Worked solution P26-Easy-1. The frequency is 16; the relative frequency is 0.3902, or 39.0%. $$\text{relative frequency}=\frac{16}{41}=0.3902.$$ Interpretation: The percentage describes the observed data set; it is not automatically a population estimate unless the collection design supports inference. Validity: Categories should be mutually exclusive for a one-response table, and the denominator must include all valid observations. Error to reject: Do not divide by the number of categories or omit a zero-count category that belongs to the defined variable.

Worked solution P26-Easy-2. The frequency is 18; the relative frequency is 0.4286, or 42.9%. $$\text{relative frequency}=\frac{18}{42}=0.4286.$$ Interpretation: The percentage describes the observed data set; it is not automatically a population estimate unless the collection design supports inference. Validity: Categories should be mutually exclusive for a one-response table, and the denominator must include all valid observations. Error to reject: Do not divide by the number of categories or omit a zero-count category that belongs to the defined variable.

Worked solution P26-Easy-3. The frequency is 27; the relative frequency is 0.5745, or 57.4%. $$\text{relative frequency}=\frac{27}{47}=0.5745.$$ Interpretation: The percentage describes the observed data set; it is not automatically a population estimate unless the collection design supports inference. Validity: Categories should be mutually exclusive for a one-response table, and the denominator must include all valid observations. Error to reject: Do not divide by the number of categories or omit a zero-count category that belongs to the defined variable.

Worked solution P26-Easy-4. The frequency is 45; the relative frequency is 0.6000, or 60.0%. $$\text{relative frequency}=\frac{45}{75}=0.6000.$$ Interpretation: The percentage describes the observed data set; it is not automatically a population estimate unless the collection design supports inference. Validity: Categories should be mutually exclusive for a one-response table, and the denominator must include all valid observations. Error to reject: Do not divide by the number of categories or omit a zero-count category that belongs to the defined variable.

Worked solution P26-Easy-5. The frequency is 55; the relative frequency is 0.5556, or 55.6%. $$\text{relative frequency}=\frac{55}{99}=0.5556.$$ Interpretation: The percentage describes the observed data set; it is not automatically a population estimate unless the collection design supports inference. Validity: Categories should be mutually exclusive for a one-response table, and the denominator must include all valid observations. Error to reject: Do not divide by the number of categories or omit a zero-count category that belongs to the defined variable.

Worked solution P26-Easy-6. The frequency is 31; the relative frequency is 0.5439, or 54.4%. $$\text{relative frequency}=\frac{31}{57}=0.5439.$$ Interpretation: The percentage describes the observed data set; it is not automatically a population estimate unless the collection design supports inference. Validity: Categories should be mutually exclusive for a one-response table, and the denominator must include all valid observations. Error to reject: Do not divide by the number of categories or omit a zero-count category that belongs to the defined variable.

Worked solution P26-Easy-7. The frequency is 41; the relative frequency is 0.5256, or 52.6%. $$\text{relative frequency}=\frac{41}{78}=0.5256.$$ Interpretation: The percentage describes the observed data set; it is not automatically a population estimate unless the collection design supports inference. Validity: Categories should be mutually exclusive for a one-response table, and the denominator must include all valid observations. Error to reject: Do not divide by the number of categories or omit a zero-count category that belongs to the defined variable.

Worked solution P26-Easy-8. The frequency is 30; the relative frequency is 0.3093, or 30.9%. $$\text{relative frequency}=\frac{30}{97}=0.3093.$$ Interpretation: The percentage describes the observed data set; it is not automatically a population estimate unless the collection design supports inference. Validity: Categories should be mutually exclusive for a one-response table, and the denominator must include all valid observations. Error to reject: Do not divide by the number of categories or omit a zero-count category that belongs to the defined variable.

Worked solution P26-Easy-9. The frequency is 38; the relative frequency is 0.5938, or 59.4%. $$\text{relative frequency}=\frac{38}{64}=0.5938.$$ Interpretation: The percentage describes the observed data set; it is not automatically a population estimate unless the collection design supports inference. Validity: Categories should be mutually exclusive for a one-response table, and the denominator must include all valid observations. Error to reject: Do not divide by the number of categories or omit a zero-count category that belongs to the defined variable.

