Question

Car Colors. In Exercises 9–12, assume that 100 cars are randomly selected. Refer to the accompanying graph, which shows the top car colors and the percentages of cars with those colors (based on PPG Industries).

White CarsFind the probability that fewer than 20 cars are white. Is 20 a significantly low number of white cars?

Short answer

The probability that fewer than 20 cars are white is 0.2033.

20 is not a significantly low number of white cars.

Step-by-step solution

Given information

The number of randomly selected cars$n=100$ and probability of success for a white car is $p=0.23$.

Then,

$$\begin{gathered} q=1-p \\ =1-0.23 \\ =0.77 \end{gathered}$$

 Step 2: Check the requirement

From the given information,

$$\begin{gathered} np=100\times 0.23 \\ =23 \\ >5 \end{gathered}$$

$$\begin{gathered} nq=100\times 0.77 \\ =77 \\ >5 \end{gathered}$$

Here both and are greater than 5. Hence the binomial probability distribution can be approximated to normal distribution.

Mean and standard deviation for normal distribution

The mean value is,

$$\begin{gathered} \mu =np \\ =100\times 0.23 \\ =23 \end{gathered}$$

The standard deviation is,

$$\begin{gathered} \sigma =\sqrt{npq} \\ =\sqrt{100\times 0.23\times 0.77} \\ =4.21 \end{gathered}$$

Continuity correction

The probability that fewer than n outcomes appear is expressed as,

$P\left(X<n\right)=P\left(X<n-0.5\right)$

In this case, define random variable X for the number of white cars.

The probability that fewer than 20 cars are white is expressed as,

$$\begin{gathered} P\left(X<20\right)=P\left(X<20-0.5\right) \\ =P\left(X<19.5\right) \end{gathered}$$

The continuity correction is $\mathbf{P}\left(\mathbf{X}\mathbf{<}\mathbf{19}\mathbf{.}\mathbf{5}\right)$.

Compute the Z-score

Obtain z-score using$x=19.5$, $\mu =23,\sigma =4.21$as follows:

$$\begin{gathered} z=\frac{x-\mu }{\sigma } \\ =\frac{19.5-23}{4.21} \\ =-0.83 \end{gathered}$$

The z-score is -0.83

Using the standard normal table, the corresponds to a cumulative right area of 0.2033.

Define significantly low

If $P\left(Xorfewer\right)⩽0.05$then it is significantly low.

In this case, 0.2033 is greater than 0.05. Hence 20 white cars are not significantly low.

Thus, 20 is not significantly a low number of white cars.