Worked solution P50-Easy-1. The sampling distribution has mean 75.000, standard error 2.4000, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=75,{\sigma }_{\overline{X}}=\frac{12}{\sqrt{25}}=2.4000,z=1.$$ 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 P50-Easy-2. The sampling distribution has mean 53.000, standard error 1.6432, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=53,{\sigma }_{\overline{X}}=\frac{9}{\sqrt{30}}=1.6432,z=1.$$ 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 P50-Easy-3. The sampling distribution has mean 59.000, standard error 2.0284, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=59,{\sigma }_{\overline{X}}=\frac{12}{\sqrt{35}}=2.0284,z=1.$$ 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 P50-Easy-4. The sampling distribution has mean 55.000, standard error 1.8974, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=55,{\sigma }_{\overline{X}}=\frac{12}{\sqrt{40}}=1.8974,z=1.$$ 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 P50-Easy-5. The sampling distribution has mean 72.000, standard error 1.9379, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=72,{\sigma }_{\overline{X}}=\frac{13}{\sqrt{45}}=1.9379,z=1.$$ 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 P50-Easy-6. The sampling distribution has mean 54.000, standard error 1.8385, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=54,{\sigma }_{\overline{X}}=\frac{13}{\sqrt{50}}=1.8385,z=1.$$ 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 P50-Easy-7. The sampling distribution has mean 80.000, standard error 1.0787, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=80,{\sigma }_{\overline{X}}=\frac{8}{\sqrt{55}}=1.0787,z=1.$$ 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 P50-Easy-8. The sampling distribution has mean 56.000, standard error 1.0328, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=56,{\sigma }_{\overline{X}}=\frac{8}{\sqrt{60}}=1.0328,z=1.$$ 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 P50-Easy-9. The sampling distribution has mean 74.000, standard error 1.6000, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=74,{\sigma }_{\overline{X}}=\frac{8}{\sqrt{25}}=1.6000,z=1.$$ 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 P50-Easy-10. The sampling distribution has mean 63.000, standard error 2.5560, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=63,{\sigma }_{\overline{X}}=\frac{14}{\sqrt{30}}=2.5560,z=1.$$ 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 P50-Easy-11. The sampling distribution has mean 72.000, standard error 1.5213, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=72,{\sigma }_{\overline{X}}=\frac{9}{\sqrt{35}}=1.5213,z=1.$$ 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 P50-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 P50-Easy-13. The sampling distribution has mean 61.000, standard error 2.0870, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=61,{\sigma }_{\overline{X}}=\frac{14}{\sqrt{45}}=2.0870,z=1.$$ 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 P50-Easy-14. 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 P50-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 P50-Tough-1. The sampling distribution has mean 59.000, standard error 2.2000, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=59,{\sigma }_{\overline{X}}=\frac{11}{\sqrt{25}}=2.2000,z=1.$$ 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 P50-Tough-2. The sampling distribution has mean 69.000, standard error 2.1909, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=69,{\sigma }_{\overline{X}}=\frac{12}{\sqrt{30}}=2.1909,z=1.$$ 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 P50-Tough-3. The sampling distribution has mean 68.000, standard error 1.3522, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=68,{\sigma }_{\overline{X}}=\frac{8}{\sqrt{35}}=1.3522,z=1.$$ 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 P50-Tough-4. The sampling distribution has mean 67.000, standard error 1.2649, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=67,{\sigma }_{\overline{X}}=\frac{8}{\sqrt{40}}=1.2649,z=1.$$ 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 P50-Tough-5. The sampling distribution has mean 51.000, standard error 2.0870, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=51,{\sigma }_{\overline{X}}=\frac{14}{\sqrt{45}}=2.0870,z=1.$$ 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 P50-Tough-6. The sampling distribution has mean 62.000, standard error 1.2728, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=62,{\sigma }_{\overline{X}}=\frac{9}{\sqrt{50}}=1.2728,z=1.$$ 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 P50-Tough-7. The sampling distribution has mean 74.000, standard error 1.2136, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=74,{\sigma }_{\overline{X}}=\frac{9}{\sqrt{55}}=1.2136,z=1.$$ 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 P50-Tough-8. The sampling distribution has mean 57.000, standard error 1.4201, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=57,{\sigma }_{\overline{X}}=\frac{11}{\sqrt{60}}=1.4201,z=1.$$ 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 P50-Tough-9. 