Worked solution P26-Easy-10. The frequency is 25; the relative frequency is 0.3165, or 31.6%. $$\text{relative frequency}=\frac{25}{79}=0.3165.$$ Interpretation: The percentage describes the observed data set; it is not automatically a population estimate unless the collection design supports inference. Validity: Categories should be mutually exclusive for a one-response table, and the denominator must include all valid observations. Error to reject: Do not divide by the number of categories or omit a zero-count category that belongs to the defined variable.

Worked solution P26-Easy-11. The frequency is 22; the relative frequency is 0.3729, or 37.3%. $$\text{relative frequency}=\frac{22}{59}=0.3729.$$ Interpretation: The percentage describes the observed data set; it is not automatically a population estimate unless the collection design supports inference. Validity: Categories should be mutually exclusive for a one-response table, and the denominator must include all valid observations. Error to reject: Do not divide by the number of categories or omit a zero-count category that belongs to the defined variable.

Worked solution P26-Easy-12. The frequency is 46; the relative frequency is 0.4694, or 46.9%. $$\text{relative frequency}=\frac{46}{98}=0.4694.$$ Interpretation: The percentage describes the observed data set; it is not automatically a population estimate unless the collection design supports inference. Validity: Categories should be mutually exclusive for a one-response table, and the denominator must include all valid observations. Error to reject: Do not divide by the number of categories or omit a zero-count category that belongs to the defined variable.

Worked solution P26-Easy-13. The frequency is 47; the relative frequency is 0.5875, or 58.8%. $$\text{relative frequency}=\frac{47}{80}=0.5875.$$ Interpretation: The percentage describes the observed data set; it is not automatically a population estimate unless the collection design supports inference. Validity: Categories should be mutually exclusive for a one-response table, and the denominator must include all valid observations. Error to reject: Do not divide by the number of categories or omit a zero-count category that belongs to the defined variable.

Worked solution P26-Easy-14. The frequency is 37; the relative frequency is 0.5441, or 54.4%. $$\text{relative frequency}=\frac{37}{68}=0.5441.$$ Interpretation: The percentage describes the observed data set; it is not automatically a population estimate unless the collection design supports inference. Validity: Categories should be mutually exclusive for a one-response table, and the denominator must include all valid observations. Error to reject: Do not divide by the number of categories or omit a zero-count category that belongs to the defined variable.

Worked solution P26-Easy-15. The frequency is 21; the relative frequency is 0.3000, or 30.0%. $$\text{relative frequency}=\frac{21}{70}=0.3000.$$ Interpretation: The percentage describes the observed data set; it is not automatically a population estimate unless the collection design supports inference. Validity: Categories should be mutually exclusive for a one-response table, and the denominator must include all valid observations. Error to reject: Do not divide by the number of categories or omit a zero-count category that belongs to the defined variable.

Worked solution P26-Easy-16. The frequency is 30; the relative frequency is 0.5556, or 55.6%. $$\text{relative frequency}=\frac{30}{54}=0.5556.$$ Interpretation: The percentage describes the observed data set; it is not automatically a population estimate unless the collection design supports inference. Validity: Categories should be mutually exclusive for a one-response table, and the denominator must include all valid observations. Error to reject: Do not divide by the number of categories or omit a zero-count category that belongs to the defined variable.

Worked solution P26-Easy-17. The frequency is 25; the relative frequency is 0.3623, or 36.2%. $$\text{relative frequency}=\frac{25}{69}=0.3623.$$ Interpretation: The percentage describes the observed data set; it is not automatically a population estimate unless the collection design supports inference. Validity: Categories should be mutually exclusive for a one-response table, and the denominator must include all valid observations. Error to reject: Do not divide by the number of categories or omit a zero-count category that belongs to the defined variable.

Worked solution P26-Easy-18. The frequency is 21; the relative frequency is 0.3182, or 31.8%. $$\text{relative frequency}=\frac{21}{66}=0.3182.$$ Interpretation: The percentage describes the observed data set; it is not automatically a population estimate unless the collection design supports inference. Validity: Categories should be mutually exclusive for a one-response table, and the denominator must include all valid observations. Error to reject: Do not divide by the number of categories or omit a zero-count category that belongs to the defined variable.

