8.01 Two way frequency tables

Describe the group of people which has exactly 401 people in it.

We should find 401 in the table and look at the row and column headings to identify what the number represents.

The number 401 lies in the row for people who chose meat and in the column for people who did not add spice. This means that there were 401 people who chose meat chili and did not add spice.

Identify whether the number of people who chose vegetarian chili is a marginal or joint frequency and state its value.

It is not specified whether this group added extra spice or not, so we are looking at all people who chose the vegetarian chili.

Since we are looking at the total number of people who chose vegetarian chili, this is a marginal frequency.

There are 168 people who chose the vegetarian chili.

Calculate the total number of people who ate chili at Chili Fest.

We can find this total in a variety of ways:

• Add all the joint frequencies
• Add the marginal frequencies for the type of chili
• Add the marginal frequencies whether they added spice or not

Let's add the marginal frequencies for the type of chili.

\text{Meat}+\text{Vegetarian}=732+168=900

There were 900 people who ate chili at Chili Fest.

We can check this by adding up all of the joint frequencies.

331+401+125+43=900

Organize the data into a two-way frequency table based on the given information about coreopsis plants.

First we should decide what our table's row and column headings should be. The two characteristics we are considering are whether or not the plants flowered and what type of soil they were planted in.

Then we can fill in the given information. Use the fact that the total number of all plants is 1000 and that the marginal frequencies are the sums of the rows and columns to calculate the missing frequencies.

Based on the information we have been given, we can already determine:

• Marginal frequency for number that flowered: 1000-136=864
• Joint frequency of peat soil/did not flower: 136-30=106

Based on the new calculated information, we can determine:

• Marginal frequency for number planted in peat soil: 394+106=500
• Then the marginal frequency for number planted in sandy soil: 1000-500=500
• Joint frequency for plants that flowered in sandy soil: 864-394=470

We can always check by adding across each row and down each column to ensure the marginal frequency is the sum of the joint frequencies.

Compare the number of plants that flowered in peat soil to the number that flowered in sandy soil. Explain any conclusions the scientist might make.

Notice that there were 500 plants planted in peat soil and 500 plants planted in sandy soil.

However, there were not the same number of plants that flowered in each type of soil. There were more plants that flowered when planted in sandy soil than in peat soil. We can see this because 470 >394.

The scientists might conclude that coreopsis plants are more likely to flower when planted in sandy soil than when planted in peat soil.

Note that we do not know under what conditions this data was collected. We do not know if this was set up as an experiment to control other variables, such as sunlight and water. Based on the minimal information given, we cannot say "Being planted in sandy soil causes the plants to flower more."

Marvis said because there were a total of 1000 plants observed and 30 plants that were planted in sandy soil did not flower, this means 1000-30=970 were planted in peat soil and did flower. Identify and correct his error.

Review the completed two-way table to determine whether the statement is reasonable.

Marvis seems to have a misconception around how joint frequencies relate to one another. He has stated that \text{sandy and flowered}+\text{peat and not flowered}=\text{Total} while it should be\text{sandy and flowered}+\text{sandy and not flowered}+\text{peat and flowered}+\text{peat and not flowered}=\\ \text{Total}

We need to consider all possible cases, not just opposite cases.

He should have said that because there were a total of 1000 plants observed and 30 plants planted in sandy soil did not flower, this means 1000-30=970 represents the plants that were planted in the peat soil plus the plants that were planted in the sandy soil and flowered.