Introduction to Probability and Statistics
Introduction to Probability and Statistics
14th Edition
ISBN: 9781133103752
Author: Mendenhall, William
Publisher: Cengage Learning
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Chapter 14, Problem 14.54SE

i.

To determine

To identify:

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

i.

Expert Solution
Check Mark

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:

    ColorSilver BlackGrayBlueRed 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 ,

  Total=19+17+17+15+12+12Total=92Total100

Now for adding other column for car colors:

  Others=10092Others=8

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.

ii.

Expert Solution
Check Mark

Answer to Problem 14.54SE

There is sufficient evidence to indicate the difference in percentage are similar for all the 7colors as considered under the H0.

Explanation of Solution

Given:

    ColorSilver BlackGrayBlueRed 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).

  Total=52+43+48+41+32+19Total=235Total250

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

  Others=250235Others=15

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,

  p1,p2,......p6,p7 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:

  H0:p1=19%,p2=17%,p3=17%,p4=15%,p5=12%,p6=12%,p7=8%

Alternative hypothesis:

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

The expected count for the ith color is defined as the Ei

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

By using n and pi , the expected count is determined as,

  Ei=n×piEi=250×pi

When i=1

  E1=n×piE1=250×0.19E1=47.5

When i=2

  E2=n×piE2=250×0.17E2=42.5

When i=3

  E3=n×piE3=250×0.17E3=42.5

When i=4

  E4=n×piE4=250×0.15E4=37.5

When i=5

  E5=n×piE5=250×0.12E5=30

When i=6

  E6=n×piE6=250×0.12E6=30

When i=7

  E7=n×piE7=250×0.08E7=20

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

  x2= ( O i E i ) 2 E i x2= ( 5247.5 )263.24+ ( 4342.5 )242.5+......+ ( 1520 )220x2=6.88729

Thus, observed value is x2=6.88729 .

As number of color are 7

That means: k=7

Then,

  df=k1df=71df=6

The degrees of freedom is 6 .

From chi-square table, degree of freedom is df=6 and observed value is x2=6.447 which is less than x0.12=10.6446 hence, appropriate value of p is p>0.1 .

As, p value is very large fail to reject the null hypothesis H0 .

There is sufficient evidence to indicate the difference in percentage are similar for all the 7 colors as considered under the H0 .

Conclusion:

Thus, There is sufficient evidence to indicate the difference in percentage are similar for all the 7 colors as considered under the H0 .

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