Worked solution P44-Easy-1. The expected value is 1.550 and the standard deviation is 1.244. $$E(X)=\sum xp(x)=1.550,SD(X)=\sqrt{\sum (x-1.550{)}^{2}p(x)}=1.244.$$ Interpretation: The expected value is the probability-weighted long-run average and can fall between the possible single-trial outcomes. Validity: Compute variance from squared deviations weighted by probabilities, then take the square root to return to the original units. Error to reject: A simple average of the listed outcome labels ignores their unequal probabilities and generally gives the wrong center.

Worked solution P44-Easy-2. The expected value is 1.550 and the standard deviation is 1.244. $$E(X)=\sum xp(x)=1.550,SD(X)=\sqrt{\sum (x-1.550{)}^{2}p(x)}=1.244.$$ Interpretation: The expected value is the probability-weighted long-run average and can fall between the possible single-trial outcomes. Validity: Compute variance from squared deviations weighted by probabilities, then take the square root to return to the original units. Error to reject: A simple average of the listed outcome labels ignores their unequal probabilities and generally gives the wrong center.

Worked solution P44-Easy-3. The expected value is 1.550 and the standard deviation is 1.244. $$E(X)=\sum xp(x)=1.550,SD(X)=\sqrt{\sum (x-1.550{)}^{2}p(x)}=1.244.$$ Interpretation: The expected value is the probability-weighted long-run average and can fall between the possible single-trial outcomes. Validity: Compute variance from squared deviations weighted by probabilities, then take the square root to return to the original units. Error to reject: A simple average of the listed outcome labels ignores their unequal probabilities and generally gives the wrong center.

Worked solution P44-Easy-4. The expected value is 1.550 and the standard deviation is 1.244. $$E(X)=\sum xp(x)=1.550,SD(X)=\sqrt{\sum (x-1.550{)}^{2}p(x)}=1.244.$$ Interpretation: The expected value is the probability-weighted long-run average and can fall between the possible single-trial outcomes. Validity: Compute variance from squared deviations weighted by probabilities, then take the square root to return to the original units. Error to reject: A simple average of the listed outcome labels ignores their unequal probabilities and generally gives the wrong center.

Worked solution P44-Easy-5. The expected value is 1.550 and the standard deviation is 1.244. $$E(X)=\sum xp(x)=1.550,SD(X)=\sqrt{\sum (x-1.550{)}^{2}p(x)}=1.244.$$ Interpretation: The expected value is the probability-weighted long-run average and can fall between the possible single-trial outcomes. Validity: Compute variance from squared deviations weighted by probabilities, then take the square root to return to the original units. Error to reject: A simple average of the listed outcome labels ignores their unequal probabilities and generally gives the wrong center.

Worked solution P44-Easy-6. The expected value is 1.550 and the standard deviation is 1.244. $$E(X)=\sum xp(x)=1.550,SD(X)=\sqrt{\sum (x-1.550{)}^{2}p(x)}=1.244.$$ Interpretation: The expected value is the probability-weighted long-run average and can fall between the possible single-trial outcomes. Validity: Compute variance from squared deviations weighted by probabilities, then take the square root to return to the original units. Error to reject: A simple average of the listed outcome labels ignores their unequal probabilities and generally gives the wrong center.

Worked solution P44-Easy-7. The expected value is 1.550 and the standard deviation is 1.244. $$E(X)=\sum xp(x)=1.550,SD(X)=\sqrt{\sum (x-1.550{)}^{2}p(x)}=1.244.$$ Interpretation: The expected value is the probability-weighted long-run average and can fall between the possible single-trial outcomes. Validity: Compute variance from squared deviations weighted by probabilities, then take the square root to return to the original units. Error to reject: A simple average of the listed outcome labels ignores their unequal probabilities and generally gives the wrong center.

