Starting in the $1970s$, medical technology has enabled babies with very low birth weight (VLBW, less than$1500$grams, or about 3.3 pounds) to survive without major handicaps. It was noticed that these children nonetheless had difficulties in school and as adults. A long-term study has followed 242 randomly selected VLBW babies to age $20$years, along with a control group of 233 randomly selected babies from the same population who had normal birth weight. $50$

a. Is this an experiment or an observational study? Why?

b. At age$20,179$ of the VLBW group and 193 of the control group had graduated from high school. Do these data provide convincing evidence at the α=0.05 significance level that the graduation rate among VLBW babies is less than for normal-birth-weight babies?

We have to tell is this an experiment or an observational study.

The experimenters have no way of knowing whether a baby would survive without major disabilities. This is an observational research project.

An observational study differs from an experimental design in that the researchers have control over the experiment.

Because the experimenters cannot control the factors that may lead to a handicapped person, they must conduct an observational study.

We have to tell the significance level that the graduation rate among VLBW babies is less than for normal-birth-weight babies.

Random, Independent (10 percent condition), and Normal are the three conditions for generating a hypothesis test for the population proportion pp (large counts).

Because the babies were chosen at random from different populations, I'm satisfied.

Independent: Content because the VLBW newborns account for less than 10% of all VLBW babies, and the normal-birth-weight babies account for less than 10% of all normal-birth-weight kids (assuming that there are more than VLBW babies and assuming that there are more than normal-birth-weight babies)

The static value:

$z=\frac{{\hat{p}}_1-{\hat{p}}_2-\left(p_1-p_2\right)}{\sqrt{{\hat{p}}_p\left(1-{\hat{p}}_p\right)}\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}}=\frac{0.7397-0.8283}{\sqrt{0.7832(1-0.7832)}\sqrt{\frac{1}{242}+\frac{1}{233}}}\approx -2.34$

$P=P(Z<-2.34)=0.0096$