Question

Does breast-feeding weaken bones? Breast-feeding mothers secrete calcium into their milk. Some of the calcium may come from their bones, so mothers may lose bone mineral. Researchers compared a random sample of $47$breast-feeding women with a random sample of $22$women of similar age who were neither pregnant nor lactating. They measured the percent change in the bone mineral content (BMC) of the women's spines over three months. Comparative boxplots and summary statistics for the data from fathroom are shown below

(a) Based on the graph and numerical summaries, write a few sentences comparing the percent changes in BMC for the two groups.

(b) Is the mean change in BMC significantly lower for the mothers who are breast-feeding? Carry out an appropriate test to support your answer.

(c) Can we conclude that breast-feeding causes a mother's bones to weaken? Why or why not?

(d) Construct and interpret a$95\%$confidence interval for the difference in mean bone mineral loss. Explain how this interval provides more information than the significance test in part (b).

Short answer

(a) Because there is a higher mean for nonpregnant and the boxplot is considerably more to the right, the Not Pregnant Centre likely to be greater than the Breastfeeding Centre. Because it has a higher standard deviation and more distance between the boxplot whiskers, the distribution seems to be larger than the distribution for the Breastfeed community Non-pregnant party. As the median of the boxplot farther to the left, both distributions appear to be right-skewed.

(b) Yes, breast-feeding mothers experience a significantly lower mean change in BMC.

(c) No, we can't deduce that breastfeeding weakens a mother's bones.

(d) $\left(-4.850,-2.943\right)$

Step-by-step solution

Part (a) Step 1: Given Information

Given in the question that,

We have to write a few sentences comparing the percent changes in BMC for the two groups.

Part (a) Step 2: Explanation

Because there is a higher mean for nonpregnant and the boxplot is considerably more to the right, the Not Pregnant Centre likely to be greater than the Breastfeeding Centre. Because it has a higher standard deviation and more distance between the boxplot whiskers, the distribution seems to be larger than the distribution for the Breastfeed community Non-pregnant party. As the median of the boxplot farther to the left, both distributions appear to be right-skewed.

Part (b) Step 1: Given Information 

Given in the question that,

$$\begin{gathered} s_1=2.50561 \\ {s}_2=1.29832 \\ {n}_1=47 \\ {n}_2=22 \end{gathered}$$

$$\begin{gathered} {\overline{x}}_1=-3.58723 \\ {\overline{x}}_2=0.309091 \end{gathered}$$

We have to determine is the mean change in BMC significantly lower for the mothers who are breast-feeding.

Part (b) Step 2: Explanation 

The formula of test statistic is

$$\begin{gathered} t=\frac{{\overline{x}}_1-{\overline{x}}_2}{\sqrt{\frac{s_1^2}{n_1}+\frac{s^2}{m_2}}} \\ =\frac{-3.58723-0.300091}{\sqrt{\frac{2.{50561}^2}{47}+\frac{1.298322}{22}}} \\ =-8.499 \end{gathered}$$

The degree of freedom is

$$\begin{gathered} df=\min \left(n_1-1,n_2-1\right) \\ =\min (47-1,22-1) \\ =21 \end{gathered}$$

The P value is

$$\begin{gathered} P<0.0005 \\ P<0.05\Rightarrow reject{H}_0 \end{gathered}$$

There is sufficient evidence to support the claim that the mean change in BMC for breast-feeding women is significantly lower

Part (c) Step 1: Given Information  

We have to conclude that breast-feeding causes a mother's bones to weaken

Part (c) Step 2: Explanation 

An experiment purposely puts specific care on persons in order to examine their reactions. An observational research attempts to obtain details without upsetting the scene they are investigating. Observational Research Because the study is not an experiment, a correlation (that breastfeeding weakens mother's bones) cannot be inferred because the study could be influenced by unknown circumstances.A lurking variable is one that has a significant impact on the relationship between variables in an analysis but is not one of the analysed explanatory factors. If you wish to prove causation, you'll need to do an experiment.

Part (d) Step 1: Given Information  

Given in the question that,