Chapter 9, Problem 31E

(a)

To determine

To create a scatterplot relating these two variables and describe the association.

Explanation of Solution

The data from the World Bank for the western hemisphere countries is used to examine the association between female life expectancy and the average number of children women gave birth to. The data is as given in the question. The scatterplot for the female life expectancy versus the number of children women give birth to is as:

  

The births per women is on the horizontal axis and the life expectancy is on the vertical axis. By looking at the scatterplot, we find out that expect for the outlier for the country Costa Rica, the remaining data appears to have a linear form in a negative direction.

(b)

To determine

To explain are there any countries that do not seem to fit the overall pattern.

Answer to Problem 31E

The country Costa Rica do not seem to fit the overall pattern.

Explanation of Solution

The data from the World Bank for the western hemisphere countries is used to examine the association between female life expectancy and the average number of children women gave birth to. We have to find out are there any countries that do not seem to fit the overall pattern. For this by looking at the scatterplot we found that in the right hand corner there is an outlier that does not fit the rest of the data. Thus, the country Costa Rica, whose data appear to be wrong with $25$ births per women is not fit to the data because it is wrong data.

(c)

To determine

To find the correlation and interpret the value of $R^2$ .

Answer to Problem 31E

The correlation of the data is $0.1682$ and $R^2=2.8\%$ .

Explanation of Solution

The data from the World Bank for the western hemisphere countries is used to examine the association between female life expectancy and the average number of children women gave birth to.To find the value of correlation, we will use the excel. Now,

The data given in the question is:

Births per Women	Life expectancy
2.3	74.6
2.3	70.5
1.7	75.4
3	71.9
3.7	64.5
2.3	70.9
1.5	79.8
2	78
2.4	72.6
24.9	78.7
2.8	67.8
2.7	74.5
2.8	71.1
4.4	67.6
3.6	68.2
2.4	70.8
2.2	75.1
3.2	70.1
2.6	75.1
3.7	71.2
2.8	70.4
1.9	77.5
2	77.4
2.1	75.2
2.7	73.7
2.2	78.6

Formula used:

The CORREL function returns the correlation coefficient of two cell ranges. And the syntax is:

CORREL(array$1$, array$2$)

Calculation:

The function is used for the data and the result will be:

Correlation

=CORREL(AD81:AD106,AE81:AE106)

Then by this formula we will get the result as:

Correlation

0.1682

Thus the correlation of the data is $0.1682$ .

Now, the value of $R^2$ can be calculated as:

  $\begin{array}{l}{R}^2=r^2\\ \Rightarrow R^2={0.1682}^2\\ \Rightarrow R^2=2.8\%\end{array}$

Thus, we have that the value of $R^2$ indicates that $2.8\%$ of the variation in life expectancy is explained by the variation in births per women. But without Costa Rica, we have that:

The function is used for the data and the result will be:

Correlation

=CORREL(AD81:AD105,AE81:AE105)

Then by this formula we will get the result as:

Correlation

-0.796

Thus the correlation of the data is $-0.796$ .

Now, the value of $R^2$ can be calculated as:

  $\begin{array}{l}{R}^2=r^2\\ \Rightarrow R^2={(}^{-}\\ \Rightarrow R^2=63.3\%\end{array}$

Thus, without Costa Rica, the value of $R^2$ indicates that $63.3\%$ of the variation in life expectancy is explained by the variation in births per women.

(d)

To determine

To find the equation of the regression line.

Answer to Problem 31E

   .

Explanation of Solution

The data from the World Bank for the western hemisphere countries is used to examine the association between female life expectancy and the average number of children women gave birth to. Thus, by the scatterplot we have the regression line for the data of life expectancy and the births per women. Therefore, the scatterplot by excel we have:

  

Here the regression line is mentioned in the plot, which is:

   .

(e)

To determine

To explain is the line an appropriate model or not and describe what you see in the residual plot.

Answer to Problem 31E

No, the model is not appropriate.

Explanation of Solution

The data from the World Bank for the western hemisphere countries is used to examine the association between female life expectancy and the average number of children women gave birth to. Thus, by the scatterplot we have the regression line for the data of life expectancy and the births per women. The model with Costa Rica is not appropriate as the residuals plot shows a distinct outlier, which is Costa Rica. When we remove this outlier Costa Rica it will give a better residual plot, suggesting that the linear equation is more appropriate.

(f)

To determine

To interpret the slope and y -intercept of the line.

Explanation of Solution

The data from the World Bank for the western hemisphere countries is used to examine the association between female life expectancy and the average number of children women gave birth to. Now, with the Costa Rica, the slope is near zero suggesting that the linear model is not very useful. The y -intercept suggest that with no births, the life expectancy is about $72.6$ years and without Costa Rica, the slope is negative, indicating that an average increase of one child per woman predicts a lower expectancy of $4.44$ years on average. The y -intercept indicates that a country without birth rate of zero would have a life expectancy of $84.5$ years and this extrapolation.

(g)

To determine

To explain if the government leaders wanted to increase life expectancy in their country than should they encourage women to have fewer children.

Answer to Problem 31E

No, they should not.

Explanation of Solution

The data from the World Bank for the western hemisphere countries is used to examine the association between female life expectancy and the average number of children women gave birth to. Thus, if the government leaders wanted to increase life expectancy in their country than they should not encourage women to have fewer children because while here is an association but there is no reason to expect causality and also in this lurking variables can be involved.