Chapter 5, Problem 95CE

a.

To determine

Calculate each probability (i-vi) and explain it meaning.

Answer to Problem 95CE

i. The probability $P\left(F-\right)$ is 0.5588 and the meaning is the probability of forecasts predicted a decline in interest rates is 0.5588.

ii. The probability $P\left(A+\right)$ is 0.5294 and the meaning is the probability of rise in interest rates is 0.5294.

iii. The probability $P\left(A-|F-\right)$ is 0.3684 and the meaning is the probability of actual decline given that the forecasters predicted a decline in interest rates is 0.3684.

iv. The probability $P\left(A+|F+\right)$ is 0.4 and the meaning is the probability of actual increase given that the forecasters predicted an increase in interest rates is 0.4.

v. The probability $P\left(A+\text{ and }F+\right)$ is 0.1765 and the meaning is the probability of both forecasted increase and actual increase in interest rates in a given year is 0.1765.

vi. The probability $P\left(A-\text{ and }F-\right)$ is 0.2059 and the meaning is the probability of both forecasted decline and actual decline in interest rates in a given year is 0.2059.

Explanation of Solution

Calculation:

The given table shows that frequencies of the actual and predicted interest rate movements.

The given contingency table is,

	Actual Change
Forecast Change	Decline (A–)	Rise (A+)	Row Total
Decline (F–)	7	12	19
Rise (F+)	9	6	15
Column Total	16	18	34

i. $P\left(F-\right)$:

The formula for finding the probability $P\left(F-\right)$ is,

$P\left(F-\right)=\frac{\text{Frequency for the class }F-}{\text{Total frequencies in the distribution}}$

Substitute 19 for ‘Frequency for the class F–’ and 34 for ‘Total frequencies in the distribution’

$\begin{array}{c}P\left(F-\right)=\frac{19}{34}\\ =0.5588\end{array}$

Therefore, the probability $P\left(F-\right)$ is 0.5588 and the meaning is the probability of forecasts predicted a decline in interest rates is 0.5588.

ii. $P\left(A+\right)$:

The formula for finding the probability $P\left(A+\right)$ is,

$P\left(A+\right)=\frac{\text{Frequency for the class }A+}{\text{Total frequencies in the distribution}}$

Substitute 18 for ‘Frequency for the class A+’ and 34 for ‘Total frequencies in the distribution’

$\begin{array}{c}P\left(A+\right)=\frac{18}{34}\\ =0.5294\end{array}$

Therefore, the probability $P\left(A+\right)$ is 0.5294 and the meaning is the probability of rise in interest rates is 0.5294.

iii. $P\left(A-|F-\right)$:

The formula for finding the probability $P\left(A-|F-\right)$ is,

$P\left(A-|F-\right)=\frac{\text{Frequency for the class }A-\text{ and }F-}{\text{Frequency for the class }F-}$

Substitute 7 for ‘Frequency for the class A– and F–’ and 19 for ‘Frequency for the class F–’,

$\begin{array}{c}P\left(A-|F-\right)=\frac{7}{19}\\ =0.3684\end{array}$

Therefore, the probability $P\left(A-|F-\right)$ is 0.3684 and the meaning is the probability of actual decline given that the forecasters predicted a decline in interest rates is 0.3684.

iv. $P\left(A+|F+\right)$:

The formula for finding the probability $P\left(A+|F+\right)$ is,

$P\left(A+|F+\right)=\frac{\text{Frequency for the class }A+\text{ and }F+}{\text{Frequency for the class }F+}$

Substitute 6 for ‘Frequency for the class A+ and F+’ and 15 for ‘Frequency for the class F+’,

$\begin{array}{c}P\left(A+|F+\right)=\frac{6}{15}\\ =0.4\end{array}$

Therefore, the probability $P\left(A+|F+\right)$ is 0.4 and the meaning is the probability of actual increase given that the forecasters predicted an increase in interest rates is 0.4.

v. $P\left(A+\text{ and }F+\right)$:

The formula for finding the probability $P\left(A+\text{ and }F+\right)$ is,

$P\left(A+\text{ and }F+\right)=\frac{\text{Frequency for the class }A+\text{ and }F+}{\text{Total frequencies in the distribution}}$

Substitute 6 for ‘Frequency for the class A+ and F+’ and 34 for ‘Total frequencies in the distribution’,

$\begin{array}{c}P\left(A+\text{ and }F+\right)=\frac{6}{34}\\ =0.1765\end{array}$

Therefore, the probability $P\left(A+\text{ and }F+\right)$ is 0.1765 and the meaning is the probability of both forecasted increase and actual increase in interest rates in a given year is 0.1765.

vi. $P\left(A+\text{ and }F+\right)$:

The formula for finding the probability $P\left(A-\text{ and }F-\right)$ is,

$P\left(A-\text{ and }F-\right)=\frac{\text{Frequency for the class }A-\text{ and }F-}{\text{Total frequencies in the distribution}}$

Substitute 7 for ‘Frequency for the class A– and F–’ and 34 for ‘Total frequencies in the distribution’,

$\begin{array}{c}P\left(A-\text{ and }F-\right)=\frac{7}{34}\\ =0.2059\end{array}$

Therefore, the probability $P\left(A-\text{ and }F-\right)$ is 0.2059 and the meaning is the probability of both forecasted decline and actual decline in interest rates in a given year is 0.2059.

b.

To determine

Explain whether the forecaster’s predictions are accurate or not.

Answer to Problem 95CE

No, the forecaster’s predictions are not accurate because the forecaster’s predications show lesser values than the actual values.

Explanation of Solution

Calculation:

From part (a), the probabilities $P\left(A-\right)=0.4705$ and $P\left(A-|F-\right)=0.3684$. Here, the probability of actual decline $P\left(A-\right)$ shows that the interest rates of actual decline is 47% of the time. But the predicted probability value of actual decline given the forecast change decline $P\left(A-|F-\right)$ is only 37% actual rate. Therefore, the forecaster’s predictions are not accurate.