Chapter 7, Problem 7.38TY

a.

To determine

The scatter plot of the given data of smokers and years.

Answer to Problem 7.38TY

The scatter plot is shown below,

Figure (1)

Explanation of Solution

Given Info:

The percentages of adult smokers in the years between 1965 and 2011 is given below,

Years	Smokers
1965	41.9
1974	37
1979	33.3
1983	31.9
1987	28.6
1990	25.3
1993	24.8
1997	24.6
2000	23.1
2002	22.5
2006	20.8
2009	20.6
2011	19

Software Procedure:

Step by step procedure to draw the scatterplot using the Excel:

Steps to draw a scatter plot of the given data are:

• Press [Ctrl]-N for a new workbook.
• Enter the years column A.
• Enter the percent of adult smokers into column B.
• Select Add ins then Megastats.
• Select Correlation/Regression and under that select scatter plot.
• Select the column 1 in horizontal axis and column 2 in vertical axis.
• Tick the plot linear regression line and press Ok.

Output using the Excel is given below:

Observation:

The scatter plot shows that, the number of smokers’ decreases as the year’s increases.

b.

To determine

The direction, form, and strength of the relationship between percent of smokers and year. Whether there are any outliers.

Explanation of Solution

From the scatterplot, it is clear that the value of smokers decreases with increase in number of year that means there is a linear relationship between smokers and years and number of smokers decreases with increase in year so, relationship is negative.

Thus, there is a strong negative linear relationship between the smokers and the year and there are no outliers.

c.

To determine

The least squares regression line that predicts percent of smokers from year and add the line to the scatter plot.

Answer to Problem 7.38TY

The equation of the least square regression line is $\widehat{y}=1003.28+\left(-0.49\right)x$.

Explanation of Solution

Given info:

The given values are,

$\overline{x}=1992,\overline{y}=27.2,s_x=14,s_y=7,r=-0.98$

Calculation:

The formula for regression equation is given below,

$\widehat{y}=a+bx$ (1)

Where,

• $b=r\times \frac{s_y}{s_x}$
• $a=\overline{y}-b\overline{x}$

The formula to calculate the value of b is,

$b=r\times \frac{s_y}{s_x}$

Substitute $-0.98$ for r, 14 for $s_x$ and 7 for $s_y$ to find the value of b.

$\begin{array}{c}b=-0.98\times \frac{7}{14}\\ =\frac{-0.98}{2}\\ =-0.49\end{array}$

The value of b is $-0.49$.

The formula to calculate the value of a is,

$a=\overline{y}-b\overline{x}$

Substitute $27.2$ for $\overline{y}$ , 1992 for $\overline{x}$ and $-0.49$ for $b$ to find the value of b.

$\begin{array}{c}a=27.2-\left(-0.49\times 1992\right)\\ =27.2+976.08\\ =1003.28\end{array}$

The value of a is $1003.28$.

Substitute $1003.28$ for a and $-0.49$ for b to find the least square regression line.

$\begin{array}{c}\widehat{y}=1003.28+\left(-0.49\right)x\\ =1003.28-0.49x\end{array}$

Thus, the equation of the least square regression line is $\widehat{y}=1003.28+\left(-0.49\right)x$.

d.

To determine

The decline in smoking per year on an average during this period.

Answer to Problem 7.38TY

The decline in smoking per year on an average during this period is $0.49$.

Explanation of Solution

Calculation:

The equation of the least square regression line is,

$\widehat{y}=1003.28-0.49x$

The slope of the regression is $-0.49$.

The coefficient value for year is $-0.49$. This coefficient value shows that there is a change in smokers for every unit decreases in additional year.

Thus, the decline in smoking per year on an average during this period is $0.49$.

e.

To determine

The percentage of the variation of adults who smoke can be explained by linear change over time.

Answer to Problem 7.38TY

The percentage of the variation of adults who smoke can be explained by linear change over time is 96.04%.

Explanation of Solution

Calculation:

The formula to find percentage of the variation of adults who smoke can be explained by linear change over time is given by squared value of r.

$\begin{array}{c}{r}^2={\left(-0.98\right)}^2\\ =0.9604\\ =96.04\%\end{array}$

Thus, the percentage of the variation of adults who smoke can be explained by linear change over time is 96.04%.

f.

To determine

The percentage of smokers in 2020.

Answer to Problem 7.38TY

The predicted percentage of adults that smoke in 2020 is 13.48%.

Explanation of Solution

Calculation:

From part (c) the equation for linear regression is given by,

$\widehat{y}=1003.28+\left(-0.49\right)x$

Where,

• x represents the year.
• $\widehat{y}$ represents the percentage of smokers in that particular year.

Substitute 2020 for x in the above equation to find the percentage of smokers in 2020.

$\begin{array}{c}\widehat{y}=1003.28+\left(-0.49\right)\left(2020\right)\\ =1003.28-989.8\\ =13.48\end{array}$

Thus, the predicted percentage of adults that smoke in 2020 is 13.48%.

g.

To determine

The percentage of smokers in 2075. And reason that it is foolishness to use this regression line equation to find the percentage of smokers in 2075.

Answer to Problem 7.38TY

The predicted percentage of adults that smoke in 2075 is 13.48%.

Explanation of Solution

Calculation:

From part (c) the equation for linear regression is given by,

$\widehat{y}=1003.28+\left(-0.49\right)x$

Where,

• x represents the year.
• $\widehat{y}$ represents the percentage of smokers in that particular year.

Substitute 2075 for x in the above equation to find the percentage of smokers in 2075.

$\begin{array}{c}\widehat{y}=1003.28+\left(-0.49\right)\left(2075\right)\\ =1003.28-1016.75\\ =-13.47\end{array}$

Thus, the predicted percentage of adults that smoke in 2020 is $-13.47\%$.

Clearly, the percentage of smokers came out to be negative in 2075 which is impossible.

Thus, it is foolishness to use this regression line equation to find the percentage of smokers in 2075.