Question

the scatterplot above shows the number of visitors to a railroad museum in pennsylvania each year from 1968 to 1980, where t is the number of years since 1968 and n is the number of visitors. a line of best fit is also shown. which of the following could be an equation of the line of best fit shown? in the given scatterplot, a line of best fit for the data is shown.

Explanation:

Step1: Identify the form of a linear - regression line

The equation of a line is in the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y - intercept. In the context of the scatter - plot, if \(y\) is the number of visitors \(n\) and \(x\) is the number of years since 1968 (\(t\)), the equation is \(n=mt + b\).

Step2: Determine the slope and y - intercept from the scatter - plot

The line is decreasing, so the slope \(m\) is negative. When \(t = 0\) (the year 1968), the y - intercept \(b\) is the number of visitors in 1968. Looking at the scatter - plot, as \(t\) increases, \(n\) decreases. A possible equation could be of the form \(n=-at + b\) where \(a>0\). For example, if we assume two points on the line of best - fit \((t_1,n_1)\) and \((t_2,n_2)\), the slope \(m=\frac{n_2 - n_1}{t_2 - t_1}<0\).

Since no specific points or options are given, a general form of a line of best - fit for a decreasing linear relationship between the number of years since 1968 (\(t\)) and the number of visitors \(n\) is \(n=-kt + c\), where \(k>0\) is the slope (representing the rate of decrease of visitors per year) and \(c\) is the number of visitors in 1968 (the y - intercept when \(t = 0\)).

Answer:

A general form of the line of best - fit is \(n=-kt + c\) (\(k>0\), \(c\) is the number of visitors in 1968)