Book edition OER 2018

Author(s) Barbara Illowsky, Susan Dean

Pages 902 pages

ISBN 9781938168208

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Short Answer

Do families and singles have the same distribution of cars? Use a level of significance of $0.05$ . Suppose that $100$ randomly selected families and $200$ randomly selected singles were asked what type of car they drove: sport, sedan, hatchback, truck, van/SUV. The results are shown in Table $11.20 .$ Do families and singles have the same distribution of cars? Test at a level of significance of $0.05$ .

The alpha value has been set at $0.05$ . Because $p - v a l u e < α$ , the null hypothesis, $H_{0}$ , will be rejected. As a result, the null hypothesis is rejected, whereas the alternative hypothesis is accepted. As a result, there is sufficient data to establish that the distribution of cars among families and singles is not equal.

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Given information

Given in the question that, use a level of significance of $0.05$ . Suppose that $100$ randomly selected families and $200$ randomly selected singles were asked what type of car they drove: sport, sedan, hatchback, truck, van/SUV. The results are shown in Table 11.20. We need to test at a $0.05$ significance level that whether the families and singles have the same distribution of cars.

Explanation

The null hypothesis is as follows:

$H_{0}$ : Car distribution is the same for families and singles.

The alternate hypothesis is as follows:

$H_{a}$ : Car ownership is not evenly distributed among families and singles.

The following formula can be used to calculate degrees of freedom:

$d f = ($ number of columns $- 1)$

$\Rightarrow d f = (5 - 1)$

$\Rightarrow d f = 4$

The table of observed values has already been provided. Let's now use the formula below to determine the predicted frequencies:

$E = \frac{(row total) (column total)}{overall total}$

Let's use Excel to determine the expected (E) values, as shown below.

Independence test statistic

The independence test statistic is provided below;

$= \sum_{i \times j} = \frac{(O - E)^{2}}{E}$ is a test statistic.

Apply the formula $= ((B 4 - B 11)^{2}) / B 11$ to cell B17 and drag the same formula up to cell F18 to get $\frac{(O - E)^{2}}{E}$ . After that, add up the totals of the columns and rows. The following is a table of test statistics:

As a result, the test statistic is $62.912$

In Excel, the p-value can be determined using the CHIDIST () formula, as illustrated below:

As a result, the $p$ value is zero.

Family Size	Sub & Compact	Mid-size	Full-size	Van & Truck
$1$	$20$	$35$	$40$	$35$
$2$	$20$	$50$	$70$	$80$
$3 - 4$	$20$	$50$	$100$	$90$
$5 +$	$20$	$30$	$70$	$70$

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