Question

managers of a theme park conducted a study of their customers for the day. the study showed that 60% of the customers were over the age of 21. the study also showed that 40% of the customers over the age of 21 had a season pass to the theme park, while 30% of customers aged 21 or younger had a season pass. based on the given information, which of the following relative frequency tables can be constructed? four relative frequency tables are shown with rows: season pass, no season pass, total; columns: over 21, 21 or younger, total, with various percentage values in each cell

Explanation:

Step1: Analyze Over 21 Total

60% of customers are over 21, so the "Over 21" row total should be 60%, "21 or Younger" 40%.

Step2: Analyze Over 21 Season Pass

40% of over - 21 customers have season passes. 40% of 60% is \(0.4\times0.6 = 24\%\) (season pass, over 21). So over - 21, no season pass is \(60\% - 24\%=36\%\).

Step3: Analyze 21 or Younger Season Pass

30% of 21 or younger customers have season passes. 30% of 40% is \(0.3\times0.4 = 12\%\)? Wait, no, wait the problem says 30% of customers aged 21 or younger had a season pass. Wait, let's re - check. Wait, the problem says: 40% of over 21 had season pass, 30% of 21 or younger had season pass.

Wait, total over 21: 60% of all customers. So over 21, season pass: \(0.4\times0.6 = 24\%\) of total customers. Over 21, no season pass: \(0.6 - 0.24=0.36 = 36\%\) of total customers.

21 or younger: 40% of all customers. 30% of them have season pass: \(0.3\times0.4 = 12\%\) of total customers. 21 or younger, no season pass: \(0.4 - 0.12 = 0.28=28\%\) of total customers.

Now let's check the tables. Let's look at the third table (from left? Wait, let's check the percentages.

Wait the third table (assuming the four tables: let's check the first column "Season Pass" and rows "Over 21", "21 or Younger", "Total".

Wait the third table (the one with Season Pass: 24% (Over 21), 12% (21 or Younger)? No, wait no. Wait 30% of 21 or younger: 21 or younger is 40% of total, so 30% of 40% is 12%? Wait no, maybe I misread. Wait the problem says "30% of customers aged 21 or younger had a season pass". So if total customers: 100%, over 21: 60%, 21 or younger: 40%.

Season pass, over 21: 40% of 60% = 24% of total.

Season pass, 21 or younger: 30% of 40% = 12% of total.

No season pass, over 21: 60% - 24% = 36% of total.

No season pass, 21 or younger: 40% - 12% = 28% of total.

Now let's check the tables. Let's look at the third table (the one with Season Pass: 24% (Over 21), 12%? No, wait the third table (the third from left) has Season Pass: 24% (Over 21), 12%? No, wait the third table (let's see the numbers):

Wait the third table (assuming the four tables, the third one) has:

Season Pass: Over 21:24%, 21 or Younger:12%? No, wait no, wait the third table (the one with Total row: 60% (Over 21), 40% (21 or Younger), and Season Pass: 24% (Over 21), 12%? No, wait no, let's calculate the totals for season pass and no season pass.

Total season pass: 24% (over 21) + 12% (21 or younger) = 36% of total.

Total no season pass: 36% (over 21) + 28% (21 or younger) = 64% of total.

Now let's check the tables. The third table (the one with Season Pass total 36% and No Season Pass total 64%) matches. Let's verify:

Over 21: 24% (season pass) + 36% (no season pass) = 60% (correct, since over 21 is 60% of total).

21 or younger: 12% (season pass) + 28% (no season pass) = 40% (correct, 21 or younger is 40% of total).

Season pass total: 24% + 12% = 36% (wait, but the problem says 40% of over 21 and 30% of 21 or younger. Wait, maybe I made a mistake. Wait, no: 40% of over 21 (which is 60% of total) is 0.4*0.6 = 24% of total. 30% of 21 or younger (40% of total) is 0.3*0.4 = 12% of total. Then season pass total is 24% + 12% = 36% of total. No season pass: over 21: 60% - 24% = 36% of total; 21 or younger: 40% - 12% = 28% of total; no season pass total: 36%+28% = 64% of total. So the table with Season Pass: 24% (Over 21), 12% (21 or Younger), Total 36%; No Season Pass: 36% (Over 21), 28% (21 or Younger), Total 64%; and Total rows: Over 21 60%, 21 or Younger 40%, Total 100% is the correct one. Looking at the given tables, the third table (the one with Season Pass: 24%, 12%? Wait no, wait the user's image: let's re - check the numbers. Wait the third table (from left) has Season Pass: Over 21 24%, 21 or Younger 12%? No, wait the numbers in the third table (as per the image) are: Season Pass: Over 21 24%, 21 or Younger 12%? Wait no, the user's image shows four tables. Let's assume the third table (the one with Season Pass: 24% (Over 21), 12%? No, wait the correct calculations lead to the table where Over 21, Season Pass is 24%, Over 21, No Season Pass is 36%, 21 or Younger, Season Pass is 12%? Wait no, no, the problem says 30% of 21 or younger had season pass. Wait, 30% of 21 or younger (40% of total) is 0.3*0.4 = 12%? No, 30% of 40% is 12%? Wait 40% * 30% = 12% of total customers. Yes. And 40% of 60% (over 21) is 24% of total customers. So the correct relative frequency table is the one where:

- Season Pass: Over 21: 24%, 21 or Younger: 12% (wait, no, that can't be, because 30% of 21 or younger is 12% of total? Wait, no, 21 or younger is 40% of total. 30% of 40% is 12% of total. Yes. And 40% of 60% is 24% of total. Then no season pass: over 21: 60% - 24% = 36% of total; 21 or younger: 40% - 12% = 28% of total. So the table with these percentages is the correct one.

Answer:

The third relative frequency table (the one with Season Pass: 24% (Over 21), 12% (21 or Younger), Total 36%; No Season Pass: 36% (Over 21), 28% (21 or Younger), Total 64%; and Total rows: Over 21 60%, 21 or Younger 40%, Total 100%)