Worked solution P44-Easy-8. The expected value is 1.550 and the standard deviation is 1.244. $$E(X)=\sum xp(x)=1.550,SD(X)=\sqrt{\sum (x-1.550{)}^{2}p(x)}=1.244.$$ Interpretation: The expected value is the probability-weighted long-run average and can fall between the possible single-trial outcomes. Validity: Compute variance from squared deviations weighted by probabilities, then take the square root to return to the original units. Error to reject: A simple average of the listed outcome labels ignores their unequal probabilities and generally gives the wrong center.

Worked solution P44-Easy-9. The expected value is 1.550 and the standard deviation is 1.244. $$E(X)=\sum xp(x)=1.550,SD(X)=\sqrt{\sum (x-1.550{)}^{2}p(x)}=1.244.$$ Interpretation: The expected value is the probability-weighted long-run average and can fall between the possible single-trial outcomes. Validity: Compute variance from squared deviations weighted by probabilities, then take the square root to return to the original units. Error to reject: A simple average of the listed outcome labels ignores their unequal probabilities and generally gives the wrong center.

Worked solution P44-Easy-10. The expected value is 1.550 and the standard deviation is 1.244. $$E(X)=\sum xp(x)=1.550,SD(X)=\sqrt{\sum (x-1.550{)}^{2}p(x)}=1.244.$$ Interpretation: The expected value is the probability-weighted long-run average and can fall between the possible single-trial outcomes. Validity: Compute variance from squared deviations weighted by probabilities, then take the square root to return to the original units. Error to reject: A simple average of the listed outcome labels ignores their unequal probabilities and generally gives the wrong center.

Worked solution P44-Easy-11. The expected value is 1.550 and the standard deviation is 1.244. $$E(X)=\sum xp(x)=1.550,SD(X)=\sqrt{\sum (x-1.550{)}^{2}p(x)}=1.244.$$ Interpretation: The expected value is the probability-weighted long-run average and can fall between the possible single-trial outcomes. Validity: Compute variance from squared deviations weighted by probabilities, then take the square root to return to the original units. Error to reject: A simple average of the listed outcome labels ignores their unequal probabilities and generally gives the wrong center.

Worked solution P44-Easy-12. The expected value is 1.550 and the standard deviation is 1.244. $$E(X)=\sum xp(x)=1.550,SD(X)=\sqrt{\sum (x-1.550{)}^{2}p(x)}=1.244.$$ Interpretation: The expected value is the probability-weighted long-run average and can fall between the possible single-trial outcomes. Validity: Compute variance from squared deviations weighted by probabilities, then take the square root to return to the original units. Error to reject: A simple average of the listed outcome labels ignores their unequal probabilities and generally gives the wrong center.

Worked solution P44-Easy-13. The expected value is 1.550 and the standard deviation is 1.244. $$E(X)=\sum xp(x)=1.550,SD(X)=\sqrt{\sum (x-1.550{)}^{2}p(x)}=1.244.$$ Interpretation: The expected value is the probability-weighted long-run average and can fall between the possible single-trial outcomes. Validity: Compute variance from squared deviations weighted by probabilities, then take the square root to return to the original units. Error to reject: A simple average of the listed outcome labels ignores their unequal probabilities and generally gives the wrong center.

Worked solution P44-Easy-14. The expected value is 1.550 and the standard deviation is 1.244. $$E(X)=\sum xp(x)=1.550,SD(X)=\sqrt{\sum (x-1.550{)}^{2}p(x)}=1.244.$$ Interpretation: The expected value is the probability-weighted long-run average and can fall between the possible single-trial outcomes. Validity: Compute variance from squared deviations weighted by probabilities, then take the square root to return to the original units. Error to reject: A simple average of the listed outcome labels ignores their unequal probabilities and generally gives the wrong center.

Worked solution P44-Easy-15. The expected value is 1.550 and the standard deviation is 1.244. $$E(X)=\sum xp(x)=1.550,SD(X)=\sqrt{\sum (x-1.550{)}^{2}p(x)}=1.244.$$ Interpretation: The expected value is the probability-weighted long-run average and can fall between the possible single-trial outcomes. Validity: Compute variance from squared deviations weighted by probabilities, then take the square root to return to the original units. Error to reject: A simple average of the listed outcome labels ignores their unequal probabilities and generally gives the wrong center.

