Problem 1
At rush-hour the number of cars passing point P in a ten-second period has a Poisson distribution with mean 3; at other (off-peak) periods the number of cars in a ten-second period has a Poisson distribution with mean 1. We have just received a data set with four observations: 5, 4, 3, 3.
| a) | Test the hypotheses H0: Data are from Off-peak vs. the alternative Ha: Data are from rush-hour. Find the significance level (P-value). |
| b) | If initially you thought the data were equally likely to be from rush-hour or off-peak hours, i.e. though the mean lambda was either 1.0 or 3.0 with probability 0.5 each, what would be the conditional probability that lambda is 1.0 after observing the four data observations above? |
Problem 2
Fitness scores for two manufacturing machines are recorded periodically; the ten scores for machine A averaged 78.0, with a sample standard deviation of 8.0, while the six scores for machine B averaged 85.0, with a sample standard deviation of 10.0. You may assume that fitness scores are approximately normally distributed. We are interested in whether or not the two machines have identical fitness.
| a) | To test for equality of fitness, should you use matched pairs or independent samples? Why? |
| b) | Should you use small sample or large sample methods? Why? |
| c) | Formulate and test an hypothesis to answer the question posed above. Be sure to: 1) state your hypothesis and its alternative; 2) give your test statistic and its distribution under the null hypothesis; 3) report the observed significance level (P-value); and 4) suggest whether or not the evidence support rejecting the hypothesis. |
| d) | If we were to learn that fitness scores always have (population) variance 100.0, which of your answers above would change? What would the P-value be then? (Do NOT use this hypothetical discovery in solving parts a)-c) above) |
Problem 9.14, 15, page 437-38
Problem 9.18a, page 443
Problem 9.28, page 449
Problem 9.34, page 456
Problem 9.35, page 456
Problem 9.72, page 485 (matched or indep samples? Why?)
Problem 9.74, page 486 (matched or indep samples? Why?).