Chapter 5, Problem 95CE

i.

To determine

Calculate probability $P\left(F-\right)$ and explain its meaning.

Answer to Problem 95CE

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.

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

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.

To determine

Calculate probability $P\left(A+\right)$ and explain its meaning.

Answer to Problem 95CE

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

Explanation of Solution

Calculation:

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.

To determine

Calculate probability $P\left(A-|F-\right)$ and explain its meaning.

Answer to Problem 95CE

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.

Explanation of Solution

Calculation:

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.

To determine

Calculate probability $P\left(A+|F+\right)$ and explain its meaning.

Answer to Problem 95CE

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.

Explanation of Solution

Calculation:

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.

To determine

Calculate probability $P\left(A+\text{ and }F+\right)$ and explain its meaning.

Answer to Problem 95CE

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.

Explanation of Solution

Calculation:

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.

To determine

Calculate probability $P\left(A-\text{ and }F-\right)$ and explain its meaning.

Answer to Problem 95CE

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 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.