Worked solution P44-Easy-16. The expected value is 1.550 and the standard deviation is 1.244. $$E(X)=\sum xp(x)=1.550,SD(X)=\sqrt{\sum (x-1.550{)}^{2}p(x)}=1.244.$$ Interpretation: The expected value is the probability-weighted long-run average and can fall between the possible single-trial outcomes. Validity: Compute variance from squared deviations weighted by probabilities, then take the square root to return to the original units. Error to reject: A simple average of the listed outcome labels ignores their unequal probabilities and generally gives the wrong center.

Worked solution P44-Easy-17. The expected value is 1.550 and the standard deviation is 1.244. $$E(X)=\sum xp(x)=1.550,SD(X)=\sqrt{\sum (x-1.550{)}^{2}p(x)}=1.244.$$ Interpretation: The expected value is the probability-weighted long-run average and can fall between the possible single-trial outcomes. Validity: Compute variance from squared deviations weighted by probabilities, then take the square root to return to the original units. Error to reject: A simple average of the listed outcome labels ignores their unequal probabilities and generally gives the wrong center.

Worked solution P44-Tough-1. The expected value is 1.550 and the standard deviation is 1.244. $$E(X)=\sum xp(x)=1.550,SD(X)=\sqrt{\sum (x-1.550{)}^{2}p(x)}=1.244.$$ Interpretation: The expected value is the probability-weighted long-run average and can fall between the possible single-trial outcomes. Validity: Compute variance from squared deviations weighted by probabilities, then take the square root to return to the original units. Error to reject: A simple average of the listed outcome labels ignores their unequal probabilities and generally gives the wrong center.

Worked solution P44-Tough-2. The expected value is 1.550 and the standard deviation is 1.244. $$E(X)=\sum xp(x)=1.550,SD(X)=\sqrt{\sum (x-1.550{)}^{2}p(x)}=1.244.$$ Interpretation: The expected value is the probability-weighted long-run average and can fall between the possible single-trial outcomes. Validity: Compute variance from squared deviations weighted by probabilities, then take the square root to return to the original units. Error to reject: A simple average of the listed outcome labels ignores their unequal probabilities and generally gives the wrong center.

Worked solution P44-Tough-3. The expected value is 1.550 and the standard deviation is 1.244. $$E(X)=\sum xp(x)=1.550,SD(X)=\sqrt{\sum (x-1.550{)}^{2}p(x)}=1.244.$$ Interpretation: The expected value is the probability-weighted long-run average and can fall between the possible single-trial outcomes. Validity: Compute variance from squared deviations weighted by probabilities, then take the square root to return to the original units. Error to reject: A simple average of the listed outcome labels ignores their unequal probabilities and generally gives the wrong center.

Worked solution P44-Tough-4. The expected value is 1.550 and the standard deviation is 1.244. $$E(X)=\sum xp(x)=1.550,SD(X)=\sqrt{\sum (x-1.550{)}^{2}p(x)}=1.244.$$ Interpretation: The expected value is the probability-weighted long-run average and can fall between the possible single-trial outcomes. Validity: Compute variance from squared deviations weighted by probabilities, then take the square root to return to the original units. Error to reject: A simple average of the listed outcome labels ignores their unequal probabilities and generally gives the wrong center.

Worked solution P44-Tough-5. The expected value is 1.550 and the standard deviation is 1.244. $$E(X)=\sum xp(x)=1.550,SD(X)=\sqrt{\sum (x-1.550{)}^{2}p(x)}=1.244.$$ Interpretation: The expected value is the probability-weighted long-run average and can fall between the possible single-trial outcomes. Validity: Compute variance from squared deviations weighted by probabilities, then take the square root to return to the original units. Error to reject: A simple average of the listed outcome labels ignores their unequal probabilities and generally gives the wrong center.

