2. Correlation is not causation. Perhaps the factors that lead to smoking, drinking, and so on also lead to body piercing.
3. Because the sample is with replacement, the size of the population doesn’t matter. Under the null hypothesis that p1 = p2, the difference in sample proportions is approximately normally distributed with a mean of 0 and a standard deviation equal to
The probability that the difference between the number of women in the two samples would be less than or equal to 1 is
The probability that Z will be between -0.0598 and 0.0589 is 0.047.
4. One minus the probability of no fours is 1 - (45/47)(44/46)(43/45) = 0.125.
5. The binomial model gives us the probability of exactly 5 heads and 5 tails:
6. If each student has a 0.2461 probability of guessing 5 heads and 5 tails (H0: p = 0.2461), then the probability that 27 or more of 56 students would guess 5 heads and 5 tails is given by the binomial distribution:
7. If you aim for the upper box first, there is a 0.5 probability of landing in that box and a 0.5 probability of landing in the lower box. If the ball lands in the upper box, you will toss your second ball into the lower box. If the first ball lands in the lower box, you will try for the upper box again.
Your probability of winning is 0.5 + 0.5(0.5) = 0.75. If, instead, you aim for the lower box first, your ball will land in it with certainty. You will then aim for the upper box and have a 0.5 probability of winning. So, it is better to aim for the upper box first.
8. This is a sample mean and it is consequently normally distributed with a mean equal to the actual time and a standard deviation equal to 10 seconds divided by the square root of 20, which is 2.24 seconds
9. There are a total of 7936 observations and, under the null hypothesis, the expected value for each of the 10 digits is 7936/10 = 793.6. The chi-square statistic is
With 10 - 1 = 9 degrees of freedom, the p value is less than 0.00000.
10. A normal approximation is
The two-sided p value is 2(0.318) = 0.636. The exact p value is given by the binomial distribution:
The two-sided p value is 2(0.326) = 0.652.
11. Using Bayes’s Rule,
12. Whether the observed differences are substantial would be gauged by subjectively judging the difference between the observed and expected values. Since the p value is not below 0.05, the results are NOT statistically significant at the 5% level.
13. This is true if the average male and female students were the same age and had fathers of the same height and mothers of the same height.
14. The probabilities are
|
number of
dice
|
probability
|
|
0
|
(5/6)(5/6)(5/6)
= 0.5787
|
|
1
|
3(1/6)(5/6)(5/6)
= 0.3472
|
|
2
|
3(1/6)(1/6)(5/6)
= 0.0694
|
|
3
|
(1/6)(1/6)(1/6)
= 0.0046
|
|
1.0000
|
The expected value of the payoff is about an 8 percent loss per dollar wagered:
0(0.5787) + 2(0.3472) + 3(0.0694) + 4(0.0046) = 0.921
15. The performance of a professional football team depends on its ability and also on chance–unpredictable variations in the opponent’s play calling, the players’ health, the officiating, and even the proverbial bounces of the football. When such chance events make observed performance an imperfect measure of ability, observed differences in performance tend to exaggerate underlying differences in abilities. Performances consequently regress toward the mean, in that those teams that perform the best at any point in time typically do not perform as well subsequently, and those teams that perform the worst at any point in time typically do not perform as poorly subsequently. Marcus recommended betting for teams that had been doing poorly and against teams that had been doing well, and it worked!
16. A simple regression of the length gain on birth weight is evidently negative because heavier babies also tend to be taller; the relatively small positive effect of birth weight on length gain is swamped by the negative effect of birth length on length gain. In fact, birth length and birth weight are positively correlated (the correlation coefficient is 0.57).
17. It is not a flaw since the test statistic takes into account the (possibly unequal) sample sizes.
18. The 0.5721 p value means that, if the population mean grades were equal for males and females, there is a 0.5721 probability that the difference in the sample mean grades would as large as actually observed: 7.500 - 7.125 = 0.375.
19. 75% of the variation in the prices is explained by the number of pages.
20. There is self-selection bias here; the people who play Bingo are much older than those who play Mortal Kombat.
