Chapter 20, Problem 20.5AYK

(a)

To determine

To define the individuals studied, the type of study and the explanatory and response variables and also find whether the variables are quantitative or categorical.

Answer to Problem 20.5AYK

The individuals studied are infants with severe respiratory distress, the type of the study is randomized experiment, the explanatory variable is treatment provided to infants and the response variables is the infant survival and also the variables are quantitative.

Explanation of Solution

In the question, it is given that there are safe and effective interventions to treat respiratory failure in new-borns but they are typically costly. Researchers compared the survival of infants with severe respiratory distress who were treated at random either with the new bCPAP machine or nasal oxygen therapy, serving as a control. The published findings report that survival rate for neonates receiving bCPAP was $71\%(\frac{44}{62})$ compared with $44\%(\frac{11}{25})$ for controls. Thus, we can see that the individuals studied are infants with severe respiratory distress, the type of the study is randomized experiment. The explanatory variable is treatment provided to infants and the response variables is the infant survival. Also, we find out that the variables are quantitative.

(b)

To determine

To state the corresponding null and alternative hypotheses using mathematical symbols.

Answer to Problem 20.5AYK

  $\begin{array}{l}{H}_0:p_{bCPAP}=p_{control}\\ H_a:p_{bCPAP}>p_{control}\end{array}$

Explanation of Solution

In the question, it is given that there are safe and effective interventions to treat respiratory failure in new-borns but they are typically costly. Researchers compared the survival of infants with severe respiratory distress who were treated at random either with the new bCPAP machine or nasal oxygen therapy, serving as a control. The published findings report that survival rate for neonates receiving bCPAP was $71\%(\frac{44}{62})$ compared with $44\%(\frac{11}{25})$ for controls. The researchers tested the hypothesis that the rate of survival was higher for babies receiving bCPAP than in control group. So, let the proportions for the bCPAP and the control group be ${\widehat{p}}_{bCPAP}\text{ and }{\widehat{p}}_{control}$ respectively. Thus, the null and alternative hypotheses using mathematical symbols can be defined as:

  $\begin{array}{l}{H}_0:p_{bCPAP}=p_{control}\\ H_a:p_{bCPAP}>p_{control}\end{array}$

(c)

To determine

To check whether the conditions are met.

Answer to Problem 20.5AYK

Yes, they are met.

Explanation of Solution

In the question, it is given that there are safe and effective interventions to treat respiratory failure in new-borns but they are typically costly. Researchers compared the survival of infants with severe respiratory distress who were treated at random either with the new bCPAP machine or nasal oxygen therapy, serving as a control. The published findings report that survival rate for neonates receiving bCPAP was $71\%(\frac{44}{62})$ compared with $44\%(\frac{11}{25})$ for controls. The researchers tested the hypothesis that the rate of survival was higher for babies receiving bCPAP than in control group. So, let the proportions for the bCPAP and the control group be ${\widehat{p}}_{bCPAP}\text{ and }{\widehat{p}}_{control}$ respectively. Thus, the null and alternative hypotheses using mathematical symbols can be defined as:

  $\begin{array}{l}{H}_0:p_{bCPAP}=p_{control}\\ H_a:p_{bCPAP}>p_{control}\end{array}$

Thus, the conditions are met because the data are reasonably random and representative of the corresponding populations. Also, the counts of successes and failures in both the samples are large enough.

(d)

To determine

To obtain the test statistics and P-value and then conclude in context.

Answer to Problem 20.5AYK

  $\begin{array}{l}z=2.36\\ P=0.0091\end{array}$

So, we have sufficient evidence to conclude that the rate of survival is higher for babies receiving bCPAP than in control group.

Explanation of Solution

In the question, it is given that there are safe and effective interventions to treat respiratory failure in new-borns but they are typically costly. Researchers compared the survival of infants with severe respiratory distress who were treated at random either with the new bCPAP machine or nasal oxygen therapy, serving as a control. The published findings report that survival rate for neonates receiving bCPAP was $71\%(\frac{44}{62})$ compared with $44\%(\frac{11}{25})$ for controls. The researchers tested the hypothesis that the rate of survival was higher for babies receiving bCPAP than in control group. So, let the proportions for the bCPAP and the control group be ${\widehat{p}}_{bCPAP}\text{ and }{\widehat{p}}_{control}$ respectively. Thus, the null and alternative hypotheses using mathematical symbols can be defined as:

  $\begin{array}{l}{H}_0:p_{bCPAP}=p_{control}\\ H_a:p_{bCPAP}>p_{control}\end{array}$

Thus, to do the mechanics of a hypothesis test for equality of population proportions we will use the calculator $\text{TI}-89$ . Then select $6:2-$ PropZTest from the STAT Tests menu. Enter the observed counts and sample sizes. Indicate what kind of test you want: one-tail upper tail, lower tail, or two-tail. And specify whether results should simply be calculated or displayed with the area corresponding to the P-value of the test shaded. Thus, we know that,

  $\begin{array}{l}{\widehat{p}}_{bCPAP}=\frac{x_1}{n_1}\\ =\frac{44}{62}\\ =0.71\\ {\widehat{p}}_{control}=\frac{x_2}{n_2}\\ =\frac{11}{25}\\ =0.44\end{array}$

Thus, by using the calculator $\text{TI}-89$ , the test statistics and the P-value is as:

  $\begin{array}{l}z=2.36\\ P=0.0091\end{array}$

As we know that if the P-value is less than or equal to the significance level then the null hypothesis is rejected, so we have,

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

Thus, we have sufficient evidence to conclude that the rate of survival is higher for babies receiving bCPAP than in control group.