# Use the technology of your choice to do the following tasks. From the International Data Base, published by the U.S. Census Bureau, we obtained data on infant mortality rate (IMR) and life expectancy (LE), in years, for a sample of 60 countries. a) Construct and interpret a scatterplot for the data. b) Decide whether finding a regression line for the data is reasonable. If so, then also do parts (c)-(f). c) Determine and interpret the regression equation. d) Make the indicated predictions. e) Compute and interpret the correlation coefficient. f) Identify potential outliers and influential observations.

Question
Scatterplots
Use the technology of your choice to do the following tasks. From the International Data Base, published by the U.S. Census Bureau, we obtained data on infant mortality rate (IMR) and life expectancy (LE), in years, for a sample of 60 countries. a) Construct and interpret a scatterplot for the data. b) Decide whether finding a regression line for the data is reasonable. If so, then also do parts (c)-(f). c) Determine and interpret the regression equation. d) Make the indicated predictions. e) Compute and interpret the correlation coefficient. f) Identify potential outliers and influential observations.

2021-02-17

Given: $$\displaystyle{n}={S}{a}\mp\le\ {s}{i}{z}{e}={60}$$ a) IMR is on the horizontal axis and LE is on the vertical axis. b) When there is no strong curvature presents in the scatterplot, then it is safe to assume that there is a linear relationship between the variables and thus it is then reasonable to find a regression line. We note that the scatterplot of part (a) does not contain strong curvature and thus is reasonable to find the regression line. c) We determine all necessary sums: $$\displaystyle\sum\ {x}_{{{i}}}={1743.1}$$
$$\displaystyle\sum\ {y}_{{{i}}}={4147.6}$$
$$\displaystyle\sum\ {x}_{{{i}}}\ {y}_{{{i}}}={106485.62}$$
$$\displaystyle\sum\ {{x}_{{{i}}}^{{{2}}}}={90242.13}$$
$$\displaystyle\sum\ {{y}_{{{i}}}^{{{2}}}}={293216.68}$$ Next, we can determine $$\displaystyle{S}_{{xx}}\ {\quad\text{and}\quad}\ {S}_{{{x}{y}}}$$
$$\displaystyle{S}_{{xx}}=\ \sum\ {{x}_{{{i}}}^{{{2}}}}={90242.13}\ -\ {\frac{{{1743.1}^{{{2}}}}}{{{60}}}}={39602.1698}$$
$$\displaystyle{S}_{{{x}{y}}}=\ \sum\ {x}_{{{i}}}\ {y}_{{{i}}}\ -\ {\frac{{{\left(\sum\ {x}_{{{i}}}\right)}\ {\left(\sum\ {y}_{{{i}}}\right)}}}{{{n}}}}={106485.62}\ -\ {\frac{{{1743.1}\ \cdot\ {4147.6}}}{{{60}}}}=\ -{14009.0727}$$ The estimate b of the slope $$\displaystyle\beta\ \text{is the ratio of}\ {S}_{{{x}{y}}}\ {\quad\text{and}\quad}\ {S}_{{xx}}:$$
$$\displaystyle{b}=\ {\frac{{{S}_{{{x}{y}}}}}{{{S}_{{xx}}=\ {\frac{{-{14009.0727}}}{{{39602.1698}}}}=\ -{0.3537}}}}$$ The mean is the sum of all values divided by the number of values: $$\displaystyle\overline{{{x}}}=\ {\frac{{\sum\ {x}_{{{i}}}}}{{{n}}}}=\ {\frac{{{1743.1}}}{{{60}}}}={29.0517}$$
$$\displaystyle\overline{{{y}}}=\ \sum\ {y}_{{{i}}}\rbrace{\left\lbrace{n}\right\rbrace}=\ {\frac{{{4147.6}}}{{{60}}}}={69.1267}$$ The estimate a of the intercept $$\displaystyle\alpha\ \text{is the average of y decresed by the product of the estimate of the slope and the average of x}$$
$$\displaystyle{a}=\ \overline{{{y}}}\ -\ {b}\overline{{{x}}}={69.1267}\ -\ {\left(-{0.3537}\right)}\ \cdot\ {29.0517}={79.4036}$$ General least-squares equation: $$\displaystyle\hat{{{y}}}=\ \alpha\ +\ \beta\ {x}.\ \text{Replace}\ \alpha\ {b}{y}\ {a}={79.4036}\ {\quad\text{and}\quad}\ \beta\ {b}{y}\ {b}=\ -{0.3537}\ \text{in the general least-squres equation}$$
$$\displaystyle\hat{{{y}}}={a}\ +\ {b}{x}={79.4036}\ -\ {0.3537}{x}$$
d) Let us evalute the regression line in part (c) at $$\displaystyle{x}={30}:$$
$$\displaystyle\hat{{{y}}}={79.4036}\ -\ {0.3537}{\left({30}\right)}={68.7926}$$ Thus the predicted life expectancy of a country with an IMA of 30 is 68.7926 years. e) We deteermine all necessary sums: $$\displaystyle\sum\ {x}_{{{i}}}={1743.1}$$
$$\displaystyle\sum\ {y}_{{{i}}}={4147.6}$$
$$\displaystyle\sum\ {x}_{{{i}}}\ {y}_{{{i}}}={106485.62}$$
$$\displaystyle{\sum_{{{i}}}^{{{2}}}}={90242.13}$$
$$\displaystyle\sum\ {{y}_{{{i}}}^{{{2}}}}={293216.68}$$ Determine the correlation coefficient: $$\displaystyle{r}=\ {\frac{{\sum\ {x}_{{{i}}}\ {y}_{{{i}}}\ -\ {\left(\sum\ {x}_{{{i}}}\right)}\ \frac{{\sum\ {y}_{{{i}}}}}{{n}}}}{{\sqrt{{{\left[\sum\ {{x}_{{{i}}}^{{{2}}}}\ -\ \frac{{\left(\sum\ {x}_{{{i}}}\right)}^{{{2}}}}{{n}}\right]}\ {\left[\sum\ {{y}_{{{i}}}^{{{2}}}}\ -\ \frac{{\left(\sum\ {y}_{{{i}}}\right)}^{{{2}}}}{{n}}\right]}}}}}}$$
$$\displaystyle=\ {\frac{{{106485.62}\ -\ {\left({1743.1}\right)}\ \frac{{{4147.6}}}{{60}}}}{{\sqrt{{{\left[{90242.13}\ -\ \frac{{1743.1}^{{{2}}}}{{60}}\right]}\ {\left[{293216.68}\ -\ \frac{{4147.6}^{{{2}}}}{{60}}\right]}}}}}}$$
$$\displaystyle\approx\ -{0.8727}$$ If r is positive, then there is a positive linear relationship. If r is negative, then there is a negative linear relationship. If $$\displaystyle{0}\ {<}\ {\left|{r}\right|}\ {<}\ {0.5},\ \text{then the linear relationship is weak. If}\ {0.5}\ {<}\ {\left|{r}\right|}\ {<}\ {0.8},\ \text{then the linear relationship is moderate. If}\ {0.8}\ {<}\ {\left|{r}\right|}\ {<}\ {1},\ \text{then the linear relationship is strong.}$$

We note that the linear correlation coefficient r is larger than 0.8 in absolute value and negative, thus there is a strong negative linear relationship between the variables. f) (45.6, 49.9) is an outlier, because this data point lies further from the regression line than all other points in the scatterplot. There do not appear to be any influential observations, because there is no single data point near the regression line in the scatterplot that lies far from the other data points in the scatterplot.

### Relevant Questions

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