Worked solution P26-Tough-1. The frequency is 19; the relative frequency is 0.3333, or 33.3%. $$\text{relative frequency}=\frac{19}{57}=0.3333.$$ Interpretation: The percentage describes the observed data set; it is not automatically a population estimate unless the collection design supports inference. Validity: Categories should be mutually exclusive for a one-response table, and the denominator must include all valid observations. Error to reject: Do not divide by the number of categories or omit a zero-count category that belongs to the defined variable.

Worked solution P26-Tough-2. The frequency is 39; the relative frequency is 0.3980, or 39.8%. $$\text{relative frequency}=\frac{39}{98}=0.3980.$$ Interpretation: The percentage describes the observed data set; it is not automatically a population estimate unless the collection design supports inference. Validity: Categories should be mutually exclusive for a one-response table, and the denominator must include all valid observations. Error to reject: Do not divide by the number of categories or omit a zero-count category that belongs to the defined variable.

Worked solution P26-Tough-3. The frequency is 41; the relative frequency is 0.4659, or 46.6%. $$\text{relative frequency}=\frac{41}{88}=0.4659.$$ Interpretation: The percentage describes the observed data set; it is not automatically a population estimate unless the collection design supports inference. Validity: Categories should be mutually exclusive for a one-response table, and the denominator must include all valid observations. Error to reject: Do not divide by the number of categories or omit a zero-count category that belongs to the defined variable.

Worked solution P26-Tough-4. The frequency is 30; the relative frequency is 0.4545, or 45.5%. $$\text{relative frequency}=\frac{30}{66}=0.4545.$$ Interpretation: The percentage describes the observed data set; it is not automatically a population estimate unless the collection design supports inference. Validity: Categories should be mutually exclusive for a one-response table, and the denominator must include all valid observations. Error to reject: Do not divide by the number of categories or omit a zero-count category that belongs to the defined variable.

Worked solution P26-Tough-5. The frequency is 36; the relative frequency is 0.4675, or 46.8%. $$\text{relative frequency}=\frac{36}{77}=0.4675.$$ Interpretation: The percentage describes the observed data set; it is not automatically a population estimate unless the collection design supports inference. Validity: Categories should be mutually exclusive for a one-response table, and the denominator must include all valid observations. Error to reject: Do not divide by the number of categories or omit a zero-count category that belongs to the defined variable.

Worked solution P26-Tough-6. The frequency is 27; the relative frequency is 0.3000, or 30.0%. $$\text{relative frequency}=\frac{27}{90}=0.3000.$$ Interpretation: The percentage describes the observed data set; it is not automatically a population estimate unless the collection design supports inference. Validity: Categories should be mutually exclusive for a one-response table, and the denominator must include all valid observations. Error to reject: Do not divide by the number of categories or omit a zero-count category that belongs to the defined variable.

Worked solution P26-Tough-7. The frequency is 20; the relative frequency is 0.3175, or 31.7%. $$\text{relative frequency}=\frac{20}{63}=0.3175.$$ Interpretation: The percentage describes the observed data set; it is not automatically a population estimate unless the collection design supports inference. Validity: Categories should be mutually exclusive for a one-response table, and the denominator must include all valid observations. Error to reject: Do not divide by the number of categories or omit a zero-count category that belongs to the defined variable.

Worked solution P26-Tough-8. The frequency is 41; the relative frequency is 0.4505, or 45.1%. $$\text{relative frequency}=\frac{41}{91}=0.4505.$$ Interpretation: The percentage describes the observed data set; it is not automatically a population estimate unless the collection design supports inference. Validity: Categories should be mutually exclusive for a one-response table, and the denominator must include all valid observations. Error to reject: Do not divide by the number of categories or omit a zero-count category that belongs to the defined variable.

Worked solution P26-Tough-9. The frequency is 16; the relative frequency is 0.3200, or 32.0%. $$\text{relative frequency}=\frac{16}{50}=0.3200.$$ Interpretation: The percentage describes the observed data set; it is not automatically a population estimate unless the collection design supports inference. Validity: Categories should be mutually exclusive for a one-response table, and the denominator must include all valid observations. Error to reject: Do not divide by the number of categories or omit a zero-count category that belongs to the defined variable.

