QUESTION IMAGE
Question
a random group of high school and college students were asked if they study with or without listening to music. the frequency table below shows the results of the poll.
| music | no music | total | |
|---|---|---|---|
| college | 85 | 51 | 136 |
| total | 153 | 116 | 269 |
which statements are true? check all that apply.
- the table shows conditional relative frequencies by column.
- the table shows conditional relative frequencies by row.
- the conditional relative frequency that someone is in high school, given that the person prefers to study without music, is about 0.56.
- if someone prefers to study with music, the probability the person is in college is about 63%.
- if someone prefers to study with music, the probability the person is in hs is about 44%.
Brief Explanations
- Conditional Relative Frequencies by Column/Row: The table has total row and column, but to check conditional relative frequencies by column, we divide each cell by column total; by row, by row total. The given table is a frequency table (not conditional relative yet), so first two statements are false.
- High School | No Music: Conditional relative frequency = High School (No Music) / Total (No Music) = \( \frac{65}{116} \approx 0.56 \)? Wait, no: High School (No Music) is 65, Total (No Music) is 116. \( \frac{65}{116} \approx 0.56 \)? Wait, 65/116 ≈ 0.56? Wait 65÷116≈0.56? Wait 116×0.56≈65, yes. Wait, but wait: the statement is "someone is in high school, given that the person prefers to study without music"—so it's P(High School | No Music) = \( \frac{\text{High School and No Music}}{\text{Total No Music}} = \frac{65}{116} \approx 0.56 \). Wait, but let's check the third statement. Wait, no: 65/116 ≈ 0.56? 65÷116≈0.56? 116×0.56=64.96≈65, yes. Wait, but let's check other statements.
- College | Music: P(College | Music) = \( \frac{\text{College and Music}}{\text{Total Music}} = \frac{85}{153} \approx 0.555... \approx 56\% \)? Wait no, 85/153 ≈ 0.555, which is ~56%, but the fourth statement says "about 63%". Wait, no: 85/153 ≈ 0.555 (55.5%), but wait 153 is total music. Wait, 85/153 ≈ 0.555, which is ~56%, but the fourth statement says "about 63%". Wait, maybe I miscalculated. Wait 85/153: divide numerator and denominator by 17: 5/9 ≈ 0.555. So fourth statement is wrong? Wait no, wait the fifth statement: "If someone prefers to study with music, the probability the person is in HS is about 44%". HS | Music: 68/153 ≈ 0.444, which is ~44%. So fifth statement is correct. Wait, let's re-express:
- Statement 1: The table is a frequency table (not conditional relative by column). Conditional relative by column would have each column cell divided by column total (e.g., 68/153, 65/116, etc.). So statement 1 is false.
- Statement 2: Similarly, conditional relative by row would be each row cell divided by row total (e.g., 68/133, 65/133, etc.). The table is frequency, not conditional, so statement 2 is false.
- Statement 3: P(High School | No Music) = 65/116 ≈ 0.56 (since 65÷116≈0.56). Wait, 65/116 ≈ 0.56? 116×0.56=64.96≈65, yes. So statement 3 is true? Wait, but wait 65/116 is approximately 0.56 (56%), so yes.
- Statement 4: P(College | Music) = 85/153 ≈ 0.555 (55.5%), not 63%. So statement 4 is false.
- Statement 5: P(High School | Music) = 68/153 ≈ 0.444 (44.4%), so "about 44%" is true.
Wait, but let's recheck statement 3: "The conditional relative frequency that someone is in high school, given that the person prefers to study without music, is about 0.56." So that's 65 (High School, No Music) divided by 116 (Total No Music) = 65/116 ≈ 0.56 (since 65 ÷ 116 ≈ 0.5603). Yes. So statement 3 is true. Statement 5: 68/153 ≈ 0.444, so "about 44%" is true. Wait, but maybe I made a mistake. Let's recalculate:
- Statement 3: 65 (HS, No Music) / 116 (Total No Music) = 65 ÷ 116 ≈ 0.5603 → ~0.56. True.
- Statement 5: 68 (HS, Music) / 153 (Total Music) = 68 ÷ 153 ≈ 0.4444 → ~44%. True.
Wait, but the options are check all that apply. Let's confirm:
- Statement 1: False (table is frequency, not conditional by column).
- Statement 2: False (table is frequency, not conditional by row).
- Statement 3: True (65/116 ≈ 0.56).
- Statement 4: False (85/153 ≈ 0.555, not 63%).
- Statement 5: True (68/153 ≈ 0.444, ~44%).
Wait, but maybe I messed up statement 3. Wait, "given that the person prefers to study without mu…
Snap & solve any problem in the app
Get step-by-step solutions on Sovi AI
Photo-based solutions with guided steps
Explore more problems and detailed explanations
- The conditional relative frequency that someone is in high school, given that the person prefers to study without music, is about 0.56.
- If someone prefers to study with music, the probability the person is in HS is about 44%.
(In boxed form with options as per the original:
- The conditional relative frequency that someone is in high school, given that the person prefers to study without music, is about 0.56.
- If someone prefers to study with music, the probability the person is in HS is about 44%.
But in the original options, they are:
- The conditional relative frequency that someone is in high school, given that the person prefers to study without music, is about 0.56. (Third option)
- If someone prefers to study with music, the probability the person is in HS is about 44%. (Fifth option)
So the answer is:
- The conditional relative frequency that someone is in high school, given that the person prefers to study without music, is about 0.56.
- If someone prefers to study with music, the probability the person is in HS is about 44%.