Question

a gym offers four fitness classes every saturday. the two - way frequency table shows one day of class attendance for members under 30 years old and members at least 30 years old.

	cycling	swimming	dance	boxing	total
at least 30 years old	42	33	15	25	115
total	96	60	27	57	240

indicate whether each statement is true or false.
calculator
statement
true
false
the value 0.25 is a marginal relative frequency that represents the proportion of members who attended swimming class.
the value 0.30 is a marginal relative frequency that represents the proportion of members who are at least 30 years old
the value 0.1125 is a joint relative frequency that represents the proportion of members who are at least 30 years old and attended boxing class.
the value 0.225 is a joint relative frequency that represents the proportion of members under 30 years old who attended cycling class

Explanation:

Step1: Analyze first statement

Marginal relative frequency for swimming: Total swimming / Total members = \( \frac{60}{240} = 0.25 \). So first statement is True.

Step2: Analyze second statement

Marginal relative frequency for at least 30: Total at least 30 / Total members = \( \frac{115}{240} \approx 0.479 eq 0.30 \). So second statement is False.

Step3: Analyze third statement

Joint relative frequency (at least 30 and boxing): 25 / 240 ≈ 0.104 ≠ 0.1125. Wait, wait, 25/240 is approximately 0.104, but wait, 15? No, boxing for at least 30 is 25. Wait, 25/240 ≈ 0.104, but 0.1125 is 27/240? No, wait, 25 divided by 240: 25/240 = 5/48 ≈ 0.104. Wait, maybe I miscalculated. Wait, 25/240: 240*0.1125 = 27. So 25≠27. Wait, no, the joint relative frequency is (frequency of category) / total. So for at least 30 and boxing: 25/240 ≈ 0.104, but 0.1125 is 27/240? No, 240*0.1125 = 27. Wait, maybe the statement is about dance? No, the statement is about boxing. Wait, maybe I made a mistake. Wait, 25/240 = 0.104, 0.1125 is 27/240? No, 27 is total dance. Wait, no, the third statement: "the proportion of members who are at least 30 years old and attended boxing class" is 25/240 ≈ 0.104, which is not 0.1125. Wait, but 15/240 = 0.0625, 42/240=0.175, 33/240=0.1375. Wait, maybe I messed up. Wait, 25 divided by 240: 25 ÷ 240 = 0.104166... So 0.1125 is 27/240? 27÷240=0.1125. Oh! Wait, no, the joint relative frequency for at least 30 and dance is 15/240=0.0625. Wait, no, 27 is total dance. Wait, no, the third statement is about boxing. Wait, maybe the user made a typo? Wait, no, the third statement: "the proportion of members who are at least 30 years old and attended boxing class" is 25/240 ≈ 0.104, but 0.1125 is 27/240. Wait, 27 is total dance. Wait, maybe I miscalculated. Wait, 240 * 0.1125 = 27. So 27 is total dance. So the joint relative frequency for dance total is 27/240=0.1125, but the statement is about boxing. So the third statement is False? Wait, no, maybe I made a mistake. Wait, let's recalculate:

First statement: Swimming total is 60, 60/240=0.25. Correct. So True.

Second statement: At least 30 total is 115, 115/240≈0.479≠0.30. False.

Third statement: At least 30 and boxing: 25/240≈0.104≠0.1125. Wait, but 25/240=0.104, 0.1125 is 27/240. 27 is total dance. So the third statement is False? Wait, no, maybe the statement is about dance? No, the statement says boxing. So False. Wait, but maybe I made a mistake. Wait, 25 divided by 240: 25/240 = 5/48 ≈ 0.104, 0.1125 is 9/80 = 0.1125. 9/80 *240=27. So 27 is total dance. So the joint relative frequency for dance is 27/240=0.1125, but that's marginal. Wait, no, joint relative frequency is for a cell. So the cell for at least 30 and boxing is 25, so 25/240≈0.104. So third statement is False.

Fourth statement: Under 30 and cycling: 54/240=0.225. Yes! 54 divided by 240: 54÷240=0.225. So fourth statement is True.

Wait, let's recheck:

1. Swimming total: 60. Total members:240. 60/240=0.25. True.

2. At least 30 total:115. 115/240≈0.479≠0.30. False.

3. At least 30 and boxing:25. 25/240≈0.104≠0.1125. False. Wait, but 0.1125*240=27. 27 is total dance. So maybe the statement was about dance? No, the statement is about boxing. So False.

4. Under 30 and cycling:54. 54/240=0.225. True.

So:

First statement: True

Second: False

Third: False

Fourth: True

But let's confirm each:

1. Marginal relative frequency for swimming: total swimming (60) / total (240) = 0.25. Correct. True.

2. Marginal relative frequency for at least 30: total at least 30 (115) / 240 ≈ 0.479. 0.30 would be 72/240. So False.

3. Joint relative frequency (at least 30 and boxing): 25/240 ≈ 0.104. 0.1125 is 27/240 (total dance). So False.

4. Joint relative frequency (under 30 and cycling):54/240=0.225. Correct. True.

Answer:

1. The value 0.25 is a marginal relative frequency that represents the proportion of members who attended swimming class: True
2. The value 0.30 is a marginal relative frequency that represents the proportion of members who are at least 30 years old: False
3. The value 0.1125 is a joint relative frequency that represents the proportion of members who are at least 30 years old and attended boxing class: False
4. The value 0.225 is a joint relative frequency that represents the proportion of members under 30 years old who attended cycling class: True

(For each statement, mark True/False as above)