6. In 2002, a team of researchers took a random sample of 242 children from Cleveland area birth records for low birth weight children (birth weight less than 2500g) born between 1977 and 1979. The children in the sample were contacted in 2002 to determine if they had graduated from high school. It was found that 74 percent of those 242 children had graduated from high school. A second random sample of 233 children was taken from Cleveland area birth records of normal birth weight children (at least 2500g) born between 1976 and 1979 and those individuals were also contacted in 2002 to determine if they had graduated from high school. It was found that 83 percent of those children had graduated from high school.

b. Estimate the difference between the proportion of the population of low birth weight children and the proportion of the population of normal birth weight children who graduate from high school. Report a standard error for your estimate.

c. Construct a 95 percent confidence interval for the difference between the proportion of the population of low birth weight children and the proportion of the population of normal birth weight children who graduate from high school.

d. In the context of this study of the possible effect of low birth weight on finishing high school, state the conditions that must be checked to ensure the proper use of the confidence interval requested in part (c). 

e. Does the confidence interval you computed in part (c) provide evidence that low birth weight may be a risk factor for not finishing high school? Use your confidence interval to test an appropriate null hypothesis. Clearly state the null and alternative hypotheses, report the the significance level imposed by the confidence interval, and state your conclusion.

f. Suppose your conclusion is incorrect. What type of error did you make (Type I or Type II)? Explain.

g. Test the null hypothesis that the proportion of low birth weight children who finish high school is the same as the proportion of normal birth weight children who finish high school against the alternative that the proportion of low birth weight children who finish high school is less than the proportion of normal birth weight children who finish high school. Use a 2-proportion z-test and a significance level (Type I error level) of 0.05. Report the value of your test statistic, the p-value, and state your conclusion in the context of the study.

h. Describe how the standard error you used to construct the test statistic in part g is different from the standard error you used to construct the confidence interval in part c.