Question

A sample of $212$commercial businesses was surveyed for recycling one commodity; a commodity here means any one type of recyclable material such as plastic or aluminum. Table $11.41$shows the business categories in the survey, the sample size of each category, and the number of businesses in each category that recycle one commodity. Based on the study, on average half of the businesses were expected to be recycling one commodity. As a result, the last column shows the expected number of businesses in each category that recycle one commodity. At the $5$% significance level, perform a hypothesis test to determine if the observed number of businesses that recycle one commodity follows the uniform distribution of the expected values.

Business Type	Number in class	Observed Number that recycles one commodity	Expected number that recycles one commodity
office	$35$	$19$	$17.5$
Retail/ Wholesale	$48$	$27$	$24$
Food/ Restaurants	$53$	$35$	$26.5$
Manufacturing/ Medical	$52$	$21$	$26$
Hotel/Mixed	$24$	$9$	$12$

Table 11.41

Short answer

H₀ : The observed number of businesses that recycle one commodity follows the uniform distribution of the expected values.

we cannot reject the null hypothesis. , There is not enough evidence to conclude that the distribution of businesses that recycle one commodity fit the expected number that recycles one commodity.

Step-by-step solution

Given Information

Given in the question a sample of $212$and a significance level of$5\%$

We want to test these hypotheses:

H₀ : The observed number of businesses that recycle one commodity follows the uniform distribution of the expected values.

H₁ : The observed number of businesses that recycle one commodity does not follow the uniform distribution of the expected values.

Explanation

Since there are $5$types of business, the number of degrees of freedom is $5-1=4$

$\text{We are using}{\chi }^2\text{distribution. Test statistic is given by}$

${\chi }^2=\frac{(19-17.5{)}^2}{17.5}+\frac{(27-24{)}^2}{24}+$

$+\frac{(35-26.5{)}^2}{26.5}+\frac{(21-26{)}^2}{26}$

$+\frac{(9-12{)}^2}{12}$

$=0.13+0.375+2.73+0.962+0.75$

$=4.947$

Explanation

$\text{Using the applet, we get that p-value is}0.29278\text{:}$