Questions: Nikita knows the following information about her food club that has 11 members: - 3 members like neither fruit nor vegetables. - 4 members like fruit but not vegetables. - 5 members in total like fruit. Can you help Nikita organize the results into a two-way frequency table? Like fruit Do not like fruit ------------------------------------------------------- Like vegetables 1 3 Do not like vegetables 4 3

Nikita knows the following information about her food club that has 11 members:
- 3 members like neither fruit nor vegetables.
- 4 members like fruit but not vegetables.
- 5 members in total like fruit.

Can you help Nikita organize the results into a two-way frequency table?

Like fruit Do not like fruit
-------------------------------------------------------
Like vegetables 1 3
Do not like vegetables 4
3

Transcript text: Nikita knows the following information about her food club that has 11 members: - 3 members like neither fruit nor vegetables. - 4 members like fruit but not vegetables. - 5 members in total like fruit. Can you help Nikita organize the results into a two-way frequency table? \begin{tabular}{|l|l|l|} \hline & Like fruit & Do not like fruit \\ \hline Like vegetables & 1 & 3 \\ \hline Do not like vegetables & 4 & \\ \cline { 2 - 3 } & & 3 \\ \hline \end{tabular}

Solution

Solution Steps

To solve this problem, we need to fill in the missing values in the two-way frequency table based on the given information. We know the total number of members and the number of members who like neither fruit nor vegetables, those who like fruit but not vegetables, and the total number who like fruit. Using this information, we can deduce the missing values by considering the relationships between the categories.

Step 1: Given Information

We have a food club with \( n = 11 \) members. The following information is provided:

\( n_{\text{neither}} = 3 \) (members who like neither fruit nor vegetables)
\( n_{\text{fruit only}} = 4 \) (members who like fruit but not vegetables)
\( n_{\text{total fruit}} = 5 \) (total members who like fruit)

Step 2: Calculate Members Who Like Both

To find the number of members who like both fruit and vegetables, we can use the equation: \[ n_{\text{both}} = n_{\text{total fruit}} - n_{\text{fruit only}} = 5 - 4 = 1 \]

Step 3: Calculate Members Who Like Vegetables Only

Next, we calculate the number of members who like vegetables but not fruit. The total number of members who like vegetables can be found by subtracting those who like neither from the total: \[ n_{\text{vegetables}} = n - n_{\text{neither}} = 11 - 3 = 8 \] Now, we can find the number of members who like vegetables only: \[ n_{\text{vegetables only}} = n_{\text{vegetables}} - n_{\text{both}} = 8 - 1 = 7 \] Since we already know that \( n_{\text{fruit only}} = 4 \), we can find the number of members who do not like vegetables: \[ n_{\text{do not like vegetables}} = n_{\text{neither}} + n_{\text{fruit only}} = 3 + 4 = 7 \]

Step 4: Organize Results into a Two-Way Frequency Table

Now we can summarize the results in a two-way frequency table:

\[ \begin{array}{|l|l|l|} \hline & \text{Like fruit} & \text{Do not like fruit} \\ \hline \text{Like vegetables} & 1 & 3 \\ \hline \text{Do not like vegetables} & 4 & 3 \\ \hline \end{array} \]

Final Answer

The completed two-way frequency table is as follows:

\[ \begin{array}{|l|l|l|} \hline & \text{Like fruit} & \text{Do not like fruit} \\ \hline \text{Like vegetables} & 1 & 3 \\ \hline \text{Do not like vegetables} & 4 & 3 \\ \hline \end{array} \]

Thus, the final answer is boxed as follows: \[ \boxed{\text{Completed table as shown above}} \]

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