91Ó°ÊÓ

To construct and interpret a scatterplot for the data.

The infant mortality rate (IMR) and life expectancy (LE) for 60 countries are shown in the table below. The given points are shown in the graph below, with IMR on the horizontal axis and LE on the vertical axis.

The scatterplot's plotted points indicate a straight-line pattern.
In addition, as the newborn mortality rate rises, life expectancy decreases.

To decide whether finding a regression line for the data is reasonable. If so, then also do parts (c)-(f).

If the scatterplot has no substantial curvature, it is reasonable to find the regression line for the data.
Because the scatterplot has no substantial curvature, it is reasonable to identify the regression line to the data.

To determine and interpret the regression equation.

The infant mortality rate is used to calculate life expectancy in the supplied data. As a result, life expectancy is the response variable, whereas infant mortality rate is the predictor variable. $n=60$ is the sample size.
The required sums are listed below.
$∑x_i=1742.7$
$∑y_i=4147.3$
$∑x_i^2=90225.6$
$∑y_i^2=293175.6$
$∑x_i{y}_i=106448.3$
Determine $s_{xy}$and $s_{xx}$ as follows:
$s_{xx}=90225.6-\frac{1742.7^2}{60}$

$=39608.8785$

$=-14010.0285$

$=29.045$

$=69.1217$

Determine the parameters as follows:
$b_1=\frac{-14010.0285}{39608.8785}$
$=-0.3537$
$b_0=69.1217-(-0.3537)×29.045$
$=79.3949$
The following is the regression equation for predicting life expectancy $\left(y\right)$given the newborn mortality rate $\left(x\right)$:
$\hat{y}=79.3949-0.3537x.$
According to the regression equation, increasing the infant mortality rate by one year reduces life expectancy by $0.3537$ years on average.

To make the indicated predictions for the life expectancy .

According to the calculation:
$\hat{y}=79.3949-0.3537x.$
The life expectancy when $x=30$ is,
$\hat{y}=79.3949-0.3537×30$
$=68.7839$
As a result, the indicated predictions is $68.7839$ years.

To compare and interpret the correlation coefficient.

The variables are negatively linearly connected if the estimated $r$is close to $-1$.
Determine the linear correlation coefficient as follows:
$r=\frac{∑x_i{y}_i-\left(∑x_i\right)\left(∑y_i\right)/n}{\sqrt{\left[∑x_i^2-{\left(∑x_i\right)}^2/n\right]\left[∑y_i^2-{\left(∑y_i\right)}^2/n\right]}},$
where $n$is the sample size.

According to part (b),
$r=\frac{106448.3-(1742.7)(4147.3)/60}{\sqrt{\left[90225.6-(1742.7{)}^2/60\right]\left[293175.6-(4147.3{)}^2/60\right]}}$
$=-0.8727$
The calculated linear correlation coefficient is close to $-1$, as can be shown. As a result, it may be concluded that the variables have a significant negative linear connection.

To identify the potential outliers and influential observations.

An outlier is a data point that is far off the regression line. An impactful observation is one where the removal of a point causes a significant change in the regression equation. That instance, removing a point creates a significant shift in the regression line's direction. The table below summarizes the anticipated values for the provided data. The given points and the fitted regression line are represented in the graph below.

According to the graph,
Because it is farther from the regression line, the point $(33.9,44.3)$ is an outlier. There is no significant change in the direction of the regression line when a point is removed, hence there are no potentially influencing observations.