91Ó°ÊÓ

$$\begin{gathered} \mathrm{μ}x=5.3 \\ \mathrm{σ}x=0.01 \\ \mathrm{μ}y=5.26 \\ \mathrm{σ}y=0.02 \end{gathered}$$

Let,

$X=$diameter of the randomly chosen case

$Y=$the diameter of a randomly chosen DVD

The average of the difference between two random variables is:

$$\begin{gathered} \mathrm{μ}X-Y=\mathrm{μ}X-\mathrm{μ}Y \\ =5.3-5.26 \\ =0.04 \end{gathered}$$

The variance of the difference of $2$random variables is as follows:

$$\begin{gathered} {\mathrm{σ}}^{2}X-Y={\mathrm{σ}}^{2}X+{\mathrm{σ}}^{2}Y \\ ={(0.01)}^2+{(0.02)}^2=0.0001+0.0004 \\ =0.0005 \end{gathered}$$

The standard deviation is as follows:

$$\begin{gathered} \mathrm{σ}X-Y=\sqrt{{\mathrm{σ}}^{2}X-Y} \\ ==\sqrt{0.0005} \\ =0.02236 \end{gathered}$$

$X-Y$is important to the DVD manufacturers, because of the difference.

$X-Y$is positive then the DVD will fit in the case but if the difference

$X-Y$ is negative, and then the DVD will not fit in the case.

$$\begin{gathered} \mathrm{μ}=0.04 \\ \mathrm{σ}=0.0005 \\ x=0 \end{gathered}$$

Formula used:

$z=\frac{x-\mathrm{μ}}{\mathrm{σ}}$

When the difference $X-Y$is positive, the DVD fits in the case.

The $Z$-score is :

$$\begin{gathered} z=\frac{x-\mathrm{μ}}{\mathrm{σ}} \\ =\frac{0-0.04}{0.02236} \\ =-1.7 \end{gathered}$$

Using the normal probability table, determine the corresponding probability. $P(z<-1.79)$is given in the standard normal probability table in the row beginning with $-1.7$and the column beginning with $0.09$

$$\begin{gathered} P(X-Y>0)=P(Z>-1.79) \\ =1-P(Z<-1.79) \\ =1-0.0367 \\ =0.9633 \\ =96.33\% \end{gathered}$$

$$\begin{gathered} n=100 \\ p=0.9633 \end{gathered}$$

Formula used:

$P(X=k)={}^{n}C_k·p^k·(1-p{)}^{n-k}$

Binomial probability is defined as:

$P(X=k)={}^{n}C_k·p^k·(1-p{)}^{n-k}$

At $k=100$find the definition of binomial probability.

role="math" localid="1654440514831" $$\begin{gathered} P(X=100)={}_{100}C^{100}·0.{9633}^{100}·{(1-0.9633)}^{100-100} \\ =0.02378 \\ =2.378\% \end{gathered}$$

There is a $2.378\%$ chance that all 100 DVDs will fit in their cases.