Question

Loser buys the pizza leona and Fred are friendly competitors in high school. Both are about to take the ACT college entrance examination, They agree that if one of them scores $5$ar more points better than the other, the loser will buy the winner a pizza. Suppose that in fact Fred and Leona have equal ability, so that each score varies Normally with mean $24$and standard deviation data-custom-editor="chemistry" $2$. (The variation is due to luck in guessing and the accident of the specific questions being familiar to the student.) The two scores are independent. What is the probability that the scores differ by $5$or more points in either direction? Follow the four-step process.

Short answer

There's a $7.64$percent probability that the scores will fluctuate by$5$points or more in either direction.

Step-by-step solution

Step-1 Given Information

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

${\mu }_X=24$

${\sigma }_X=2$

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

Standardize D

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 $P\left(\right|D|\ge 5)$

According to the information the two scores are independent

The mean of $D$: ${\mu }_D=0$

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

Let's standardize $D=-5$

localid="1649863584672" $$\begin{gathered} z=\frac{x-{\mu }_D}{{\sigma }_D} \\ =\frac{-5-0}{2.828} \\ =-1.77 \end{gathered}$$

Now we standardize $D=5$

localid="1649863591289" $$\begin{gathered} z=\frac{x-{\mu }_D}{{\sigma }_D} \\ =\frac{5-0}{2.828} \\ =1.77 \end{gathered}$$

Calculate the probability

Using a table of typical normal probabilities as a guide,

$P(D\le -5)\text{or}P(D\ge 5)$

$$\begin{gathered} =P(z\le -1.77)+P(z\ge 1.77) \\ =P(z\ge 1.77)+P(z\ge 1.77) \\ =2P(z\le 1.77) \\ =2[1-p(Z\le 1.77\left)\right] \\ =2[1-0.9618] \\ =0.0764 \end{gathered}$$