Chapter 25, Problem 25.16CRE

(a)

To determine

To explain is the graduation rate among the VLBW group significantly lower than for the normal birth weight controls.

Answer to Problem 25.16CRE

Yes, the graduation rate among the VLBW group is significantly lower than for the normal birth weight controls.

Explanation of Solution

In the question, it is given that a long-term study followed very low birth weight babies (VBLW) to age twenty along with a control group of normal birth weight babies from the same population. Thus, $179$ of the VLBW group and $193$ from the control group had graduated from high school. Thus, the proportion for the same is calculated as follows:

  $\begin{array}{l}{\widehat{p}}_1=\frac{x_1}{n_1}\\ =\frac{179}{242}\\ =0.74\\ {\widehat{p}}_2=\frac{x_2}{n_2}\\ =\frac{193}{233}\\ =0.83\end{array}$

Thus, to do the mechanics of a hypothesis test for equality of population proportions we will use the calculator $\text{TI}-89$ . Then select $6:2-$ PropZTestfrom 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, the null and alternative hypotheses is defined as:

  $\begin{array}{l}{H}_0:p_V=p_C\\ H_a:p_V<p_C\end{array}$

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

  $\begin{array}{l}z=-2.38\\ P=0.0086\end{array}$

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 graduation rate among the VLBW group is significantly lower than for the normal birth weight controls.

(b)

To determine

To explain is there good evidence that mean IQ is lower among VLBW men than among controls from similar background.

Answer to Problem 25.16CRE

Yes, there is good evidence that mean IQ is lower among VLBW men than among controls from similar background.

Explanation of Solution

In the question, it is given that a long-term study followed very low birth weight babies (VBLW) to age twenty along with a control group of normal birth weight babies from the same population. Thus, we have,

	VLBW	Control
Mean	87.6	94.7
S.D.	15.1	14.9
n	113	106

Thus, for testing the hypothesis we will use the calculator $\text{TI}-89$ as, first we will go in the STAT TESTS menu, choose $4:2-$ SampTTest. Then you must specify if you are using data stored in two lists or if you will enter the means, standard deviations, and sizes of both samples. You must also indicate whether to pool the variances (when in doubt, say no) and specify whether the test is to be two-tail, lower-tail, or upper-tail. So, the null and alternative hypotheses is defined as:

  $\begin{array}{l}{H}_0:{\mu }_V={\mu }_C\\ H_a:{\mu }_V<{\mu }_C\end{array}$

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

  $\begin{array}{l}t=-3.50\\ P=0.0003\end{array}$

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 thatmean IQ is lower among VLBW men than among controls from similar background.

(c)

To determine

To compute a $95\%$ confidence interval for the difference in mean IQ between VLBW women and normal birth weight women from similar background and what can you conclude from this interval.

Answer to Problem 25.16CRE

We are $95\%$ confident that for the difference in mean IQ between VLBW women and normal birth weight women from similar background is between $(-7.0274,-0.1726)$ .

Explanation of Solution

In the question, it is given that a long-term study followed very low birth weight babies (VBLW) to age twenty along with a control group of normal birth weight babies from the same population. Thus, we have,

	VLBD women	Control
Mean	86.2	89.8
S.D.	13.4	14
n	123	125

Thus, to find the confidence interval for mean we will use the calculator $\text{TI}-89$ . In the STAT Intsmenu, choose $4:2-$ SampTInt. You must specify if you are using data stored in two lists or if you will enter the means, standard deviations, and sizes of both samples. You must also indicate whether to pool the variances (when in doubt, say no) and specify the desired level of confidence. Thus, the confidence interval will be:

  $(-7.0274,-0.1726)$ .

Thus, we can conclude form this interval that we are $95\%$ confident that for the difference in mean IQ between VLBW women and normal birth weight women from similar background is between $(-7.0274,-0.1726)$ .

(d)

To determine

To explain can we conclude from this study that being born with a very low weight causes lower IQ and lower graduation rates by age twenty.

Answer to Problem 25.16CRE

Yes, we can conclude from this study that being born with a very low weight causes lower IQ and lower graduation rates by age twenty.

Explanation of Solution

In the question, it is given that a long-term study followed very low birth weight babies (VBLW) to age twenty along with a control group of normal birth weight babies from the same population. From part (a), we have sufficient evidence to conclude that the graduation rate among the VLBW group is significantly lower than for the normal birth weight controls. And from part (b), we have sufficient evidence to conclude that mean IQ is lower among VLBW men than among controls from similar background. And even in part (c), the confidence interval shows the lower IQ scores. Thus, we can conclude from this study that being born with a very low weight causes lower IQ and lower graduation rates by age twenty