Question

A NUMMI assembly line, which has been operating since $1984$, has built an average of $6,000$ cars and trucks a week. Generally, $10\%$ of the cars were defective coming off the assembly line. Suppose we draw a random sample of $n=100$ cars. Let $X$ represent the number of defective cars in the sample. What can we say about $X$ in regard to the $68-95-99.7$ empirical rule (one standard deviation, two standard deviations and three standard deviations from the mean are being referred to)? Assume a normal distribution for the defective cars in the sample.

Short answer

$68\%$of the values of $X$lie between $7$and $13,95\%$of the values of $X$lie between $4$and $16$, and $99.7\%$of the values of $X$lie between $2$and $19$.

Step-by-step solution

Given Information

The number of defective autos in a sample is represented by a regularly distributed random variable called $X$.

The sample's car count, $n=100$.

Car defect probability, $\%=0.1,p=10\%$.

Explanation

Mean, $\mu =np$

$=100*0.1$

$=10$

Standard deviation, $\sigma =\sqrt{100*0.1*(1-0.1)}$

$=3$

i. For $z=\pm 1,$

$x_1=\mu -z\sigma$

and

$x_2=\mu +z\sigma$

$\Rightarrow x_1=10-1*3$

$=7$

$x_2=10+1*3$

$=13$

$X$values range from $7$ to $13$ in $68\%$ of cases.

Explanation

ii. For $z=\pm 2,$

$x_1=\mu -z\sigma$

and $x_2=\mu +z\sigma$

$\Rightarrow x_1=10-2*3$

$=4$

and

$x_2=10+2*3$

$=16$

$95\%$of $X$values are between $4$and $16$.

iii. For $z=\pm 3,$

$x_1=\mu -z\sigma$

and $x_2=\mu +z\sigma$$\Rightarrow x_1=10-3*3$

$=1$

$x_2=10+3*3$

$=19$

$99.7\%$of the values of $X$lie between$1$and$19$.