Worked solution P51-Easy-1. The distribution has mean 0.460, standard error 0.0498, and the requested probability is approximately 0.1587. $${\mu }_{\widehat{p}}=0.46,{\sigma }_{\widehat{p}}=\sqrt{\frac{0.46(0.54)}{100}}=0.0498,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Check independence with a population at least $10n=1000$, and large counts: $np=46.0$, $n(1-p)=54.0$. Error to reject: The spread is not p(1-p)/n; that is the variance. Standard error is its square root.

Worked solution P51-Easy-2. The distribution has mean 0.430, standard error 0.0472, and the requested probability is approximately 0.1587. $${\mu }_{\widehat{p}}=0.43,{\sigma }_{\widehat{p}}=\sqrt{\frac{0.43(0.57)}{110}}=0.0472,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Check independence with a population at least $10n=1100$, and large counts: $np=47.3$, $n(1-p)=62.7$. Error to reject: The spread is not p(1-p)/n; that is the variance. Standard error is its square root.

Worked solution P51-Easy-3. The distribution has mean 0.410, standard error 0.0449, and the requested probability is approximately 0.1587. $${\mu }_{\widehat{p}}=0.41,{\sigma }_{\widehat{p}}=\sqrt{\frac{0.41(0.59)}{120}}=0.0449,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Check independence with a population at least $10n=1200$, and large counts: $np=49.2$, $n(1-p)=70.8$. Error to reject: The spread is not p(1-p)/n; that is the variance. Standard error is its square root.

Worked solution P51-Easy-4. The distribution has mean 0.360, standard error 0.0421, and the requested probability is approximately 0.1587. $${\mu }_{\widehat{p}}=0.36,{\sigma }_{\widehat{p}}=\sqrt{\frac{0.36(0.64)}{130}}=0.0421,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Check independence with a population at least $10n=1300$, and large counts: $np=46.8$, $n(1-p)=83.2$. Error to reject: The spread is not p(1-p)/n; that is the variance. Standard error is its square root.

Worked solution P51-Easy-5. The distribution has mean 0.450, standard error 0.0420, and the requested probability is approximately 0.1587. $${\mu }_{\widehat{p}}=0.45,{\sigma }_{\widehat{p}}=\sqrt{\frac{0.45(0.55)}{140}}=0.0420,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Check independence with a population at least $10n=1400$, and large counts: $np=63.0$, $n(1-p)=77.0$. Error to reject: The spread is not p(1-p)/n; that is the variance. Standard error is its square root.

Worked solution P51-Easy-6. The distribution has mean 0.390, standard error 0.0398, and the requested probability is approximately 0.1587. $${\mu }_{\widehat{p}}=0.39,{\sigma }_{\widehat{p}}=\sqrt{\frac{0.39(0.61)}{150}}=0.0398,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Check independence with a population at least $10n=1500$, and large counts: $np=58.5$, $n(1-p)=91.5$. Error to reject: The spread is not p(1-p)/n; that is the variance. Standard error is its square root.

Worked solution P51-Easy-7. The distribution has mean 0.370, standard error 0.0382, and the requested probability is approximately 0.1587. $${\mu }_{\widehat{p}}=0.37,{\sigma }_{\widehat{p}}=\sqrt{\frac{0.37(0.63)}{160}}=0.0382,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Check independence with a population at least $10n=1600$, and large counts: $np=59.2$, $n(1-p)=100.8$. Error to reject: The spread is not p(1-p)/n; that is the variance. Standard error is its square root.

Worked solution P51-Easy-8. The distribution has mean 0.390, standard error 0.0374, and the requested probability is approximately 0.1587. $${\mu }_{\widehat{p}}=0.39,{\sigma }_{\widehat{p}}=\sqrt{\frac{0.39(0.61)}{170}}=0.0374,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Check independence with a population at least $10n=1700$, and large counts: $np=66.3$, $n(1-p)=103.7$. Error to reject: The spread is not p(1-p)/n; that is the variance. Standard error is its square root.

