Answer :
We are given data where the independent variable is the number of years since 1900 (for example, the year 1935 corresponds to [tex]$t = 35$[/tex]) and the dependent variable is the lightning fatality rate (average number of lightning deaths per million people per year). The data points are as follows:
[tex]$$
\begin{array}{cc}
t\ (\text{years since } 1900) & f(t)\ (\text{fatality rate}) \\
35 & 3.4 \\
45 & 1.8 \\
55 & 1.4 \\
65 & 0.7 \\
75 & 0.6 \\
85 & 0.4 \\
95 & 0.2 \\
\end{array}
$$[/tex]
We want to find a linear regression model of the form
[tex]$$
f(t) = a + b\,t,
$$[/tex]
where [tex]$b$[/tex] is the slope and [tex]$a$[/tex] is the intercept.
Step 1. Compute the Mean Values
First, calculate the mean of the [tex]$t$[/tex]-values and the [tex]$f(t)$[/tex]-values.
The mean of the [tex]$t$[/tex]-values is
[tex]$$
\overline{t} = \frac{35 + 45 + 55 + 65 + 75 + 85 + 95}{7} = \frac{455}{7} = 65.
$$[/tex]
The mean of the [tex]$f(t)$[/tex]-values is
[tex]$$
\overline{f} = \frac{3.4 + 1.8 + 1.4 + 0.7 + 0.6 + 0.4 + 0.2}{7} \approx 1.214.
$$[/tex]
Step 2. Compute the Slope, [tex]$b$[/tex]
The formula for the slope of the regression line is
[tex]$$
b = \frac{\sum (t_i - \overline{t})(f_i - \overline{f})}{\sum (t_i - \overline{t})^2}.
$$[/tex]
After carrying out these summations with the data, we find
[tex]$$
b \approx -0.047.
$$[/tex]
This indicates that as time increases by 1 year (since 1900), the fatality rate decreases by about 0.047 (in the given units).
Step 3. Compute the Intercept, [tex]$a$[/tex]
The intercept is found using
[tex]$$
a = \overline{f} - b\,\overline{t}.
$$[/tex]
Substitute the computed means and the slope:
[tex]$$
a \approx 1.214 - (-0.047)(65) \approx 1.214 + 3.065 \approx 4.279.
$$[/tex]
Step 4. Write the Regression Equation
Putting the intercept and slope into the equation of the line gives
[tex]$$
f(t) = 4.279 + (-0.047)t.
$$[/tex]
This can also be written as
[tex]$$
f(t) = 4.279 - 0.047t.
$$[/tex]
Final Answer:
[tex]$$
f(t) = 4.279 - 0.047t.
$$[/tex]
This is the regression equation for the lightning fatality rate as a function of the number of years since 1900.
[tex]$$
\begin{array}{cc}
t\ (\text{years since } 1900) & f(t)\ (\text{fatality rate}) \\
35 & 3.4 \\
45 & 1.8 \\
55 & 1.4 \\
65 & 0.7 \\
75 & 0.6 \\
85 & 0.4 \\
95 & 0.2 \\
\end{array}
$$[/tex]
We want to find a linear regression model of the form
[tex]$$
f(t) = a + b\,t,
$$[/tex]
where [tex]$b$[/tex] is the slope and [tex]$a$[/tex] is the intercept.
Step 1. Compute the Mean Values
First, calculate the mean of the [tex]$t$[/tex]-values and the [tex]$f(t)$[/tex]-values.
The mean of the [tex]$t$[/tex]-values is
[tex]$$
\overline{t} = \frac{35 + 45 + 55 + 65 + 75 + 85 + 95}{7} = \frac{455}{7} = 65.
$$[/tex]
The mean of the [tex]$f(t)$[/tex]-values is
[tex]$$
\overline{f} = \frac{3.4 + 1.8 + 1.4 + 0.7 + 0.6 + 0.4 + 0.2}{7} \approx 1.214.
$$[/tex]
Step 2. Compute the Slope, [tex]$b$[/tex]
The formula for the slope of the regression line is
[tex]$$
b = \frac{\sum (t_i - \overline{t})(f_i - \overline{f})}{\sum (t_i - \overline{t})^2}.
$$[/tex]
After carrying out these summations with the data, we find
[tex]$$
b \approx -0.047.
$$[/tex]
This indicates that as time increases by 1 year (since 1900), the fatality rate decreases by about 0.047 (in the given units).
Step 3. Compute the Intercept, [tex]$a$[/tex]
The intercept is found using
[tex]$$
a = \overline{f} - b\,\overline{t}.
$$[/tex]
Substitute the computed means and the slope:
[tex]$$
a \approx 1.214 - (-0.047)(65) \approx 1.214 + 3.065 \approx 4.279.
$$[/tex]
Step 4. Write the Regression Equation
Putting the intercept and slope into the equation of the line gives
[tex]$$
f(t) = 4.279 + (-0.047)t.
$$[/tex]
This can also be written as
[tex]$$
f(t) = 4.279 - 0.047t.
$$[/tex]
Final Answer:
[tex]$$
f(t) = 4.279 - 0.047t.
$$[/tex]
This is the regression equation for the lightning fatality rate as a function of the number of years since 1900.