Worked solution P44-Tough-6. The expected value is 1.550 and the standard deviation is 1.244. $$E(X)=\sum xp(x)=1.550,SD(X)=\sqrt{\sum (x-1.550{)}^{2}p(x)}=1.244.$$ Interpretation: The expected value is the probability-weighted long-run average and can fall between the possible single-trial outcomes. Validity: Compute variance from squared deviations weighted by probabilities, then take the square root to return to the original units. Error to reject: A simple average of the listed outcome labels ignores their unequal probabilities and generally gives the wrong center.

Worked solution P44-Tough-7. The expected value is 1.550 and the standard deviation is 1.244. $$E(X)=\sum xp(x)=1.550,SD(X)=\sqrt{\sum (x-1.550{)}^{2}p(x)}=1.244.$$ Interpretation: The expected value is the probability-weighted long-run average and can fall between the possible single-trial outcomes. Validity: Compute variance from squared deviations weighted by probabilities, then take the square root to return to the original units. Error to reject: A simple average of the listed outcome labels ignores their unequal probabilities and generally gives the wrong center.

Worked solution P44-Tough-8. The expected value is 1.550 and the standard deviation is 1.244. $$E(X)=\sum xp(x)=1.550,SD(X)=\sqrt{\sum (x-1.550{)}^{2}p(x)}=1.244.$$ Interpretation: The expected value is the probability-weighted long-run average and can fall between the possible single-trial outcomes. Validity: Compute variance from squared deviations weighted by probabilities, then take the square root to return to the original units. Error to reject: A simple average of the listed outcome labels ignores their unequal probabilities and generally gives the wrong center.

Worked solution P44-Tough-9. The expected value is 1.550 and the standard deviation is 1.244. $$E(X)=\sum xp(x)=1.550,SD(X)=\sqrt{\sum (x-1.550{)}^{2}p(x)}=1.244.$$ Interpretation: The expected value is the probability-weighted long-run average and can fall between the possible single-trial outcomes. Validity: Compute variance from squared deviations weighted by probabilities, then take the square root to return to the original units. Error to reject: A simple average of the listed outcome labels ignores their unequal probabilities and generally gives the wrong center.

Worked solution P44-Tough-10. The expected value is 1.550 and the standard deviation is 1.244. $$E(X)=\sum xp(x)=1.550,SD(X)=\sqrt{\sum (x-1.550{)}^{2}p(x)}=1.244.$$ Interpretation: The expected value is the probability-weighted long-run average and can fall between the possible single-trial outcomes. Validity: Compute variance from squared deviations weighted by probabilities, then take the square root to return to the original units. Error to reject: A simple average of the listed outcome labels ignores their unequal probabilities and generally gives the wrong center.

Worked solution P44-Tough-11. The expected value is 1.550 and the standard deviation is 1.244. $$E(X)=\sum xp(x)=1.550,SD(X)=\sqrt{\sum (x-1.550{)}^{2}p(x)}=1.244.$$ Interpretation: The expected value is the probability-weighted long-run average and can fall between the possible single-trial outcomes. Validity: Compute variance from squared deviations weighted by probabilities, then take the square root to return to the original units. Error to reject: A simple average of the listed outcome labels ignores their unequal probabilities and generally gives the wrong center.

Worked solution P44-Tough-12. The expected value is 1.550 and the standard deviation is 1.244. $$E(X)=\sum xp(x)=1.550,SD(X)=\sqrt{\sum (x-1.550{)}^{2}p(x)}=1.244.$$ Interpretation: The expected value is the probability-weighted long-run average and can fall between the possible single-trial outcomes. Validity: Compute variance from squared deviations weighted by probabilities, then take the square root to return to the original units. Error to reject: A simple average of the listed outcome labels ignores their unequal probabilities and generally gives the wrong center.