Worked solution P51-Easy-9. The distribution has mean 0.510, standard error 0.0500, and the requested probability is approximately 0.1587. $${\mu }_{\widehat{p}}=0.51,{\sigma }_{\widehat{p}}=\sqrt{\frac{0.51(0.49)}{100}}=0.0500,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Check independence with a population at least $10n=1000$, and large counts: $np=51.0$, $n(1-p)=49.0$. Error to reject: The spread is not p(1-p)/n; that is the variance. Standard error is its square root.

Worked solution P51-Easy-10. The distribution has mean 0.410, standard error 0.0469, and the requested probability is approximately 0.1587. $${\mu }_{\widehat{p}}=0.41,{\sigma }_{\widehat{p}}=\sqrt{\frac{0.41(0.59)}{110}}=0.0469,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Check independence with a population at least $10n=1100$, and large counts: $np=45.1$, $n(1-p)=64.9$. Error to reject: The spread is not p(1-p)/n; that is the variance. Standard error is its square root.

Worked solution P51-Easy-11. The distribution has mean 0.420, standard error 0.0451, and the requested probability is approximately 0.1587. $${\mu }_{\widehat{p}}=0.42,{\sigma }_{\widehat{p}}=\sqrt{\frac{0.42(0.58)}{120}}=0.0451,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Check independence with a population at least $10n=1200$, and large counts: $np=50.4$, $n(1-p)=69.6$. Error to reject: The spread is not p(1-p)/n; that is the variance. Standard error is its square root.

Worked solution P51-Easy-12. The distribution has mean 0.370, standard error 0.0423, and the requested probability is approximately 0.1587. $${\mu }_{\widehat{p}}=0.37,{\sigma }_{\widehat{p}}=\sqrt{\frac{0.37(0.63)}{130}}=0.0423,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Check independence with a population at least $10n=1300$, and large counts: $np=48.1$, $n(1-p)=81.9$. Error to reject: The spread is not p(1-p)/n; that is the variance. Standard error is its square root.

Worked solution P51-Easy-13. The distribution has mean 0.520, standard error 0.0422, and the requested probability is approximately 0.1587. $${\mu }_{\widehat{p}}=0.52,{\sigma }_{\widehat{p}}=\sqrt{\frac{0.52(0.48)}{140}}=0.0422,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Check independence with a population at least $10n=1400$, and large counts: $np=72.8$, $n(1-p)=67.2$. Error to reject: The spread is not p(1-p)/n; that is the variance. Standard error is its square root.

Worked solution P51-Easy-14. The distribution has mean 0.470, standard error 0.0408, and the requested probability is approximately 0.1587. $${\mu }_{\widehat{p}}=0.47,{\sigma }_{\widehat{p}}=\sqrt{\frac{0.47(0.53)}{150}}=0.0408,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Check independence with a population at least $10n=1500$, and large counts: $np=70.5$, $n(1-p)=79.5$. Error to reject: The spread is not p(1-p)/n; that is the variance. Standard error is its square root.

Worked solution P51-Easy-15. The distribution has mean 0.390, standard error 0.0386, and the requested probability is approximately 0.1587. $${\mu }_{\widehat{p}}=0.39,{\sigma }_{\widehat{p}}=\sqrt{\frac{0.39(0.61)}{160}}=0.0386,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Check independence with a population at least $10n=1600$, and large counts: $np=62.4$, $n(1-p)=97.6$. Error to reject: The spread is not p(1-p)/n; that is the variance. Standard error is its square root.

Worked solution P51-Easy-16. The distribution has mean 0.410, standard error 0.0377, and the requested probability is approximately 0.1587. $${\mu }_{\widehat{p}}=0.41,{\sigma }_{\widehat{p}}=\sqrt{\frac{0.41(0.59)}{170}}=0.0377,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Check independence with a population at least $10n=1700$, and large counts: $np=69.7$, $n(1-p)=100.3$. Error to reject: The spread is not p(1-p)/n; that is the variance. Standard error is its square root.