Worked solution P26-Tough-10. The frequency is 15; the relative frequency is 0.3750, or 37.5%. $$\text{relative frequency}=\frac{15}{40}=0.3750.$$ Interpretation: The percentage describes the observed data set; it is not automatically a population estimate unless the collection design supports inference. Validity: Categories should be mutually exclusive for a one-response table, and the denominator must include all valid observations. Error to reject: Do not divide by the number of categories or omit a zero-count category that belongs to the defined variable.

Worked solution P26-Tough-11. The frequency is 49; the relative frequency is 0.5052, or 50.5%. $$\text{relative frequency}=\frac{49}{97}=0.5052.$$ Interpretation: The percentage describes the observed data set; it is not automatically a population estimate unless the collection design supports inference. Validity: Categories should be mutually exclusive for a one-response table, and the denominator must include all valid observations. Error to reject: Do not divide by the number of categories or omit a zero-count category that belongs to the defined variable.

Worked solution P26-Tough-12. The frequency is 20; the relative frequency is 0.3175, or 31.7%. $$\text{relative frequency}=\frac{20}{63}=0.3175.$$ Interpretation: The percentage describes the observed data set; it is not automatically a population estimate unless the collection design supports inference. Validity: Categories should be mutually exclusive for a one-response table, and the denominator must include all valid observations. Error to reject: Do not divide by the number of categories or omit a zero-count category that belongs to the defined variable.

Worked solution P26-Tough-13. The frequency is 30; the relative frequency is 0.4918, or 49.2%. $$\text{relative frequency}=\frac{30}{61}=0.4918.$$ Interpretation: The percentage describes the observed data set; it is not automatically a population estimate unless the collection design supports inference. Validity: Categories should be mutually exclusive for a one-response table, and the denominator must include all valid observations. Error to reject: Do not divide by the number of categories or omit a zero-count category that belongs to the defined variable.

Worked solution P26-Tough-14. The frequency is 30; the relative frequency is 0.3000, or 30.0%. $$\text{relative frequency}=\frac{30}{100}=0.3000.$$ Interpretation: The percentage describes the observed data set; it is not automatically a population estimate unless the collection design supports inference. Validity: Categories should be mutually exclusive for a one-response table, and the denominator must include all valid observations. Error to reject: Do not divide by the number of categories or omit a zero-count category that belongs to the defined variable.

Worked solution P26-Tough-15. The frequency is 24; the relative frequency is 0.3429, or 34.3%. $$\text{relative frequency}=\frac{24}{70}=0.3429.$$ Interpretation: The percentage describes the observed data set; it is not automatically a population estimate unless the collection design supports inference. Validity: Categories should be mutually exclusive for a one-response table, and the denominator must include all valid observations. Error to reject: Do not divide by the number of categories or omit a zero-count category that belongs to the defined variable.

Worked solution P26-Tough-16. The frequency is 28; the relative frequency is 0.4444, or 44.4%. $$\text{relative frequency}=\frac{28}{63}=0.4444.$$ Interpretation: The percentage describes the observed data set; it is not automatically a population estimate unless the collection design supports inference. Validity: Categories should be mutually exclusive for a one-response table, and the denominator must include all valid observations. Error to reject: Do not divide by the number of categories or omit a zero-count category that belongs to the defined variable.

Worked solution P26-Tough-17. The frequency is 22; the relative frequency is 0.3729, or 37.3%. $$\text{relative frequency}=\frac{22}{59}=0.3729.$$ Interpretation: The percentage describes the observed data set; it is not automatically a population estimate unless the collection design supports inference. Validity: Categories should be mutually exclusive for a one-response table, and the denominator must include all valid observations. Error to reject: Do not divide by the number of categories or omit a zero-count category that belongs to the defined variable.

Worked solution P26-Tough-18. The frequency is 32; the relative frequency is 0.3596, or 36.0%. $$\text{relative frequency}=\frac{32}{89}=0.3596.$$ Interpretation: The percentage describes the observed data set; it is not automatically a population estimate unless the collection design supports inference. Validity: Categories should be mutually exclusive for a one-response table, and the denominator must include all valid observations. Error to reject: Do not divide by the number of categories or omit a zero-count category that belongs to the defined variable.

