Chapter 6.2, Problem 72E

To determine

Probability for the scores differ by 5 or more points in either direction.

Answer to Problem 72E

Probability for the scores differ by 5 or more points in either direction is 0.0768.

Explanation of Solution

Given information:

Mean and standard deviation,

For Fred:

  ${\mu }_{X_1}=24,{\sigma }_{X_1}=2$

For Leona:

  ${\mu }_{X_2}=24,{\sigma }_{X_2}=2$

Calculations:

Let

X₁ : Fred’s score

X₂ : Leona’s score

Then

X₁ - X₂ is the difference in scores of Fred and Leona

X₁ and X₂ are independent.

Since X₁ and X₂ have a Normal distribution, their difference ( X₁ - X₂ ) also has a Normal distribution.

Now,

For any two variables,

The mean of the difference is equal to the difference of their means.

  ${\mu }_{X_1-X_2}={\mu }_{X_1}-{\mu }_{X_2}=24-24=0$

When the random variables are independent, the variance of the difference is equal to the sum of their variances.

  ${\sigma }_{X_1-X_2}^2={\sigma }_{X_1}^2+{\sigma }_{X_2}^2={\left(2\right)}^2+{\left(2\right)}^2=4+4=8$

We know that

The standard deviation is the square root of the variance.

  ${\sigma }_{X_1-X_2}=\sqrt{{\sigma }_{X_1-X_2}^2}=\sqrt{8}\approx 2.8284$

Thus,

The difference ( X₂ - X₁ ) follows a Normal distribution.

Now,

For Probability:

Calculate the z − score,

  $z=\frac{-5-0}{2.8284}\approx -1.77$

Or

  $z=\frac{5-0}{2.8284}\approx 1.77$

Use normal probability table in the appendix, to find the corresponding probability.

See the row that starts with -1.7 and the column that starts with .07 of the standard normal probability table for $P\left(z<-1.77\right)$ .

  $\begin{array}{l}P\left(X_1-X_2<-5\text{or}{X}_1-X_2>5\right)=P\left(z<-1.77\text{or}z>1.77\right)\\ =2\left(P\left(z<-1.77\right)\right)\\ =2\left(0.0384\right)\\ =0.0768\end{array}$

Thus,

Probability that the scores differ by 5 or more points in either direction is 0.0768.