Worked solution P52-Easy-1. The sampling distribution has mean 66.000, standard error 2.0000, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=66,{\sigma }_{\overline{X}}=\frac{10}{\sqrt{25}}=2.0000,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Use an independent random sample; if sampling without replacement, the population should be at least $10n=250$. Normality is exact for a normal population and approximate through CLT otherwise. Error to reject: Increasing n reduces the standard error but does not change the population standard deviation or make the raw observations normal.

Worked solution P52-Easy-2. The sampling distribution has mean 80.000, standard error 1.4606, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=80,{\sigma }_{\overline{X}}=\frac{8}{\sqrt{30}}=1.4606,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Use an independent random sample; if sampling without replacement, the population should be at least $10n=300$. Normality is exact for a normal population and approximate through CLT otherwise. Error to reject: Increasing n reduces the standard error but does not change the population standard deviation or make the raw observations normal.

Worked solution P52-Easy-3. The sampling distribution has mean 73.000, standard error 1.6903, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=73,{\sigma }_{\overline{X}}=\frac{10}{\sqrt{35}}=1.6903,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Use an independent random sample; if sampling without replacement, the population should be at least $10n=350$. Normality is exact for a normal population and approximate through CLT otherwise. Error to reject: Increasing n reduces the standard error but does not change the population standard deviation or make the raw observations normal.

Worked solution P52-Easy-4. The sampling distribution has mean 66.000, standard error 2.2136, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=66,{\sigma }_{\overline{X}}=\frac{14}{\sqrt{40}}=2.2136,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Use an independent random sample; if sampling without replacement, the population should be at least $10n=400$. Normality is exact for a normal population and approximate through CLT otherwise. Error to reject: Increasing n reduces the standard error but does not change the population standard deviation or make the raw observations normal.

Worked solution P52-Easy-5. The sampling distribution has mean 80.000, standard error 1.1926, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=80,{\sigma }_{\overline{X}}=\frac{8}{\sqrt{45}}=1.1926,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Use an independent random sample; if sampling without replacement, the population should be at least $10n=450$. Normality is exact for a normal population and approximate through CLT otherwise. Error to reject: Increasing n reduces the standard error but does not change the population standard deviation or make the raw observations normal.

Worked solution P52-Easy-6. The sampling distribution has mean 61.000, standard error 1.5556, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=61,{\sigma }_{\overline{X}}=\frac{11}{\sqrt{50}}=1.5556,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Use an independent random sample; if sampling without replacement, the population should be at least $10n=500$. Normality is exact for a normal population and approximate through CLT otherwise. Error to reject: Increasing n reduces the standard error but does not change the population standard deviation or make the raw observations normal.

Worked solution P52-Easy-7. The sampling distribution has mean 63.000, standard error 1.3484, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=63,{\sigma }_{\overline{X}}=\frac{10}{\sqrt{55}}=1.3484,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Use an independent random sample; if sampling without replacement, the population should be at least $10n=550$. Normality is exact for a normal population and approximate through CLT otherwise. Error to reject: Increasing n reduces the standard error but does not change the population standard deviation or make the raw observations normal.

Worked solution P52-Easy-8. The sampling distribution has mean 64.000, standard error 1.4201, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=64,{\sigma }_{\overline{X}}=\frac{11}{\sqrt{60}}=1.4201,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Use an independent random sample; if sampling without replacement, the population should be at least $10n=600$. Normality is exact for a normal population and approximate through CLT otherwise. Error to reject: Increasing n reduces the standard error but does not change the population standard deviation or make the raw observations normal.

Worked solution P52-Easy-9. The sampling distribution has mean 65.000, standard error 1.6000, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=65,{\sigma }_{\overline{X}}=\frac{8}{\sqrt{25}}=1.6000,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Use an independent random sample; if sampling without replacement, the population should be at least $10n=250$. Normality is exact for a normal population and approximate through CLT otherwise. Error to reject: Increasing n reduces the standard error but does not change the population standard deviation or make the raw observations normal.