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 P50-Tough-10. The sampling distribution has mean 71.000, standard error 1.4606, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=71,{\sigma }_{\overline{X}}=\frac{8}{\sqrt{30}}=1.4606,z=1.$$ 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 P50-Tough-11. The sampling distribution has mean 55.000, standard error 2.1974, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=55,{\sigma }_{\overline{X}}=\frac{13}{\sqrt{35}}=2.1974,z=1.$$ 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 P50-Tough-12. The sampling distribution has mean 77.000, standard error 1.4230, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=77,{\sigma }_{\overline{X}}=\frac{9}{\sqrt{40}}=1.4230,z=1.$$ 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 P50-Tough-13. The sampling distribution has mean 63.000, standard error 1.3416, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=63,{\sigma }_{\overline{X}}=\frac{9}{\sqrt{45}}=1.3416,z=1.$$ 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 P50-Tough-14. The sampling distribution has mean 67.000, standard error 1.1314, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=67,{\sigma }_{\overline{X}}=\frac{8}{\sqrt{50}}=1.1314,z=1.$$ 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 P50-Tough-15. The sampling distribution has mean 69.000, standard error 1.7529, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=69,{\sigma }_{\overline{X}}=\frac{13}{\sqrt{55}}=1.7529,z=1.$$ 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 P50-Toughest-1. The sampling distribution has mean 55.000, standard error 1.6000, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=55,{\sigma }_{\overline{X}}=\frac{8}{\sqrt{25}}=1.6000,z=1.$$ 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 P50-Toughest-2. The sampling distribution has mean 54.000, standard error 1.4606, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=54,{\sigma }_{\overline{X}}=\frac{8}{\sqrt{30}}=1.4606,z=1.$$ 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 P50-Toughest-3. The sampling distribution has mean 79.000, standard error 1.5213, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=79,{\sigma }_{\overline{X}}=\frac{9}{\sqrt{35}}=1.5213,z=1.$$ 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 P50-Toughest-4. The sampling distribution has mean 80.000, standard error 1.2649, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=80,{\sigma }_{\overline{X}}=\frac{8}{\sqrt{40}}=1.2649,z=1.$$ 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 P50-Toughest-5. The sampling distribution has mean 60.000, standard error 1.6398, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=60,{\sigma }_{\overline{X}}=\frac{11}{\sqrt{45}}=1.6398,z=1.$$ 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 P50-Toughest-6. The sampling distribution has mean 63.000, standard error 1.9799, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=63,{\sigma }_{\overline{X}}=\frac{14}{\sqrt{50}}=1.9799,z=1.$$ 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 P50-Toughest-7. The sampling distribution has mean 69.000, standard error 1.0787, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=69,{\sigma }_{\overline{X}}=\frac{8}{\sqrt{55}}=1.0787,z=1.$$ 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 P50-Toughest-8. The sampling distribution has mean 75.000, standard error 1.8074, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=75,{\sigma }_{\overline{X}}=\frac{14}{\sqrt{60}}=1.8074,z=1.$$ 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 P50-Toughest-9. The sampling distribution has mean 75.000, standard error 2.8000, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=75,{\sigma }_{\overline{X}}=\frac{14}{\sqrt{25}}=2.8000,z=1.$$ 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 P50-Toughest-10. The sampling distribution has mean 60.000, standard error 1.6432, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=60,{\sigma }_{\overline{X}}=\frac{9}{\sqrt{30}}=1.6432,z=1.$$ 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 P50-Toughest-11. The sampling distribution has mean 78.000, standard error 1.5213, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=78,{\sigma }_{\overline{X}}=\frac{9}{\sqrt{35}}=1.5213,z=1.$$ 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 P50-Toughest-12. The sampling distribution has mean 50.000, standard error 1.5811, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=50,{\sigma }_{\overline{X}}=\frac{10}{\sqrt{40}}=1.5811,z=1.$$ 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 P50-Toughest-13. The sampling distribution has mean 79.000, standard error 2.0870, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=79,{\sigma }_{\overline{X}}=\frac{14}{\sqrt{45}}=2.0870,z=1.$$ 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 P50-Toughest-14. The sampling distribution has mean 53.000, standard error 1.9799, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=53,{\sigma }_{\overline{X}}=\frac{14}{\sqrt{50}}=1.9799,z=1.$$ 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 P50-Toughest-15. The sampling distribution has mean 60.000, standard error 1.0787, and the probability is approximately 0.1587 under a justified normal approximation. $${\mu }_{\overline{X}}=60,{\sigma }_{\overline{X}}=\frac{8}{\sqrt{55}}=1.0787,z=1.$$ 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.