Chapter 10, Problem AP3.32CPT

(a)

To determine

To find out that is this an experiment or an observational study and explain why.

Answer to Problem AP3.32CPT

Observational study.

Explanation of Solution

In this question, the researchers have studied about the difference in the normal weight month babies with the very low birth weight (VLBW) babies.

Thus, the experimenters do not have control on whether or not a baby will survive with no major handicaps. So, this is an observational study. As, the difference between an observational study and experimental design is whether or not the researchers have control over the experiment. Therefore this is an observational study since the experiments cannot control the factors that may lead to a handicapped person.

(b)

To determine

To explain do these data provide convincing evidence at the $\alpha =0.05$ significance level that the graduation rate among VLBW babies is less than for a normal month weight babies.

Answer to Problem AP3.32CPT

There is a convincing evidence that the graduation rate among VLBW babies is less than for the graduation rate for normal birth weight babies.

Explanation of Solution

It is given in the question that:

  $\begin{array}{l}{n}_1=242\\ n_2=233\\ x_1=179\\ x_2=193\\ \alpha =0.05\end{array}$

Now, the claim that is given is: Proportion is less for VLBW babies.

The claim is either null hypothesis or alternative hypothesis. Thus, we will define the hypotheses for this case as:

  $\begin{array}{l}{H}_0:p_1=p_2\\ H_a:p_1<p_2\end{array}$

Where,

  $p_1$ is the graduation rate among VLBW babies and

  $p_2$ is the graduation rate among normal birth weight babies.

Now, we will conduct the hypotheses testing as follows:

The sample proportion will be:

  $\begin{array}{l}{\widehat{p}}_p=\frac{x_1+x_2}{n_1+n_2}\\ =\frac{179+193}{242+233}\\ =\frac{372}{475}\\ =0.7832\end{array}$

Thus, the value of test statistics will be then,

  $\begin{array}{l}z=\frac{{\widehat{p}}_1-{\widehat{p}}_2-(p_1-p_2)}{\sqrt{{\widehat{p}}_p(1-{\widehat{p}}_p)}\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}}\\ =\frac{0.7397-0.8283-0}{\sqrt{0.7832(1-0.7832)}\sqrt{\frac{1}{242}+\frac{1}{233}}}\\ =-2.34\end{array}$

The P-value is the probability of obtaining the value of test statistic or a value more extreme assuming the null hypothesis is true.

  $\begin{array}{l}P=P(Z<-2.34)\\ =0.0096\end{array}$

As we know that if the P-value is less than the significance level then reject the null hypothesis. So,

  $P<0.05\Rightarrow \text{Reject }{H}_0$

Thus, we conclude that there is a convincing evidence that the graduation rate among VLBW babies is less than for the graduation rate for normal birth weight babies.