j)}{125},{\chi }^2=6.912,df=2.$$ Interpretation: A small p-value gives evidence that the two categorical variables are associated in the population represented by the one-sample design. Validity: One probability sample supplies independent observational units; every unit contributes to exactly one cell, and expected counts support the approximation. Error to reject: The independence conclusion concerns association between two variables in one population, not equality of several separately sampled populations.

Worked solution P69-Toughest-9. For the independence test, ${\chi }^2=6.385$, df=2, and p=0.0411; expected counts model no association within the population. $$E_{ij}=\frac{({\text{row total}}_i)({\text{column total}}_j)}{123},{\chi }^2=6.385,df=2.$$ Interpretation: A small p-value gives evidence that the two categorical variables are associated in the population represented by the one-sample design. Validity: One probability sample supplies independent observational units; every unit contributes to exactly one cell, and expected counts support the approximation. Error to reject: The independence conclusion concerns association between two variables in one population, not equality of several separately sampled populations.

Worked solution P69-Toughest-10. For the independence test, ${\chi }^2=7.189$, df=2, and p=0.0275; expected counts model no association within the population. $$E_{ij}=\frac{({\text{row total}}_i)({\text{column total}}_j)}{125},{\chi }^2=7.189,df=2.$$ Interpretation: A small p-value gives evidence that the two categorical variables are associated in the population represented by the one-sample design. Validity: One probability sample supplies independent observational units; every unit contributes to exactly one cell, and expected counts support the approximation. Error to reject: The independence conclusion concerns association between two variables in one population, not equality of several separately sampled populations.

Worked solution P69-Toughest-11. For the independence test, ${\chi }^2=5.622$, df=2, and p=0.0602; expected counts model no association within the population. $$E_{ij}=\frac{({\text{row total}}_i)({\text{column total}}_j)}{122},{\chi }^2=5.622,df=2.$$ Interpretation: A small p-value gives evidence that the two categorical variables are associated in the population represented by the one-sample design. Validity: One probability sample supplies independent observational units; every unit contributes to exactly one cell, and expected counts support the approximation. Error to reject: The independence conclusion concerns association between two variables in one population, not equality of several separately sampled populations.

Worked solution P69-Toughest-12. For the independence test, ${\chi }^2=6.418$, df=2, and p=0.0404; expected counts model no association within the population. $$E_{ij}=\frac{({\text{row total}}_i)({\text{column total}}_j)}{124},{\chi }^2=6.418,df=2.$$ Interpretation: A small p-value gives evidence that the two categorical variables are associated in the population represented by the one-sample design. Validity: One probability sample supplies independent observational units; every unit contributes to exactly one cell, and expected counts support the approximation. Error to reject: The independence conclusion concerns association between two variables in one population, not equality of several separately sampled populations.

Worked solution P69-Toughest-13. For the independence test, ${\chi }^2=5.931$, df=2, and p=0.0515; expected counts model no association within the population. $$E_{ij}=\frac{({\text{row total}}_i)({\text{column total}}_j)}{122},{\chi }^2=5.931,df=2.$$ Interpretation: A small p-value gives evidence that the two categorical variables are associated in the population represented by the one-sample design. Validity: One probability sample supplies independent observational units; every unit contributes to exactly one cell, and expected counts support the approximation. Error to reject: The independence conclusion concerns association between two variables in one population, not equality of several separately sampled populations.

Worked solution P69-Toughest-14. For the independence test, ${\chi }^2=6.715$, df=2, and p=0.0348; expected counts model no association within the population. $$E_{ij}=\frac{({\text{row total}}_i)({\text{column total}}_j)}{124},{\chi }^2=6.715,df=2.$$ Interpretation: A small p-value gives evidence that the two categorical variables are associated in the population represented by the one-sample design. Validity: One probability sample supplies independent observational units; every unit contributes to exactly one cell, and expected counts support the approximation. Error to reject: The independence conclusion concerns association between two variables in one population, not equality of several separately sampled populations.

Worked solution P69-Toughest-15. For the independence test, ${\chi }^2=7.546$, df=2, and p=0.0230; expected counts model no association within the population. $$E_{ij}=\frac{({\text{row total}}_i)({\text{column total}}_j)}{126},{\chi }^2=7.546,df=2.$$ Interpretation: A small p-value gives evidence that the two categorical variables are associated in the population represented by the one-sample design. Validity: One probability sample supplies independent observational units; every unit contributes to exactly one cell, and expected counts support the approximation. Error to reject: The independence conclusion concerns association between two variables in one population, not equality of several separately sampled populations.