Chapter 9, Problem 21E

(a)

To determine

To find out what is the correlation between rate and year.

Answer to Problem 21E

The correlation between the rate and the year is $0.88$ .

Explanation of Solution

In the question, it is shown a plot showing a federal rate on three-month treasury bill from $1950\text{ to 1980}$ and a regression model fit to the relationship between rate in percentage and the years. The dependent variable is rate. And it is also given in question that:

  $\begin{array}{l}{R}^2=77.4\%\\ s=1.239\\ \text{Intercept}=0.640282\\ \text{Year}-1950=0.247637\end{array}$

As we know that the formula for calculation correlation from the R -squared is:

  $r^2=R^2$

This implies we have,

  $\begin{array}{l}{r}^2=R^2\\ =0.774\\ \Rightarrow r=\sqrt{0.774}\\ \Rightarrow r=0.88\end{array}$

Thus, the correlation between the rate and the year is $0.88$ .

(b)

To determine

To interpret the slope and intercept.

Explanation of Solution

In the question, it is shown a plot showing a federal rate on three-month treasury bill from $1950\text{ to 1980}$ and a regression model fit to the relationship between rate in percentage and the years. The dependent variable is rate. And it is also given in question that:

  $\begin{array}{l}{R}^2=77.4\%\\ s=1.239\\ \text{Intercept}=0.640282\\ \text{Year}-1950=0.247637\end{array}$

Thus we interpret from the slope and intercept that interest rates during this period grew at about $0.25\%$ per year starting from an interest rate of about $0.64\%$ .

(c)

To determine

To explain what does this model predict for the interest rate in the year $2000$ .

Explanation of Solution

In the question, it is shown a plot showing a federal rate on three-month treasury bill from $1950\text{ to 1980}$ and a regression model fit to the relationship between rate in percentage and the years. The dependent variable is rate. And it is also given in question that:

  $\begin{array}{l}{R}^2=77.4\%\\ s=1.239\\ \text{Intercept}=0.640282\\ \text{Year}-1950=0.247637\end{array}$

Therefore, this model predict for the interest rate in the year $2000$ that if we substitute $50$ in the model than the model yields a prediction of about $13\%$ .

(d)

To determine

To explain would you expect this prediction to have been accurate or not.

Answer to Problem 21E

No, it is not accurate.

Explanation of Solution

In the question, it is shown a plot showing a federal rate on three-month treasury bill from $1950\text{ to 1980}$ and a regression model fit to the relationship between rate in percentage and the years. The dependent variable is rate. And it is also given in question that:

  $\begin{array}{l}{R}^2=77.4\%\\ s=1.239\\ \text{Intercept}=0.640282\\ \text{Year}-1950=0.247637\end{array}$

Thus, from above we would not expect this prediction to have been accurate because extrapolating $20$ years beyond the end of these data would be dangerous and unlikely to be accurate.