Worked solution P26-Toughest-1. The frequency is 58; the relative frequency is 0.5918, or 59.2%. $$\text{relative frequency}=\frac{58}{98}=0.5918.$$ Interpretation: The percentage describes the observed data set; it is not automatically a population estimate unless the collection design supports inference. Validity: Categories should be mutually exclusive for a one-response table, and the denominator must include all valid observations. Error to reject: Do not divide by the number of categories or omit a zero-count category that belongs to the defined variable.

Worked solution P26-Toughest-2. The frequency is 39; the relative frequency is 0.4815, or 48.1%. $$\text{relative frequency}=\frac{39}{81}=0.4815.$$ Interpretation: The percentage describes the observed data set; it is not automatically a population estimate unless the collection design supports inference. Validity: Categories should be mutually exclusive for a one-response table, and the denominator must include all valid observations. Error to reject: Do not divide by the number of categories or omit a zero-count category that belongs to the defined variable.

Worked solution P26-Toughest-3. The frequency is 31; the relative frequency is 0.4079, or 40.8%. $$\text{relative frequency}=\frac{31}{76}=0.4079.$$ Interpretation: The percentage describes the observed data set; it is not automatically a population estimate unless the collection design supports inference. Validity: Categories should be mutually exclusive for a one-response table, and the denominator must include all valid observations. Error to reject: Do not divide by the number of categories or omit a zero-count category that belongs to the defined variable.

Worked solution P26-Toughest-4. The frequency is 28; the relative frequency is 0.3544, or 35.4%. $$\text{relative frequency}=\frac{28}{79}=0.3544.$$ Interpretation: The percentage describes the observed data set; it is not automatically a population estimate unless the collection design supports inference. Validity: Categories should be mutually exclusive for a one-response table, and the denominator must include all valid observations. Error to reject: Do not divide by the number of categories or omit a zero-count category that belongs to the defined variable.

Worked solution P26-Toughest-5. The frequency is 28; the relative frequency is 0.3784, or 37.8%. $$\text{relative frequency}=\frac{28}{74}=0.3784.$$ Interpretation: The percentage describes the observed data set; it is not automatically a population estimate unless the collection design supports inference. Validity: Categories should be mutually exclusive for a one-response table, and the denominator must include all valid observations. Error to reject: Do not divide by the number of categories or omit a zero-count category that belongs to the defined variable.

Worked solution P26-Toughest-6. The frequency is 16; the relative frequency is 0.3200, or 32.0%. $$\text{relative frequency}=\frac{16}{50}=0.3200.$$ Interpretation: The percentage describes the observed data set; it is not automatically a population estimate unless the collection design supports inference. Validity: Categories should be mutually exclusive for a one-response table, and the denominator must include all valid observations. Error to reject: Do not divide by the number of categories or omit a zero-count category that belongs to the defined variable.

Worked solution P26-Toughest-7. The frequency is 32; the relative frequency is 0.5333, or 53.3%. $$\text{relative frequency}=\frac{32}{60}=0.5333.$$ Interpretation: The percentage describes the observed data set; it is not automatically a population estimate unless the collection design supports inference. Validity: Categories should be mutually exclusive for a one-response table, and the denominator must include all valid observations. Error to reject: Do not divide by the number of categories or omit a zero-count category that belongs to the defined variable.

Worked solution P26-Toughest-8. The frequency is 25; the relative frequency is 0.4630, or 46.3%. $$\text{relative frequency}=\frac{25}{54}=0.4630.$$ Interpretation: The percentage describes the observed data set; it is not automatically a population estimate unless the collection design supports inference. Validity: Categories should be mutually exclusive for a one-response table, and the denominator must include all valid observations. Error to reject: Do not divide by the number of categories or omit a zero-count category that belongs to the defined variable.

Worked solution P26-Toughest-9. The frequency is 49; the relative frequency is 0.5904, or 59.0%. $$\text{relative frequency}=\frac{49}{83}=0.5904.$$ Interpretation: The percentage describes the observed data set; it is not automatically a population estimate unless the collection design supports inference. Validity: Categories should be mutually exclusive for a one-response table, and the denominator must include all valid observations. Error to reject: Do not divide by the number of categories or omit a zero-count category that belongs to the defined variable.