Worked solution P44-Tough-13. The expected value is 1.550 and the standard deviation is 1.244. $$E(X)=\sum xp(x)=1.550,SD(X)=\sqrt{\sum (x-1.550{)}^{2}p(x)}=1.244.$$ Interpretation: The expected value is the probability-weighted long-run average and can fall between the possible single-trial outcomes. Validity: Compute variance from squared deviations weighted by probabilities, then take the square root to return to the original units. Error to reject: A simple average of the listed outcome labels ignores their unequal probabilities and generally gives the wrong center.

Worked solution P44-Tough-14. The expected value is 1.550 and the standard deviation is 1.244. $$E(X)=\sum xp(x)=1.550,SD(X)=\sqrt{\sum (x-1.550{)}^{2}p(x)}=1.244.$$ Interpretation: The expected value is the probability-weighted long-run average and can fall between the possible single-trial outcomes. Validity: Compute variance from squared deviations weighted by probabilities, then take the square root to return to the original units. Error to reject: A simple average of the listed outcome labels ignores their unequal probabilities and generally gives the wrong center.

Worked solution P44-Tough-15. The expected value is 1.550 and the standard deviation is 1.244. $$E(X)=\sum xp(x)=1.550,SD(X)=\sqrt{\sum (x-1.550{)}^{2}p(x)}=1.244.$$ Interpretation: The expected value is the probability-weighted long-run average and can fall between the possible single-trial outcomes. Validity: Compute variance from squared deviations weighted by probabilities, then take the square root to return to the original units. Error to reject: A simple average of the listed outcome labels ignores their unequal probabilities and generally gives the wrong center.

Worked solution P44-Tough-16. The expected value is 1.550 and the standard deviation is 1.244. $$E(X)=\sum xp(x)=1.550,SD(X)=\sqrt{\sum (x-1.550{)}^{2}p(x)}=1.244.$$ Interpretation: The expected value is the probability-weighted long-run average and can fall between the possible single-trial outcomes. Validity: Compute variance from squared deviations weighted by probabilities, then take the square root to return to the original units. Error to reject: A simple average of the listed outcome labels ignores their unequal probabilities and generally gives the wrong center.

Worked solution P44-Tough-17. The expected value is 1.550 and the standard deviation is 1.244. $$E(X)=\sum xp(x)=1.550,SD(X)=\sqrt{\sum (x-1.550{)}^{2}p(x)}=1.244.$$ Interpretation: The expected value is the probability-weighted long-run average and can fall between the possible single-trial outcomes. Validity: Compute variance from squared deviations weighted by probabilities, then take the square root to return to the original units. Error to reject: A simple average of the listed outcome labels ignores their unequal probabilities and generally gives the wrong center.

Worked solution P44-Toughest-1. The expected value is 1.550 and the standard deviation is 1.244. $$E(X)=\sum xp(x)=1.550,SD(X)=\sqrt{\sum (x-1.550{)}^{2}p(x)}=1.244.$$ Interpretation: The expected value is the probability-weighted long-run average and can fall between the possible single-trial outcomes. Validity: Compute variance from squared deviations weighted by probabilities, then take the square root to return to the original units. Error to reject: A simple average of the listed outcome labels ignores their unequal probabilities and generally gives the wrong center.

Worked solution P44-Toughest-2. The expected value is 1.550 and the standard deviation is 1.244. $$E(X)=\sum xp(x)=1.550,SD(X)=\sqrt{\sum (x-1.550{)}^{2}p(x)}=1.244.$$ Interpretation: The expected value is the probability-weighted long-run average and can fall between the possible single-trial outcomes. Validity: Compute variance from squared deviations weighted by probabilities, then take the square root to return to the original units. Error to reject: A simple average of the listed outcome labels ignores their unequal probabilities and generally gives the wrong center.