Worked solution P52-Easy-10. The sampling distribution has mean 53.000, standard error 2.0083, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=53,{\sigma }_{\overline{X}}=\frac{11}{\sqrt{30}}=2.0083,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Use an independent random sample; if sampling without replacement, the population should be at least $10n=300$. Normality is exact for a normal population and approximate through CLT otherwise. Error to reject: Increasing n reduces the standard error but does not change the population standard deviation or make the raw observations normal.

Worked solution P52-Easy-11. The sampling distribution has mean 75.000, standard error 2.3664, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=75,{\sigma }_{\overline{X}}=\frac{14}{\sqrt{35}}=2.3664,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Use an independent random sample; if sampling without replacement, the population should be at least $10n=350$. Normality is exact for a normal population and approximate through CLT otherwise. Error to reject: Increasing n reduces the standard error but does not change the population standard deviation or make the raw observations normal.

Worked solution P52-Easy-12. The sampling distribution has mean 80.000, standard error 1.4230, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=80,{\sigma }_{\overline{X}}=\frac{9}{\sqrt{40}}=1.4230,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Use an independent random sample; if sampling without replacement, the population should be at least $10n=400$. Normality is exact for a normal population and approximate through CLT otherwise. Error to reject: Increasing n reduces the standard error but does not change the population standard deviation or make the raw observations normal.

Worked solution P52-Easy-13. The sampling distribution has mean 67.000, standard error 1.7889, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=67,{\sigma }_{\overline{X}}=\frac{12}{\sqrt{45}}=1.7889,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Use an independent random sample; if sampling without replacement, the population should be at least $10n=450$. Normality is exact for a normal population and approximate through CLT otherwise. Error to reject: Increasing n reduces the standard error but does not change the population standard deviation or make the raw observations normal.

Worked solution P52-Easy-14. The sampling distribution has mean 61.000, standard error 1.1314, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=61,{\sigma }_{\overline{X}}=\frac{8}{\sqrt{50}}=1.1314,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Use an independent random sample; if sampling without replacement, the population should be at least $10n=500$. Normality is exact for a normal population and approximate through CLT otherwise. Error to reject: Increasing n reduces the standard error but does not change the population standard deviation or make the raw observations normal.

Worked solution P52-Easy-15. The sampling distribution has mean 58.000, standard error 1.7529, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=58,{\sigma }_{\overline{X}}=\frac{13}{\sqrt{55}}=1.7529,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Use an independent random sample; if sampling without replacement, the population should be at least $10n=550$. Normality is exact for a normal population and approximate through CLT otherwise. Error to reject: Increasing n reduces the standard error but does not change the population standard deviation or make the raw observations normal.

Worked solution P52-Tough-1. The sampling distribution has mean 55.000, standard error 2.2000, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=55,{\sigma }_{\overline{X}}=\frac{11}{\sqrt{25}}=2.2000,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Use an independent random sample; if sampling without replacement, the population should be at least $10n=250$. Normality is exact for a normal population and approximate through CLT otherwise. Error to reject: Increasing n reduces the standard error but does not change the population standard deviation or make the raw observations normal.

Worked solution P52-Tough-2. The sampling distribution has mean 62.000, standard error 2.0083, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=62,{\sigma }_{\overline{X}}=\frac{11}{\sqrt{30}}=2.0083,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Use an independent random sample; if sampling without replacement, the population should be at least $10n=300$. Normality is exact for a normal population and approximate through CLT otherwise. Error to reject: Increasing n reduces the standard error but does not change the population standard deviation or make the raw observations normal.

Worked solution P52-Tough-3. The sampling distribution has mean 66.000, standard error 1.3522, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=66,{\sigma }_{\overline{X}}=\frac{8}{\sqrt{35}}=1.3522,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Use an independent random sample; if sampling without replacement, the population should be at least $10n=350$. Normality is exact for a normal population and approximate through CLT otherwise. Error to reject: Increasing n reduces the standard error but does not change the population standard deviation or make the raw observations normal.

