91Ó°ÊÓ

According to the information, the mean and standard deviation of$X$ are

${\mathrm{μ}}_X=24$

${\mathrm{σ}}_X=2$

We have to find the probability the scores differ by $5$or more points on either direction

Consider $X_1$and $X_2$as the random variable, which showing the scores of Leon and Fred.

Define the variable $D=X_1−X_2$

We have to find $P\left(\right|D|â‰\yen 5)$

According to the information the two scores are independent

The mean of $D$: ${\mathrm{μ}}_D=0$

The standard deviation of $D$:${\mathrm{σ}}_D=2.828$

Let's standardize $D=-5$

localid="1649863584672" $$\begin{gathered} z=\frac{x-{\mathrm{μ}}_D}{{\mathrm{σ}}_D} \\ =\frac{−5−0}{2.828} \\ =−1.77 \end{gathered}$$

Now we standardize $D=5$

localid="1649863591289" $$\begin{gathered} z=\frac{x-{\mathrm{μ}}_D}{{\mathrm{σ}}_D} \\ =\frac{5−0}{2.828} \\ =1.77 \end{gathered}$$

Using a table of typical normal probabilities as a guide,

$P(D≤−5)\text{or}P(Dâ‰\yen 5)$

$$\begin{gathered} =P(z≤−1.77)+P(zâ‰\yen 1.77) \\ =P(zâ‰\yen 1.77)+P(zâ‰\yen 1.77) \\ =2P(z≤1.77) \\ =2[1−p(Z≤1.77\left)\right] \\ =2[1−0.9618] \\ =0.0764 \end{gathered}$$