Worked solution P44-Toughest-3. The expected value is 1.550 and the standard deviation is 1.244. $$E(X)=\sum xp(x)=1.550,SD(X)=\sqrt{\sum (x-1.550{)}^{2}p(x)}=1.244.$$ Interpretation: The expected value is the probability-weighted long-run average and can fall between the possible single-trial outcomes. Validity: Compute variance from squared deviations weighted by probabilities, then take the square root to return to the original units. Error to reject: A simple average of the listed outcome labels ignores their unequal probabilities and generally gives the wrong center.

Worked solution P44-Toughest-4. The expected value is 1.550 and the standard deviation is 1.244. $$E(X)=\sum xp(x)=1.550,SD(X)=\sqrt{\sum (x-1.550{)}^{2}p(x)}=1.244.$$ Interpretation: The expected value is the probability-weighted long-run average and can fall between the possible single-trial outcomes. Validity: Compute variance from squared deviations weighted by probabilities, then take the square root to return to the original units. Error to reject: A simple average of the listed outcome labels ignores their unequal probabilities and generally gives the wrong center.

Worked solution P44-Toughest-5. The expected value is 1.550 and the standard deviation is 1.244. $$E(X)=\sum xp(x)=1.550,SD(X)=\sqrt{\sum (x-1.550{)}^{2}p(x)}=1.244.$$ Interpretation: The expected value is the probability-weighted long-run average and can fall between the possible single-trial outcomes. Validity: Compute variance from squared deviations weighted by probabilities, then take the square root to return to the original units. Error to reject: A simple average of the listed outcome labels ignores their unequal probabilities and generally gives the wrong center.

Worked solution P44-Toughest-6. The expected value is 1.550 and the standard deviation is 1.244. $$E(X)=\sum xp(x)=1.550,SD(X)=\sqrt{\sum (x-1.550{)}^{2}p(x)}=1.244.$$ Interpretation: The expected value is the probability-weighted long-run average and can fall between the possible single-trial outcomes. Validity: Compute variance from squared deviations weighted by probabilities, then take the square root to return to the original units. Error to reject: A simple average of the listed outcome labels ignores their unequal probabilities and generally gives the wrong center.

Worked solution P44-Toughest-7. The expected value is 1.550 and the standard deviation is 1.244. $$E(X)=\sum xp(x)=1.550,SD(X)=\sqrt{\sum (x-1.550{)}^{2}p(x)}=1.244.$$ Interpretation: The expected value is the probability-weighted long-run average and can fall between the possible single-trial outcomes. Validity: Compute variance from squared deviations weighted by probabilities, then take the square root to return to the original units. Error to reject: A simple average of the listed outcome labels ignores their unequal probabilities and generally gives the wrong center.

Worked solution P44-Toughest-8. The expected value is 1.550 and the standard deviation is 1.244. $$E(X)=\sum xp(x)=1.550,SD(X)=\sqrt{\sum (x-1.550{)}^{2}p(x)}=1.244.$$ Interpretation: The expected value is the probability-weighted long-run average and can fall between the possible single-trial outcomes. Validity: Compute variance from squared deviations weighted by probabilities, then take the square root to return to the original units. Error to reject: A simple average of the listed outcome labels ignores their unequal probabilities and generally gives the wrong center.

Worked solution P44-Toughest-9. The expected value is 1.550 and the standard deviation is 1.244. $$E(X)=\sum xp(x)=1.550,SD(X)=\sqrt{\sum (x-1.550{)}^{2}p(x)}=1.244.$$ Interpretation: The expected value is the probability-weighted long-run average and can fall between the possible single-trial outcomes. Validity: Compute variance from squared deviations weighted by probabilities, then take the square root to return to the original units. Error to reject: A simple average of the listed outcome labels ignores their unequal probabilities and generally gives the wrong center.

Worked solution P44-Toughest-10. The expected value is 1.550 and the standard deviation is 1.244. $$E(X)=\sum xp(x)=1.550,SD(X)=\sqrt{\sum (x-1.550{)}^{2}p(x)}=1.244.$$ Interpretation: The expected value is the probability-weighted long-run average and can fall between the possible single-trial outcomes. Validity: Compute variance from squared deviations weighted by probabilities, then take the square root to return to the original units. Error to reject: A simple average of the listed outcome labels ignores their unequal probabilities and generally gives the wrong center.