Worked solution P52-Tough-4. The sampling distribution has mean 56.000, standard error 1.7393, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=56,{\sigma }_{\overline{X}}=\frac{11}{\sqrt{40}}=1.7393,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Use an independent random sample; if sampling without replacement, the population should be at least $10n=400$. Normality is exact for a normal population and approximate through CLT otherwise. Error to reject: Increasing n reduces the standard error but does not change the population standard deviation or make the raw observations normal.

Worked solution P52-Tough-5. The sampling distribution has mean 57.000, standard error 1.7889, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=57,{\sigma }_{\overline{X}}=\frac{12}{\sqrt{45}}=1.7889,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Use an independent random sample; if sampling without replacement, the population should be at least $10n=450$. Normality is exact for a normal population and approximate through CLT otherwise. Error to reject: Increasing n reduces the standard error but does not change the population standard deviation or make the raw observations normal.

Worked solution P52-Tough-6. The sampling distribution has mean 67.000, standard error 1.9799, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=67,{\sigma }_{\overline{X}}=\frac{14}{\sqrt{50}}=1.9799,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Use an independent random sample; if sampling without replacement, the population should be at least $10n=500$. Normality is exact for a normal population and approximate through CLT otherwise. Error to reject: Increasing n reduces the standard error but does not change the population standard deviation or make the raw observations normal.

Worked solution P52-Tough-7. The sampling distribution has mean 63.000, standard error 1.2136, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=63,{\sigma }_{\overline{X}}=\frac{9}{\sqrt{55}}=1.2136,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Use an independent random sample; if sampling without replacement, the population should be at least $10n=550$. Normality is exact for a normal population and approximate through CLT otherwise. Error to reject: Increasing n reduces the standard error but does not change the population standard deviation or make the raw observations normal.

Worked solution P52-Tough-8. The sampling distribution has mean 53.000, standard error 1.6783, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=53,{\sigma }_{\overline{X}}=\frac{13}{\sqrt{60}}=1.6783,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Use an independent random sample; if sampling without replacement, the population should be at least $10n=600$. Normality is exact for a normal population and approximate through CLT otherwise. Error to reject: Increasing n reduces the standard error but does not change the population standard deviation or make the raw observations normal.

Worked solution P52-Tough-9. The sampling distribution has mean 62.000, standard error 1.6000, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=62,{\sigma }_{\overline{X}}=\frac{8}{\sqrt{25}}=1.6000,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Use an independent random sample; if sampling without replacement, the population should be at least $10n=250$. Normality is exact for a normal population and approximate through CLT otherwise. Error to reject: Increasing n reduces the standard error but does not change the population standard deviation or make the raw observations normal.

Worked solution P52-Tough-10. The sampling distribution has mean 71.000, standard error 2.3735, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=71,{\sigma }_{\overline{X}}=\frac{13}{\sqrt{30}}=2.3735,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Use an independent random sample; if sampling without replacement, the population should be at least $10n=300$. Normality is exact for a normal population and approximate through CLT otherwise. Error to reject: Increasing n reduces the standard error but does not change the population standard deviation or make the raw observations normal.

Worked solution P52-Tough-11. The sampling distribution has mean 62.000, standard error 1.8593, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=62,{\sigma }_{\overline{X}}=\frac{11}{\sqrt{35}}=1.8593,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Use an independent random sample; if sampling without replacement, the population should be at least $10n=350$. Normality is exact for a normal population and approximate through CLT otherwise. Error to reject: Increasing n reduces the standard error but does not change the population standard deviation or make the raw observations normal.

Worked solution P52-Tough-12. The sampling distribution has mean 65.000, standard error 1.4230, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=65,{\sigma }_{\overline{X}}=\frac{9}{\sqrt{40}}=1.4230,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Use an independent random sample; if sampling without replacement, the population should be at least $10n=400$. Normality is exact for a normal population and approximate through CLT otherwise. Error to reject: Increasing n reduces the standard error but does not change the population standard deviation or make the raw observations normal.