Worked solution P51-Tough-1. The distribution has mean 0.460, standard error 0.0498, and the requested probability is approximately 0.1587. $${\mu }_{\widehat{p}}=0.46,{\sigma }_{\widehat{p}}=\sqrt{\frac{0.46(0.54)}{100}}=0.0498,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Check independence with a population at least $10n=1000$, and large counts: $np=46.0$, $n(1-p)=54.0$. Error to reject: The spread is not p(1-p)/n; that is the variance. Standard error is its square root.

Worked solution P51-Tough-2. The distribution has mean 0.400, standard error 0.0467, and the requested probability is approximately 0.1587. $${\mu }_{\widehat{p}}=0.40,{\sigma }_{\widehat{p}}=\sqrt{\frac{0.40(0.60)}{110}}=0.0467,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Check independence with a population at least $10n=1100$, and large counts: $np=44.0$, $n(1-p)=66.0$. Error to reject: The spread is not p(1-p)/n; that is the variance. Standard error is its square root.

Worked solution P51-Tough-3. The distribution has mean 0.440, standard error 0.0453, and the requested probability is approximately 0.1587. $${\mu }_{\widehat{p}}=0.44,{\sigma }_{\widehat{p}}=\sqrt{\frac{0.44(0.56)}{120}}=0.0453,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Check independence with a population at least $10n=1200$, and large counts: $np=52.8$, $n(1-p)=67.2$. Error to reject: The spread is not p(1-p)/n; that is the variance. Standard error is its square root.

Worked solution P51-Tough-4. The distribution has mean 0.550, standard error 0.0436, and the requested probability is approximately 0.1587. $${\mu }_{\widehat{p}}=0.55,{\sigma }_{\widehat{p}}=\sqrt{\frac{0.55(0.45)}{130}}=0.0436,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Check independence with a population at least $10n=1300$, and large counts: $np=71.5$, $n(1-p)=58.5$. Error to reject: The spread is not p(1-p)/n; that is the variance. Standard error is its square root.

Worked solution P51-Tough-5. The distribution has mean 0.420, standard error 0.0417, and the requested probability is approximately 0.1587. $${\mu }_{\widehat{p}}=0.42,{\sigma }_{\widehat{p}}=\sqrt{\frac{0.42(0.58)}{140}}=0.0417,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Check independence with a population at least $10n=1400$, and large counts: $np=58.8$, $n(1-p)=81.2$. Error to reject: The spread is not p(1-p)/n; that is the variance. Standard error is its square root.

Worked solution P51-Tough-6. The distribution has mean 0.510, standard error 0.0408, and the requested probability is approximately 0.1587. $${\mu }_{\widehat{p}}=0.51,{\sigma }_{\widehat{p}}=\sqrt{\frac{0.51(0.49)}{150}}=0.0408,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Check independence with a population at least $10n=1500$, and large counts: $np=76.5$, $n(1-p)=73.5$. Error to reject: The spread is not p(1-p)/n; that is the variance. Standard error is its square root.

Worked solution P51-Tough-7. The distribution has mean 0.500, standard error 0.0395, and the requested probability is approximately 0.1587. $${\mu }_{\widehat{p}}=0.50,{\sigma }_{\widehat{p}}=\sqrt{\frac{0.50(0.50)}{160}}=0.0395,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Check independence with a population at least $10n=1600$, and large counts: $np=80.0$, $n(1-p)=80.0$. Error to reject: The spread is not p(1-p)/n; that is the variance. Standard error is its square root.

Worked solution P51-Tough-8. The distribution has mean 0.520, standard error 0.0383, and the requested probability is approximately 0.1587. $${\mu }_{\widehat{p}}=0.52,{\sigma }_{\widehat{p}}=\sqrt{\frac{0.52(0.48)}{170}}=0.0383,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Check independence with a population at least $10n=1700$, and large counts: $np=88.4$, $n(1-p)=81.6$. Error to reject: The spread is not p(1-p)/n; that is the variance. Standard error is its square root.

