Question

question 2 (multiple choice worth 2 points) (02.01h mc) students (ages 10–18) were surveyed to choose a favorite free-time activity. | | playing sports | watching movies/tv | total | middle school | | 16 | 21 | high school | 11 | | | total | | 19 | | which category has the lowest marginal frequency? middle school students who chose playing sports high school students who chose watching movie/tv students that are in high school students who chose playing sports

Explanation:

Step1: Complete the table

First, find the number of middle school students who chose playing sports: Middle School total is 21, and watching movies/TV is 16, so playing sports for middle school is \(21 - 16 = 5\).

Then, find the number of high school students who chose watching movies/TV: Total for watching movies/TV is 19, middle school is 16, so high school is \(19 - 16 = 3\).

High school total: Playing sports is 11, watching movies/TV is 3, so total is \(11 + 3 = 14\)? Wait, no, wait. Wait, middle school total is 21, high school total? Wait, maybe I made a mistake. Wait, let's re - calculate.

Middle School: Playing Sports + Watching Movies/TV = Total. So Playing Sports (Middle School) = \(21 - 16 = 5\).

Watching Movies/TV total is 19, so High School Watching Movies/TV = \(19 - 16 = 3\).

High School total: Playing Sports (11) + Watching Movies/TV (3) = 14? Wait, but maybe the total of middle and high school? Wait, no, let's find the totals for each row and column.

Row totals: Middle School: 21, High School: \(11 + 3 = 14\)? Wait, no, maybe the total of all students is \(21+14 = 35\)? Wait, no, let's do column totals.

Playing Sports total: Middle School (5) + High School (11) = 16.

Watching Movies/TV total: 19 (given).

Now, let's find the marginal frequencies:

1. Middle school students who chose playing sports: 5 (from \(21 - 16\)).

2. High school students who chose watching movie/TV: 3 (from \(19 - 16\)).

3. Students that are in high school: High School total is \(11 + 3 = 14\) (since playing sports is 11 and watching is 3).

4. Students who chose playing sports: \(5 + 11 = 16\).

Now compare the values: 3, 5, 14, 16. The smallest is 3, which is "High school students who chose watching movie/TV"? Wait, no, wait the options:

Wait the options are:

- Middle school students who chose playing sports: 5

- High school students who chose watching movie/TV: 3

- Students that are in high school: Let's recalculate high school total. Wait, middle school total is 21, watching movies/TV total is 19, playing sports total is \(5 + 11 = 16\). Total students: \(21 +\) high school total. High school playing sports is 11, watching is \(19 - 16 = 3\), so high school total is \(11 + 3 = 14\). So students in high school: 14.

Students who chose playing sports: \(5 + 11 = 16\).

Now, the values: 5 (middle school sports), 3 (high school movies), 14 (high school total), 16 (sports total). So the lowest is 3, which is "High school students who chose watching movie/TV"? Wait, but let's check again.

Wait, maybe I made a mistake in high school total. Wait, middle school total is 21, watching movies/TV for middle school is 16, so playing sports middle school is 5. High school playing sports is 11, so playing sports total is \(5 + 11 = 16\). Watching movies/TV total is 19, so high school watching is \(19 - 16 = 3\). High school total is \(11 + 3 = 14\). So the marginal frequencies:

- Middle school sports: 5

- High school movies: 3

- High school total: 14

- Sports total: 16

So the lowest is 3, which is "High school students who chose watching movie/TV"? Wait, but let's check the options again.

Wait the options are:

1. Middle school students who chose playing sports: 5

2. High school students who chose watching movie/TV: 3

3. Students that are in high school: 14 (high school total)

4. Students who chose playing sports: 16 (playing sports total)

So the lowest is 3, which is the second option. But wait, maybe I made a mistake. Wait, let's re - do the table:

| | Playing Sports | Watching Movies/TV | Total |
|----------------|----------------|--------------------|-------|
| Middle School | \(21 - 16 = 5\) | 16 | 21 |
| High School | 11 | \(19 - 16 = 3\) | \(11 + 3 = 14\) |
| Total | \(5 + 11 = 16\) | 19 | \(21+14 = 35\) |

Now, marginal frequencies:

- For "Middle school students who chose playing sports": 5 (cell frequency, but marginal frequency for the row or column? Wait, marginal frequency is the total of a row or column. Wait, maybe I confused joint and marginal frequencies.

Wait, marginal frequency is the total of a row or a column. So:

- Row marginal frequencies: Middle School total = 21, High School total = 14.

- Column marginal frequencies: Playing Sports total = 16, Watching Movies/TV total = 19.

- Joint frequencies: Middle - Sports: 5, Middle - Movies:16, High - Sports:11, High - Movies:3.

Wait, the question is about marginal frequency. So:

1. "Middle school students who chose playing sports" is a joint frequency (5), not marginal. Wait, maybe the question has a mis - term, but let's check the options.

Wait the options:

- Middle school students who chose playing sports: 5 (joint)

- High school students who chose watching movie/TV: 3 (joint)

- Students that are in high school: High School row total = 14 (marginal)

- Students who chose playing sports: Playing Sports column total = 16 (marginal)

Wait, maybe the question is using "marginal frequency" to include joint frequencies? No, marginal frequency is the total of a row or column. But maybe in the question's context, they consider the totals of each category (row or column) as marginal.

Wait, let's re - evaluate:

Option 1: Middle school students who chose playing sports: 5 (joint)

Option 2: High school students who chose watching movie/TV: 3 (joint)

Option 3: Students that are in high school: High School total = 14 (marginal)

Option 4: Students who chose playing sports: Playing Sports total = 16 (marginal)

Now, among 5, 3, 14, 16, the smallest is 3, which is option 2: "High school students who chose watching movie/TV". But wait, maybe I made a mistake in high school watching movies. Wait, watching movies total is 19, middle school is 16, so high school is \(19 - 16 = 3\), correct. High school playing sports is 11, so high school total is \(11 + 3 = 14\), correct. Middle school playing sports is \(21 - 16 = 5\), correct. Playing sports total is \(5 + 11 = 16\), correct.

So the lowest value is 3, which is "High school students who chose watching movie/TV".

Answer:

High school students who chose watching movie/TV