Note: for many of these questions you need to perform a hypothesis test.  Be sure to state clearly each step, including giving the significance level, stating
the null and alternative hypotheses, giving the test statistic, calculating the observed value, etc.

1.  Download the Lead data.  These data were taken from 33 pairs of children.  One set of children had parents whose worked exposed them to lead.  Lead can accumulate in the bloodstream and cause serious medical and developmental problems for children.  The researchers were concerned that parents who work around lead inadvertantly bring lead dust home and expose their children to health risks. For each of these 33 children, the researchers found a "matched" child of the same age and in the same neighborhood whose parents did NOT work around lead.  The lead levels in the bloodstreams of all children were measured.  The data set you are downloading has two columns. The first contains the lead levels of the children whose parents work around lead.  The second column contains the lead levels of the "controls."  Our goal is to determine if the exposed children have a significantly higher lead level than the controls, or if any observed difference might just be due to chance.

a. Consider the control group a random sample of "healthy" children.  Find a 95% confidence interval for the mean  lead level in the bloodstream of healthy children.
b. Choose any graphical summary you like to compare the lead levels of the two groups.  What do you think about the differences between the groups?
c.  Most doctors agree that children with lead levels above 40 micrograms need medical treatment.  Estimate the probability that a child will need medical treatment if they have a parent whose work exposes them to lead.  Estimate the same probability for the control children.
d.  How many of the 33 exposed children are higher than the highest control child?  Do you find this an alarmingly high number?
e. Find the difference in lead levels for each pair of exposed/control children.  In how many pairs is the exposed child higher than the control child?  Do you find this number alarmingly high?
f. Assuming these 33 pairs are representative of a larger population, estimate the mean difference in blood lead levels.
g.  Medical experts believe those who work around lead expose their children to dangerous levels of lead.  Others say that they don't.  Perform a hypothesis test to test this on these data.  State the p-value of your observed test statistic.
 

2.  Consider the Class Data.  Is there evidence that women watch a different amount of TV then men?  That their drinking behavior is different?  In both cases, give the p-value.  Would you generalize these results to all college age men and women?
 

3. Download the blood pressure data.
a) Is there evidence that the blood pressure changed?  Give the p-value.  If so, would you say the drug caused this?  Explain.
b) Is there evidence that the blood pressure has lowered?  Give the p-value.

4. A researcher is testing his hypothesis that children who experienced a certain pre-natal condition are shorter at age 3 than children who did not.  He assumes in his test that children's heights are normally distributed with mean 2 inches.   He has a random sample of 12 children who experienced this condition.  His decision rule is that he will reject the null hypothesis if his test statistic is smaller than -1.96.
a) What test statistic is he using?
b) What significance level is he using?

5.  Suppose that a Coke bottling machine puts X ounces of Coke in a bottle, where X is normally distributed with mean 12 oz. and SD 0.1 oz.  To test whether the machine is working, we take a random sample of n bottles and perform a hypothesis test: H0: mu = 12  vs. H1: mu <> 12.
a) Why is it incorrect to state that the null hypothesis is "Xbar = 12"?
b) What critical value will we use if our significance level is alpha = 0.05?
c)  Suppose n=9.  Also, suppose the machine really *is* off and is actually dispensing a mean amount of 12.05 ounces of Coke.  What's the probability that our test will detect this?  (This probability is the "Power"  of the test to detect an "effect" of size .05 ounces.)
d) What happens to the this probability if we increase n?  Explain.
e) What happens to this probability if the "effect" size gets bigger?  Smaller?  Explain.