Worked solution P52-Tough-13. The sampling distribution has mean 61.000, standard error 1.7889, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=61,{\sigma }_{\overline{X}}=\frac{12}{\sqrt{45}}=1.7889,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Use an independent random sample; if sampling without replacement, the population should be at least $10n=450$. Normality is exact for a normal population and approximate through CLT otherwise. Error to reject: Increasing n reduces the standard error but does not change the population standard deviation or make the raw observations normal.

Worked solution P52-Tough-14. The sampling distribution has mean 74.000, standard error 1.1314, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=74,{\sigma }_{\overline{X}}=\frac{8}{\sqrt{50}}=1.1314,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Use an independent random sample; if sampling without replacement, the population should be at least $10n=500$. Normality is exact for a normal population and approximate through CLT otherwise. Error to reject: Increasing n reduces the standard error but does not change the population standard deviation or make the raw observations normal.

Worked solution P52-Tough-15. The sampling distribution has mean 66.000, standard error 1.0787, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=66,{\sigma }_{\overline{X}}=\frac{8}{\sqrt{55}}=1.0787,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Use an independent random sample; if sampling without replacement, the population should be at least $10n=550$. Normality is exact for a normal population and approximate through CLT otherwise. Error to reject: Increasing n reduces the standard error but does not change the population standard deviation or make the raw observations normal.

Worked solution P52-Toughest-1. The sampling distribution has mean 77.000, standard error 2.0000, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=77,{\sigma }_{\overline{X}}=\frac{10}{\sqrt{25}}=2.0000,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Use an independent random sample; if sampling without replacement, the population should be at least $10n=250$. Normality is exact for a normal population and approximate through CLT otherwise. Error to reject: Increasing n reduces the standard error but does not change the population standard deviation or make the raw observations normal.

Worked solution P52-Toughest-2. The sampling distribution has mean 50.000, standard error 2.5560, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=50,{\sigma }_{\overline{X}}=\frac{14}{\sqrt{30}}=2.5560,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Use an independent random sample; if sampling without replacement, the population should be at least $10n=300$. Normality is exact for a normal population and approximate through CLT otherwise. Error to reject: Increasing n reduces the standard error but does not change the population standard deviation or make the raw observations normal.

Worked solution P52-Toughest-3. The sampling distribution has mean 71.000, standard error 2.1974, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=71,{\sigma }_{\overline{X}}=\frac{13}{\sqrt{35}}=2.1974,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Use an independent random sample; if sampling without replacement, the population should be at least $10n=350$. Normality is exact for a normal population and approximate through CLT otherwise. Error to reject: Increasing n reduces the standard error but does not change the population standard deviation or make the raw observations normal.

Worked solution P52-Toughest-4. The sampling distribution has mean 65.000, standard error 2.2136, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=65,{\sigma }_{\overline{X}}=\frac{14}{\sqrt{40}}=2.2136,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Use an independent random sample; if sampling without replacement, the population should be at least $10n=400$. Normality is exact for a normal population and approximate through CLT otherwise. Error to reject: Increasing n reduces the standard error but does not change the population standard deviation or make the raw observations normal.

Worked solution P52-Toughest-5. The sampling distribution has mean 67.000, standard error 1.9379, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=67,{\sigma }_{\overline{X}}=\frac{13}{\sqrt{45}}=1.9379,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Use an independent random sample; if sampling without replacement, the population should be at least $10n=450$. Normality is exact for a normal population and approximate through CLT otherwise. Error to reject: Increasing n reduces the standard error but does not change the population standard deviation or make the raw observations normal.

Worked solution P52-Toughest-6. The sampling distribution has mean 59.000, standard error 1.6971, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=59,{\sigma }_{\overline{X}}=\frac{12}{\sqrt{50}}=1.6971,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Use an independent random sample; if sampling without replacement, the population should be at least $10n=500$. Normality is exact for a normal population and approximate through CLT otherwise. Error to reject: Increasing n reduces the standard error but does not change the population standard deviation or make the raw observations normal.

