A survey was designed to study how business operations vary by size. Companies were classified as small, medium, or large. Questionnaires were sent to 200 randomly selected businesses of each size. Because not all questionnaires are returned, researchers decided to investigate the relationship between the response rate and the size of the business. The data are given in the following two-way table.

Which of the following conclusions seems to be supported by the data

a. There are more small companies than large companies in the survey.

b. Small companies appear to have a higher response rate than medium or big companies.

c. Exactly the same number of companies responded as didn’t respond.

d. Overall, more than half of companies responded to the survey.

e. If we combined the medium and large companies, then their response rate would be equal to that of the small companies.

a. There are more small companies than large companies in the survey.

b. Small companies appear to have a higher response rate than medium or big companies.

c. Exactly the same number of companies responded as didn’t respond.

d. Overall, more than half of companies responded to the survey.

e. If we combined the medium and large companies, then their response rate would be equal to that of the small companies.

(a) The data do not support this conclusion because 200 questionnaires were sent to each type of business, resulting in 200 small businesses and 200 large businesses in the survey.

(b) The number of responses for small businesses is given in the table's row with "Yes" and column with "Small," so there were 125 responses for small businesses.

As a result, 125 people responded to the 200 questionnaires.

$$\begin{gathered} \frac{125}{200}=\frac{5}{8} \\ \frac{5}{8}=0.625 \\ 0.625=62.5\% \end{gathered}$$

As a result, we can see that the response rate for small businesses is 62.5 percent.

The number of responses for medium companies is given in the table's row with "Yes" and column with "Medium," so there were 81 responses for medium companies.

On the 200 questionnaires, 81 responses were received.

$$\begin{gathered} \frac{81}{200}=0.405 \\ 0.405=40.5\% \end{gathered}$$

As a result, we can see that the response rate for medium-sized businesses is 40.5 percent.

The number of responses for large companies is given in the table's row with "Yes" and column with "Large," so there were 40 responses for large companies.

On the 200 questionnaires, 81 responses were received.

$$\begin{gathered} \frac{40}{200}=\frac{1}{5} \\ \frac{1}{5}=0.20 \\ 0.20=20\% \end{gathered}$$

As a result, we can see that the response rate for large corporations is 20%.

We then observe that the response rate for small businesses (62.5 percent) is higher than the response rate for medium (40.5 percent) and larger businesses (40.5 percent) (20 percent ). As a result, the data supports the conclusion.

(c) The number of responses for each type of company is given in the table's row "Yes," so we can conclude that 125 +81 +40 = 246 companies responded.

The number of non-responses for each type of company is given in the table's row "No," so we can conclude that 75+ 119 +160 = 354 companies did not respond.

Because 246 and 354 are not the same, a different number of companies responded than did not respond, and thus the conclusion is not supported by the data.

(d) The number of responses for each type of company is given in the table's row "Yes," so we can conclude that 125 +81 +40=246 companies responded.

Each type of business received 200 questionnaires, so there are 200 small businesses in the survey, 200 medium businesses in the survey, and 200 large businesses in the survey. In total, 200+ 200+200=600 questionnaires were sent.

As a result, 246 of the 600 companies responded.

$$\begin{gathered} \frac{246}{600}=\frac{41}{100} \\ \frac{41}{100}=0.41 \\ 0.41=41\% \end{gathered}$$

Because 41% is less than 50%, less than half of the companies responded to the survey, and thus the conclusion is not supported by the data.

(e) The data do not support the conclusion, because we concluded that small businesses have a higher response rate than medium businesses, and small businesses have a higher response rate than large businesses (b). Small company response rates will then be higher than medium and large company response rates (when the two groups are combined), because medium and large company response rates will be in between their two individual response rates.

As a result:

(b) Small companies have larger response rate than medium/large companies.