Question

the scatter plot shows the price, in cents, of a postage stamp used to mail a letter in the united states for the years from 1958 to 2014. also shown is a line of fit to model the data.
postage stamp prices
the equation of the line of fit is ( y = - 0.71+0.86x ), where ( y ) represents the predicted price, in cents, of a stamp and ( x ) represents the number of years since 1958.
part a
which statement best describes the model?
a. the model estimates an increase, on average, of 0.71 cent per year in the price of a stamp.
b. the model estimates a decrease, on average, of 0.71 cent per year in the price of a stamp.
c. the model estimates an increase, on average, of 0.86 cent per year in the price of a stamp.
d. the model estimates a decrease, on average, of 0.86 cent per year in the price of a stamp.
part b
in 1958, the price of a postage stamp was 4 cents. the price remained the same until 1963, when the price increased. based on the information shown in the scatter plot, what is true about the model for the years from 1958 to 1963?
a. the model overpredicts the actual price of a stamp for the time period from 1958 to 1963.
b. the model underpredicts the actual price of a stamp for the time period from 1958 to 1963.
c. the model overpredicts the actual price of a stamp for 1958 and underpredicts the actual price of a stamp for 1963.
d. the model underpredicts the actual price of a stamp for 1958 and overpredicts the actual price of a stamp for 1963.

Explanation:

Part A

Step1: Recall the slope - intercept form of a line

The equation of a line is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y - intercept. In the given equation \(y=-0.71 + 0.86x\), the slope \(m = 0.86\).

Step2: Interpret the slope

The slope \(m\) represents the rate of change. Since \(m=0.86>0\), it means that for each unit increase in \(x\) (each year), \(y\) (the price of the stamp) increases by \(0.86\) cents on average.

Part B

Step1: Calculate the predicted price for \(x = 0\) (1958)

Substitute \(x = 0\) into the equation \(y=-0.71+0.86x\). Then \(y=-0.71 + 0.86\times0=-0.71\) (negative value, which is a prediction). The actual price in 1958 is \(y = 4\) cents. So, the model underpredicts for 1958.

Step2: Calculate the predicted price for \(x = 5\) (1963)

Substitute \(x = 5\) into the equation \(y=-0.71+0.86x\). Then \(y=-0.71+0.86\times5=-0.71 + 4.3=3.59\). The actual price in 1963 is greater than the price in 1958 (since the price increased in 1963). So, the model underpredicts for the period 1958 - 1963.

Answer:

Part A: C. The model estimates an increase, on average, of \(0.86\) cent per year in the price of a stamp.
Part B: B. The model underpredicts the actual price of a stamp for the time period from 1958 to 1963.