Chapter 14, Problem 14.54SE

i.

To determine

To identify:

Category that missing in the classification and no of vehicles that belongs to category.

Answer to Problem 14.54SE

One category is missing having any other color which is not in the top colors and $8\%$ of the vehicles belong to the other’s category.

Explanation of Solution

Given:

Color	Silver	Black	Gray	Blue	Red	White
Percent	$19$	$17$	$17$	$15$	$12$	$12$

Calculation:

The percentage of market shares are given for the top size colors cars.

And the total is equals to the $100$ ,

  $\begin{array}{l}Total=19+17+17+15+12+12\\ Total=92\\ Total\ne 100\end{array}$

Now for adding other column for car colors:

  $\begin{array}{l}Others=100-92\\ Others=8\end{array}$

Thus, one category is missing having any other color which is not in the top colors. And the eight percent of the vehicles belong to the other’s category.

Conclusion:

Thus, one category is missing having any other color which is not in the top colors. And the eight percent of the vehicles belong to the other’s category.

ii.

To determine

To identify:

The $p-$ value and the provided data is sufficient evidence to indicate the percentages of the colors for compact or sports cars are differing from given.

Answer to Problem 14.54SE

There is sufficient evidence to indicate the difference in percentage are similar for all the $7$colors as considered under the $H_0$.

Explanation of Solution

Given:

Color	Silver	Black	Gray	Blue	Red	White
Percent	$19$	$17$	$17$	$15$	$12$	$12$

Calculation:

There is need to add the new category “others”, in order to maintain the sample size as $250$ from part (a).

  $\begin{array}{l}Total=52+43+48+41+32+19\\ Total=235\\ Total\ne 250\end{array}$

Now there is need to add other column for the car colors.

  $\begin{array}{l}Others=250-235\\ Others=15\end{array}$

Thus, one category is missing having any other color, that is not in the top colors and $15$ no of vehicles belongs to the other’s category.

For testing the whether the percentages of the colors for the compact or the sport car’s which are differ with the given true observations or not null and alternative hypothesis are defined.

Let,

  $p_1,p_{2,\mathrm{......}}{p}_6,p_7$ are the percentages of the market for silver, black, gray, blue, red, white and any other colored compact cars.

Let’s consider the null and alternative hypothesis as below:

Null hypothesis:

  $H_0:p_1=19\%,p_2=17\%,p_3=17\%,p_4=15\%,p_5=12\%,p_6=12\%,p_7=8\%$

Alternative hypothesis:

  $H_a:$ Minimum one out of seven probabilities is differing from the specified value.

The expected count for the $i^{th}$ color is defined as the $E_i$

And total of all observed counts for $6$ colors is $n$ , so $n=250$

By using $n$ and $p_i$ , the expected count is determined as,

  $\begin{array}{l}{E}_i=n\times p_i\\ E_i=250\times p_i\end{array}$

When $i=1$

  $\begin{array}{l}{E}_1=n\times p_i\\ E_1=250\times 0.19\\ E_1=47.5\end{array}$

When $i=2$

  $\begin{array}{l}{E}_2=n\times p_i\\ E_2=250\times 0.17\\ E_2=42.5\end{array}$

When $i=3$

  $\begin{array}{l}{E}_3=n\times p_i\\ E_3=250\times 0.17\\ E_3=42.5\end{array}$

When $i=4$

  $\begin{array}{l}{E}_4=n\times p_i\\ E_4=250\times 0.15\\ E_4=37.5\end{array}$

When $i=5$

  $\begin{array}{l}{E}_5=n\times p_i\\ E_5=250\times 0.12\\ E_5=30\end{array}$

When $i=6$

  $\begin{array}{l}{E}_6=n\times p_i\\ E_6=250\times 0.12\\ E_6=30\end{array}$

When $i=7$

  $\begin{array}{l}{E}_7=n\times p_i\\ E_7=250\times 0.08\\ E_7=20\end{array}$

The observed value is determined using observed count and expected count as below:

  $\begin{array}{l}{x}^2={\displaystyle \sum \frac{{\left(O_i-E_i\right)}^2}{E_i}}\\ x^2=\frac{{\left(52-47.5\right)}^2}{63.24}+\frac{{\left(43-42.5\right)}^2}{42.5}+\mathrm{......}+\frac{{\left(15-20\right)}^2}{20}\\ x^2=6.88729\end{array}$

Thus, observed value is $x^2=6.88729$ .

As number of color are $7$

That means: $k=7$

Then,

  $\begin{array}{l}df=k-1\\ df=7-1\\ df=6\end{array}$

The degrees of freedom is $6$ .

From chi-square table, degree of freedom is $df=6$ and observed value is $x^2=6.447$ which is less than $x_{0.1}^2=10.6446$ hence, appropriate value of $p$ is $p>0.1$ .

As, $p-$ value is very large fail to reject the null hypothesis $H_0$ .

There is sufficient evidence to indicate the difference in percentage are similar for all the $7$ colors as considered under the $H_0$ .

Conclusion:

Thus, There is sufficient evidence to indicate the difference in percentage are similar for all the $7$ colors as considered under the $H_0$ .