Starting in the $1970s$, medical technology allowed 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 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.

(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. Is the graduation rate among the VLBW group significantly lower than for the normal-birth-weight controls? Give appropriate statistical evidence to justify your answer.

We are given that a long 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 and

we have to find out whether it is observational study or not.

The difference between an observational study and experimental study is that whether the researchers have control over the experiment or not.

But as the experiments do not have a control whether baby will survive with no major handicaps or not.

Hence, it is an observational study.

We are given that at age $20$,$179$of the VLBW group and $193$of the control group had graduated from high school.

We have to find out that the graduation rate among the VLBW group significantly lower than for the normal-birth-weight controls.

Now, Find out the null and alternative hypothesis,

$H_0:P_1=P_{2}and{H}_a:P_1<P_2$

Now as we have a large sample sizes so we will use central limit theorem

We get, $Z=\frac{P_1-P_2}{\sqrt{P(1-P)({\displaystyle \frac{1}{n_1}}+{\displaystyle \frac{1}{n_2}})}}$

where $P=\frac{X_1+X_2}{n_1+n_2}$,$P_1=\frac{X_1}{n_1}$and $P_2=\frac{X_2}{n_2}$

here $X_1=179,n_1=242,X_2=193and{n}_2=233$

Put these values in the respective equations which gives,

$P=0.78316,P_1=0.73967and{P}_2=0.82833$.

Now, calculating Z

we get, $Z=-2.344$

For the critical region as we are not given $\alpha$so assume it as $0.05$

which gives, $Z_{\alpha }=-1.645$.

As the test static falls beyond the critical value, we will neglect the null hypothesis.

Hence, there is sufficient evidence that the graduation rate among the VLBW group significantly lower than for the normal-birth-weight controls.