Worked solution P26-Toughest-10. The frequency is 28; the relative frequency is 0.4000, or 40.0%. $$\text{relative frequency}=\frac{28}{70}=0.4000.$$ Interpretation: The percentage describes the observed data set; it is not automatically a population estimate unless the collection design supports inference. Validity: Categories should be mutually exclusive for a one-response table, and the denominator must include all valid observations. Error to reject: Do not divide by the number of categories or omit a zero-count category that belongs to the defined variable.

Worked solution P26-Toughest-11. The frequency is 39; the relative frequency is 0.4105, or 41.1%. $$\text{relative frequency}=\frac{39}{95}=0.4105.$$ Interpretation: The percentage describes the observed data set; it is not automatically a population estimate unless the collection design supports inference. Validity: Categories should be mutually exclusive for a one-response table, and the denominator must include all valid observations. Error to reject: Do not divide by the number of categories or omit a zero-count category that belongs to the defined variable.

Worked solution P26-Toughest-12. The frequency is 29; the relative frequency is 0.3867, or 38.7%. $$\text{relative frequency}=\frac{29}{75}=0.3867.$$ Interpretation: The percentage describes the observed data set; it is not automatically a population estimate unless the collection design supports inference. Validity: Categories should be mutually exclusive for a one-response table, and the denominator must include all valid observations. Error to reject: Do not divide by the number of categories or omit a zero-count category that belongs to the defined variable.

Worked solution P26-Toughest-13. The frequency is 22; the relative frequency is 0.5500, or 55.0%. $$\text{relative frequency}=\frac{22}{40}=0.5500.$$ Interpretation: The percentage describes the observed data set; it is not automatically a population estimate unless the collection design supports inference. Validity: Categories should be mutually exclusive for a one-response table, and the denominator must include all valid observations. Error to reject: Do not divide by the number of categories or omit a zero-count category that belongs to the defined variable.

Worked solution P26-Toughest-14. The frequency is 14; the relative frequency is 0.2979, or 29.8%. $$\text{relative frequency}=\frac{14}{47}=0.2979.$$ Interpretation: The percentage describes the observed data set; it is not automatically a population estimate unless the collection design supports inference. Validity: Categories should be mutually exclusive for a one-response table, and the denominator must include all valid observations. Error to reject: Do not divide by the number of categories or omit a zero-count category that belongs to the defined variable.

Worked solution P26-Toughest-15. The frequency is 28; the relative frequency is 0.3457, or 34.6%. $$\text{relative frequency}=\frac{28}{81}=0.3457.$$ Interpretation: The percentage describes the observed data set; it is not automatically a population estimate unless the collection design supports inference. Validity: Categories should be mutually exclusive for a one-response table, and the denominator must include all valid observations. Error to reject: Do not divide by the number of categories or omit a zero-count category that belongs to the defined variable.

Worked solution P26-Toughest-16. The frequency is 32; the relative frequency is 0.3636, or 36.4%. $$\text{relative frequency}=\frac{32}{88}=0.3636.$$ Interpretation: The percentage describes the observed data set; it is not automatically a population estimate unless the collection design supports inference. Validity: Categories should be mutually exclusive for a one-response table, and the denominator must include all valid observations. Error to reject: Do not divide by the number of categories or omit a zero-count category that belongs to the defined variable.

Worked solution P26-Toughest-17. The frequency is 32; the relative frequency is 0.3765, or 37.6%. $$\text{relative frequency}=\frac{32}{85}=0.3765.$$ Interpretation: The percentage describes the observed data set; it is not automatically a population estimate unless the collection design supports inference. Validity: Categories should be mutually exclusive for a one-response table, and the denominator must include all valid observations. Error to reject: Do not divide by the number of categories or omit a zero-count category that belongs to the defined variable.

Worked solution P26-Toughest-18. The frequency is 29; the relative frequency is 0.4328, or 43.3%. $$\text{relative frequency}=\frac{29}{67}=0.4328.$$ Interpretation: The percentage describes the observed data set; it is not automatically a population estimate unless the collection design supports inference. Validity: Categories should be mutually exclusive for a one-response table, and the denominator must include all valid observations. Error to reject: Do not divide by the number of categories or omit a zero-count category that belongs to the defined variable.