Worked solution P52-Toughest-7. The sampling distribution has mean 59.000, standard error 1.8878, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=59,{\sigma }_{\overline{X}}=\frac{14}{\sqrt{55}}=1.8878,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Use an independent random sample; if sampling without replacement, the population should be at least $10n=550$. Normality is exact for a normal population and approximate through CLT otherwise. Error to reject: Increasing n reduces the standard error but does not change the population standard deviation or make the raw observations normal.

Worked solution P52-Toughest-8. The sampling distribution has mean 59.000, standard error 1.1619, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=59,{\sigma }_{\overline{X}}=\frac{9}{\sqrt{60}}=1.1619,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Use an independent random sample; if sampling without replacement, the population should be at least $10n=600$. Normality is exact for a normal population and approximate through CLT otherwise. Error to reject: Increasing n reduces the standard error but does not change the population standard deviation or make the raw observations normal.

Worked solution P52-Toughest-9. The sampling distribution has mean 56.000, standard error 1.6000, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=56,{\sigma }_{\overline{X}}=\frac{8}{\sqrt{25}}=1.6000,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Use an independent random sample; if sampling without replacement, the population should be at least $10n=250$. Normality is exact for a normal population and approximate through CLT otherwise. Error to reject: Increasing n reduces the standard error but does not change the population standard deviation or make the raw observations normal.

Worked solution P52-Toughest-10. The sampling distribution has mean 77.000, standard error 1.4606, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=77,{\sigma }_{\overline{X}}=\frac{8}{\sqrt{30}}=1.4606,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Use an independent random sample; if sampling without replacement, the population should be at least $10n=300$. Normality is exact for a normal population and approximate through CLT otherwise. Error to reject: Increasing n reduces the standard error but does not change the population standard deviation or make the raw observations normal.

Worked solution P52-Toughest-11. The sampling distribution has mean 66.000, standard error 1.8593, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=66,{\sigma }_{\overline{X}}=\frac{11}{\sqrt{35}}=1.8593,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Use an independent random sample; if sampling without replacement, the population should be at least $10n=350$. Normality is exact for a normal population and approximate through CLT otherwise. Error to reject: Increasing n reduces the standard error but does not change the population standard deviation or make the raw observations normal.

Worked solution P52-Toughest-12. The sampling distribution has mean 70.000, standard error 1.4230, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=70,{\sigma }_{\overline{X}}=\frac{9}{\sqrt{40}}=1.4230,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Use an independent random sample; if sampling without replacement, the population should be at least $10n=400$. Normality is exact for a normal population and approximate through CLT otherwise. Error to reject: Increasing n reduces the standard error but does not change the population standard deviation or make the raw observations normal.

Worked solution P52-Toughest-13. The sampling distribution has mean 55.000, standard error 1.7889, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=55,{\sigma }_{\overline{X}}=\frac{12}{\sqrt{45}}=1.7889,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Use an independent random sample; if sampling without replacement, the population should be at least $10n=450$. Normality is exact for a normal population and approximate through CLT otherwise. Error to reject: Increasing n reduces the standard error but does not change the population standard deviation or make the raw observations normal.

Worked solution P52-Toughest-14. The sampling distribution has mean 66.000, standard error 1.5556, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=66,{\sigma }_{\overline{X}}=\frac{11}{\sqrt{50}}=1.5556,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Use an independent random sample; if sampling without replacement, the population should be at least $10n=500$. Normality is exact for a normal population and approximate through CLT otherwise. Error to reject: Increasing n reduces the standard error but does not change the population standard deviation or make the raw observations normal.

Worked solution P52-Toughest-15. The sampling distribution has mean 56.000, standard error 1.8878, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=56,{\sigma }_{\overline{X}}=\frac{14}{\sqrt{55}}=1.8878,z=1.$$ Interpretation: The sampling distribution describes variation among statistics from repeated samples, not variation among individual observations. Validity: Use an independent random sample; if sampling without replacement, the population should be at least $10n=550$. Normality is exact for a normal population and approximate through CLT otherwise. Error to reject: Increasing n reduces the standard error but does not change the population standard deviation or make the raw observations normal.