# Analysis of Variance Test

It has been reported that 10.3% of U.S. households do not own a vehicle, with 34.2% owning 1 vehicle, 38.4% owning 2 vehicles, and 17.1% owning 3 or more vehicles. The data for a random sample of 100 households in a resort community are summarized in the frequency distribution below. At the 0.05 level of significance, can we reject the possibility that the vehicle-ownership distribution in this community differs from that of the nation as a whole?

# of Vehicles Owned # of Households

0 20

1 35

2 23

3 or more 22

100

13.34 A research organization has collected the following data on household size and telephone ownership for 200 U.S. households. At the 0.05 level, are the two variables independent? Based on the chi-square table, what is the most accurate statement that can be made about the p-value for the test?

Telephones Owned

≤ 2 ≥ 3

Person ≤ 2 49 18 13 80

In the 3-4 40 27 21 88

Household ≥5 11 13 8 32

100 58 42 200

15.10 The following data represent x = boat sales and y = boat trailer sales from 1995 through 2000.

Year Boat Sales Boat Trailer Sales

Year (Thousands) (Thousands)

1995 649 207

1996 619 194

1997 596 181

1998 576 174

1999 585 168

2000 574 159

a. Determine the least-squares regression line and interpret its slope.

b. Estimate, for a year during which 500,000 boats are sold, the number of boat trailers that would be sold.

c. What reasons might explain why the number of boat trailers sold per year is less than the number of boats sold

15.38 The following data show U.S. production of motor vehicles versus tons of domestic steel shipped for motor vehicle manufacture.

X = U. S. Production y = Tons of

Year of Motor Vehicles Domestic Steel

2000 12.83 million 16.06 million

2001 11.52 14.06

2002 12.33 14.00

2003 12.15 15.88

2004 12.02 13.86

a. Determine the lest-squares regression line and calculate r.

b. What proportion of the variability in steel shipments for motor vehicles is explained by the regression equation?

c. During a year in which U.S. production of motor vehicles is 12.0 million, what would be the prediction for the number of tons of domestic steel used for vehicle production?

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#### Solution Preview

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Problem 1.

We are given that the distribution of the vehicle- ownership of the nation as a whole as follows

No. of vehicle owned Percentage

0 10.3

1 34.2

2 38.4

3 or more 17.1

Total 100

The distribution of the sample of 100 households

No. of vehicle owned Frequency

0 20.0

1 35.0

2 23.0

3 or more 22.0

Total 100

In order to examine whether there is significant difference in the distribution of vehicle-ownership of the sampled community and nation as a whole, we have to test the 'Goodness of fit" of the sample distribution. For this we use Chi-Square test for Goodness of fit. The test procedure is as follows.

The null hypothesis to be tested is H0: There is no significant difference between the distributions of the sample and population.

Alternative hypothesis H1: There is significant difference between the distributions.

Let Oi denotes the observed frequency (i.e., frequency of the ith class of the sample) and Ei denotes the theoretical frequency (given percentage of the ith class of the distribution of the nation).

The test statistic is given by , where n denotes the number of classes (4 in the present problem). This statistic follows Chi-square distribution with n-1(=3) degrees of freedom. Hence, the p value of the test can be taken from the chi-square tables with 3 degrees of freedom.

The Chi-square statistic can be calculated as follows

Class Oi Ei

0 20 10.3 94.09 9.13

1 35 34.2 0.64 0.02

2 23 38.4 237.16 6.18

3 or more 22 17.1 24.01 1.40

16.73

From Chi-square tables with 3 d.f we get the p-value as 0.0008.

(Note that the p value can be obtained by using CHIDIST(16.73,3) ...

#### Solution Summary

An analysis of variance test is determined in the solution.