Worked solution P51-Tough-9. The distribution has mean 0.380, standard error 0.0485, and the requested probability is approximately 0.1587. $${\mu }_{\widehat{p}}=0.38,{\sigma }_{\widehat{p}}=\sqrt{\frac{0.38(0.62)}{100}}=0.0485,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Check independence with a population at least $10n=1000$, and large counts: $np=38.0$, $n(1-p)=62.0$. Error to reject: The spread is not p(1-p)/n; that is the variance. Standard error is its square root.

Worked solution P51-Tough-10. The distribution has mean 0.470, standard error 0.0476, and the requested probability is approximately 0.1587. $${\mu }_{\widehat{p}}=0.47,{\sigma }_{\widehat{p}}=\sqrt{\frac{0.47(0.53)}{110}}=0.0476,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Check independence with a population at least $10n=1100$, and large counts: $np=51.7$, $n(1-p)=58.3$. Error to reject: The spread is not p(1-p)/n; that is the variance. Standard error is its square root.

Worked solution P51-Tough-11. The distribution has mean 0.470, standard error 0.0456, and the requested probability is approximately 0.1587. $${\mu }_{\widehat{p}}=0.47,{\sigma }_{\widehat{p}}=\sqrt{\frac{0.47(0.53)}{120}}=0.0456,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Check independence with a population at least $10n=1200$, and large counts: $np=56.4$, $n(1-p)=63.6$. Error to reject: The spread is not p(1-p)/n; that is the variance. Standard error is its square root.

Worked solution P51-Tough-12. The distribution has mean 0.460, standard error 0.0437, and the requested probability is approximately 0.1587. $${\mu }_{\widehat{p}}=0.46,{\sigma }_{\widehat{p}}=\sqrt{\frac{0.46(0.54)}{130}}=0.0437,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Check independence with a population at least $10n=1300$, and large counts: $np=59.8$, $n(1-p)=70.2$. Error to reject: The spread is not p(1-p)/n; that is the variance. Standard error is its square root.

Worked solution P51-Tough-13. The distribution has mean 0.510, standard error 0.0422, and the requested probability is approximately 0.1587. $${\mu }_{\widehat{p}}=0.51,{\sigma }_{\widehat{p}}=\sqrt{\frac{0.51(0.49)}{140}}=0.0422,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Check independence with a population at least $10n=1400$, and large counts: $np=71.4$, $n(1-p)=68.6$. Error to reject: The spread is not p(1-p)/n; that is the variance. Standard error is its square root.

Worked solution P51-Tough-14. The distribution has mean 0.450, standard error 0.0406, and the requested probability is approximately 0.1587. $${\mu }_{\widehat{p}}=0.45,{\sigma }_{\widehat{p}}=\sqrt{\frac{0.45(0.55)}{150}}=0.0406,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Check independence with a population at least $10n=1500$, and large counts: $np=67.5$, $n(1-p)=82.5$. Error to reject: The spread is not p(1-p)/n; that is the variance. Standard error is its square root.

Worked solution P51-Tough-15. The distribution has mean 0.420, standard error 0.0390, and the requested probability is approximately 0.1587. $${\mu }_{\widehat{p}}=0.42,{\sigma }_{\widehat{p}}=\sqrt{\frac{0.42(0.58)}{160}}=0.0390,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Check independence with a population at least $10n=1600$, and large counts: $np=67.2$, $n(1-p)=92.8$. Error to reject: The spread is not p(1-p)/n; that is the variance. Standard error is its square root.

Worked solution P51-Tough-16. The distribution has mean 0.420, standard error 0.0379, and the requested probability is approximately 0.1587. $${\mu }_{\widehat{p}}=0.42,{\sigma }_{\widehat{p}}=\sqrt{\frac{0.42(0.58)}{170}}=0.0379,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Check independence with a population at least $10n=1700$, and large counts: $np=71.4$, $n(1-p)=98.6$. Error to reject: The spread is not p(1-p)/n; that is the variance. Standard error is its square root.