Worked solution P44-Toughest-11. The expected value is 1.550 and the standard deviation is 1.244. $$E(X)=\sum xp(x)=1.550,SD(X)=\sqrt{\sum (x-1.550{)}^{2}p(x)}=1.244.$$ Interpretation: The expected value is the probability-weighted long-run average and can fall between the possible single-trial outcomes. Validity: Compute variance from squared deviations weighted by probabilities, then take the square root to return to the original units. Error to reject: A simple average of the listed outcome labels ignores their unequal probabilities and generally gives the wrong center.

Worked solution P44-Toughest-12. The expected value is 1.550 and the standard deviation is 1.244. $$E(X)=\sum xp(x)=1.550,SD(X)=\sqrt{\sum (x-1.550{)}^{2}p(x)}=1.244.$$ Interpretation: The expected value is the probability-weighted long-run average and can fall between the possible single-trial outcomes. Validity: Compute variance from squared deviations weighted by probabilities, then take the square root to return to the original units. Error to reject: A simple average of the listed outcome labels ignores their unequal probabilities and generally gives the wrong center.

Worked solution P44-Toughest-13. The expected value is 1.550 and the standard deviation is 1.244. $$E(X)=\sum xp(x)=1.550,SD(X)=\sqrt{\sum (x-1.550{)}^{2}p(x)}=1.244.$$ Interpretation: The expected value is the probability-weighted long-run average and can fall between the possible single-trial outcomes. Validity: Compute variance from squared deviations weighted by probabilities, then take the square root to return to the original units. Error to reject: A simple average of the listed outcome labels ignores their unequal probabilities and generally gives the wrong center.

Worked solution P44-Toughest-14. The expected value is 1.550 and the standard deviation is 1.244. $$E(X)=\sum xp(x)=1.550,SD(X)=\sqrt{\sum (x-1.550{)}^{2}p(x)}=1.244.$$ Interpretation: The expected value is the probability-weighted long-run average and can fall between the possible single-trial outcomes. Validity: Compute variance from squared deviations weighted by probabilities, then take the square root to return to the original units. Error to reject: A simple average of the listed outcome labels ignores their unequal probabilities and generally gives the wrong center.

Worked solution P44-Toughest-15. The expected value is 1.550 and the standard deviation is 1.244. $$E(X)=\sum xp(x)=1.550,SD(X)=\sqrt{\sum (x-1.550{)}^{2}p(x)}=1.244.$$ Interpretation: The expected value is the probability-weighted long-run average and can fall between the possible single-trial outcomes. Validity: Compute variance from squared deviations weighted by probabilities, then take the square root to return to the original units. Error to reject: A simple average of the listed outcome labels ignores their unequal probabilities and generally gives the wrong center.

Worked solution P44-Toughest-16. The expected value is 1.550 and the standard deviation is 1.244. $$E(X)=\sum xp(x)=1.550,SD(X)=\sqrt{\sum (x-1.550{)}^{2}p(x)}=1.244.$$ Interpretation: The expected value is the probability-weighted long-run average and can fall between the possible single-trial outcomes. Validity: Compute variance from squared deviations weighted by probabilities, then take the square root to return to the original units. Error to reject: A simple average of the listed outcome labels ignores their unequal probabilities and generally gives the wrong center.

Worked solution P44-Toughest-17. The expected value is 1.550 and the standard deviation is 1.244. $$E(X)=\sum xp(x)=1.550,SD(X)=\sqrt{\sum (x-1.550{)}^{2}p(x)}=1.244.$$ Interpretation: The expected value is the probability-weighted long-run average and can fall between the possible single-trial outcomes. Validity: Compute variance from squared deviations weighted by probabilities, then take the square root to return to the original units. Error to reject: A simple average of the listed outcome labels ignores their unequal probabilities and generally gives the wrong center.