1. When working properly, a cereal packaging machine is supposed to place 16 ounces of cereal in each box. A random sample of 10 boxes gave the following weights: 15.3, 16.1, 16.3, 15.2, 15.6, 16.4, 15.9, 15.0, 14.8, 15.1. Assuming the data are from a normal distribution with mean $\mu$, use R to test the null hypothesis $H_0 : \mu = 16$, against the alternative hypotheses (a) $H_1 : \mu < 16$ and (b) $H_1 : \mu \ne 16$.
2. Female killdeers usually lay four eggs each spring. It is suspected that the average weight of the first-hatched birds will exceed the average weight of the last-hatched birds. Suppose that in eight broods, the following differences (first-hatched minus last-hatched) in weights are observed: 0.02, -0.10, 0.06, 0.23, 0.14, 0.14, -0.04, 0.29. Assume the data are from a normal distribution with mean $\mu$. (a) State an appropriate null hypothesis. (b) What alternative hypothesis is of interest to the investigators? (c) Use R to test the hypotheses.
3. R. A. Fisher investigated the results of testing two sleeping drugs on ten patients.The additional hours of sleep obtained from drug A instead of drug B were reported to be: 1.2, 2.4, 1.3, 1.3, 0.0, 1.0, 1.8, 0.8, 4.6, 1.3. Assume the data are from a normal distribution with mean $\mu$. Using R, test the hypothesis that the mean number of additional hours of sleep is 0. Should the alternative hypothesis be one-sided or two-sided?
4. To determine the effect, if any, of red light on mosquitoes, 958 mosquitoes were placed in the center of a metal tube which had a red light source at one end and a white light source at the other. After 2.5 minutes, 642 mosquitoes were found in the red end of the tube. If $p$ is the true probability that a mosquito will go toward the red light, test the null hypothesis $H_0 : p = \frac{1}{2}$ (use both tests for proportions in R). Should the alternative hypothesis be one-sided or two-sided? Is anything being assumed about the independence of the behavior of the mosquitoes?
5. There is a theory that the anticipation of a birthday can prolong a person's life. In a study set up to examine that notion statistically, it was found that only 60 of 747 people whose obituaries were published in Salt Lake City in 1975 died in the three-month period preceding their birthday. Let $p$ be the probability that a person dies in the three month period prior to her or his birthday. (a) What should $p$ be if the birthdays have no affect on prolonging a person's life? (b) Set up appropriate null and alternative hypotheses. Should the alternative hypothesis be one-sided or two-sided? (c) Test the hypotheses using both tests of proportions in R.

Answers:

1. (a) Test statistic: t = -2.3707; p-value = 0.02093 (b) Same test statistic; p-value = 0.04186
2. (a) $H_0 : \mu = 0$ (b) $H_1 : \mu > 0$ (c) Test statistic: t = 1.9709, p-value == 0.04469
3. Test statistic: t = 4.0298, p-value = 0.002974, with a two-sided alternative
4. With a two-sided alternative, the p-value is less than $2.2 \times 10^{-16}$ for both tests. We are assuming that the mosquitoes are acting independently of one another.
5. (a) $p = \frac{1}{4}$ (b) $H_0 : p = \frac{1}{4}$, $H_1 : p < \frac{1}{4}$ (c) p-value is less than $2.2 \times 10^{-16}$ for both tests