Worked solution P51-Toughest-1. The distribution has mean 0.440, standard error 0.0496, and the requested probability is approximately 0.1587. $${\mu }_{\widehat{p}}=0.44,{\sigma }_{\widehat{p}}=\sqrt{\frac{0.44(0.56)}{100}}=0.0496,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Check independence with a population at least $10n=1000$, and large counts: $np=44.0$, $n(1-p)=56.0$. Error to reject: The spread is not p(1-p)/n; that is the variance. Standard error is its square root.

Worked solution P51-Toughest-2. The distribution has mean 0.530, standard error 0.0476, and the requested probability is approximately 0.1587. $${\mu }_{\widehat{p}}=0.53,{\sigma }_{\widehat{p}}=\sqrt{\frac{0.53(0.47)}{110}}=0.0476,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Check independence with a population at least $10n=1100$, and large counts: $np=58.3$, $n(1-p)=51.7$. Error to reject: The spread is not p(1-p)/n; that is the variance. Standard error is its square root.

Worked solution P51-Toughest-3. The distribution has mean 0.380, standard error 0.0443, and the requested probability is approximately 0.1587. $${\mu }_{\widehat{p}}=0.38,{\sigma }_{\widehat{p}}=\sqrt{\frac{0.38(0.62)}{120}}=0.0443,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Check independence with a population at least $10n=1200$, and large counts: $np=45.6$, $n(1-p)=74.4$. Error to reject: The spread is not p(1-p)/n; that is the variance. Standard error is its square root.

Worked solution P51-Toughest-4. The distribution has mean 0.350, standard error 0.0418, and the requested probability is approximately 0.1587. $${\mu }_{\widehat{p}}=0.35,{\sigma }_{\widehat{p}}=\sqrt{\frac{0.35(0.65)}{130}}=0.0418,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Check independence with a population at least $10n=1300$, and large counts: $np=45.5$, $n(1-p)=84.5$. Error to reject: The spread is not p(1-p)/n; that is the variance. Standard error is its square root.

Worked solution P51-Toughest-5. The distribution has mean 0.410, standard error 0.0416, and the requested probability is approximately 0.1587. $${\mu }_{\widehat{p}}=0.41,{\sigma }_{\widehat{p}}=\sqrt{\frac{0.41(0.59)}{140}}=0.0416,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Check independence with a population at least $10n=1400$, and large counts: $np=57.4$, $n(1-p)=82.6$. Error to reject: The spread is not p(1-p)/n; that is the variance. Standard error is its square root.

Worked solution P51-Toughest-6. The distribution has mean 0.360, standard error 0.0392, and the requested probability is approximately 0.1587. $${\mu }_{\widehat{p}}=0.36,{\sigma }_{\widehat{p}}=\sqrt{\frac{0.36(0.64)}{150}}=0.0392,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Check independence with a population at least $10n=1500$, and large counts: $np=54.0$, $n(1-p)=96.0$. Error to reject: The spread is not p(1-p)/n; that is the variance. Standard error is its square root.

Worked solution P51-Toughest-7. The distribution has mean 0.450, standard error 0.0393, and the requested probability is approximately 0.1587. $${\mu }_{\widehat{p}}=0.45,{\sigma }_{\widehat{p}}=\sqrt{\frac{0.45(0.55)}{160}}=0.0393,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Check independence with a population at least $10n=1600$, and large counts: $np=72.0$, $n(1-p)=88.0$. Error to reject: The spread is not p(1-p)/n; that is the variance. Standard error is its square root.

Worked solution P51-Toughest-8. The distribution has mean 0.480, standard error 0.0383, and the requested probability is approximately 0.1587. $${\mu }_{\widehat{p}}=0.48,{\sigma }_{\widehat{p}}=\sqrt{\frac{0.48(0.52)}{170}}=0.0383,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Check independence with a population at least $10n=1700$, and large counts: $np=81.6$, $n(1-p)=88.4$. Error to reject: The spread is not p(1-p)/n; that is the variance. Standard error is its square root.

Worked solution P51-Toughest-9. The distribution has mean 0.370, standard error 0.0483, and the requested probability is approximately 0.1587. $${\mu }_{\widehat{p}}=0.37,{\sigma }_{\widehat{p}}=\sqrt{\frac{0.37(0.63)}{100}}=0.0483,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Check independence with a population at least $10n=1000$, and large counts: $np=37.0$, $n(1-p)=63.0$. Error to reject: The spread is not p(1-p)/n; that is the variance. Standard error is its square root.

Worked solution P51-Toughest-10. The distribution has mean 0.500, standard error 0.0477, and the requested probability is approximately 0.1587. $${\mu }_{\widehat{p}}=0.50,{\sigma }_{\widehat{p}}=\sqrt{\frac{0.50(0.50)}{110}}=0.0477,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Check independence with a population at least $10n=1100$, and large counts: $np=55.0$, $n(1-p)=55.0$. Error to reject: The spread is not p(1-p)/n; that is the variance. Standard error is its square root.

Worked solution P51-Toughest-11. The distribution has mean 0.490, standard error 0.0456, and the requested probability is approximately 0.1587. $${\mu }_{\widehat{p}}=0.49,{\sigma }_{\widehat{p}}=\sqrt{\frac{0.49(0.51)}{120}}=0.0456,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Check independence with a population at least $10n=1200$, and large counts: $np=58.8$, $n(1-p)=61.2$. Error to reject: The spread is not p(1-p)/n; that is the variance. Standard error is its square root.

Worked solution P51-Toughest-12. The distribution has mean 0.430, standard error 0.0434, and the requested probability is approximately 0.1587. $${\mu }_{\widehat{p}}=0.43,{\sigma }_{\widehat{p}}=\sqrt{\frac{0.43(0.57)}{130}}=0.0434,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Check independence with a population at least $10n=1300$, and large counts: $np=55.9$, $n(1-p)=74.1$. Error to reject: The spread is not p(1-p)/n; that is the variance. Standard error is its square root.

Worked solution P51-Toughest-13. The distribution has mean 0.400, standard error 0.0414, and the requested probability is approximately 0.1587. $${\mu }_{\widehat{p}}=0.40,{\sigma }_{\widehat{p}}=\sqrt{\frac{0.40(0.60)}{140}}=0.0414,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Check independence with a population at least $10n=1400$, and large counts: $np=56.0$, $n(1-p)=84.0$. Error to reject: The spread is not p(1-p)/n; that is the variance. Standard error is its square root.

Worked solution P51-Toughest-14. The distribution has mean 0.530, standard error 0.0408, and the requested probability is approximately 0.1587. $${\mu }_{\widehat{p}}=0.53,{\sigma }_{\widehat{p}}=\sqrt{\frac{0.53(0.47)}{150}}=0.0408,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Check independence with a population at least $10n=1500$, and large counts: $np=79.5$, $n(1-p)=70.5$. Error to reject: The spread is not p(1-p)/n; that is the variance. Standard error is its square root.

Worked solution P51-Toughest-15. The distribution has mean 0.430, standard error 0.0391, and the requested probability is approximately 0.1587. $${\mu }_{\widehat{p}}=0.43,{\sigma }_{\widehat{p}}=\sqrt{\frac{0.43(0.57)}{160}}=0.0391,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Check independence with a population at least $10n=1600$, and large counts: $np=68.8$, $n(1-p)=91.2$. Error to reject: The spread is not p(1-p)/n; that is the variance. Standard error is its square root.

Worked solution P51-Toughest-16. The distribution has mean 0.510, standard error 0.0383, and the requested probability is approximately 0.1587. $${\mu }_{\widehat{p}}=0.51,{\sigma }_{\widehat{p}}=\sqrt{\frac{0.51(0.49)}{170}}=0.0383,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Check independence with a population at least $10n=1700$, and large counts: $np=86.7$, $n(1-p)=83.3$. Error to reject: The spread is not p(1-p)/n; that is the variance. Standard error is its square root.