Question

for a certain class, the following data table summarizes the amount of sleep of students and whether or not they have an a in the class. what is the probability that a student chosen randomly from the class slept 6 hours or less?

	slept more than 6 hours	slept 6 hours or less
does not have an a	2	6

Explanation:

Step1: Find total number of students

To find the total number of students, we sum up all the values in the table. The values are 5 (Has an A, Slept more than 6 hours), 15 (Has an A, Slept 6 hours or less), 2 (Does not have an A, Slept more than 6 hours), and 6 (Does not have an A, Slept 6 hours or less). So total students \(= 5 + 15 + 2 + 6\).
\(5+15 = 20\), \(2 + 6 = 8\), then \(20+8=28\).

Step2: Find number of students who slept 6 hours or less

The number of students who slept 6 hours or less is the sum of those who have an A and slept 6 hours or less, and those who do not have an A and slept 6 hours or less. So that's \(15 + 6\).
\(15+6 = 21\).

Step3: Calculate the probability

Probability is the number of favorable outcomes (students who slept 6 hours or less) divided by the total number of outcomes (total students). So probability \(=\frac{\text{Number of students who slept 6 hours or less}}{\text{Total number of students}}=\frac{21}{28}\).
Simplify \(\frac{21}{28}\) by dividing numerator and denominator by 7, we get \(\frac{3}{4}\)? Wait, no, wait: \(15 + 6 = 21\)? Wait, no, wait the table: "Has an A, Slept 6 hours or less" is 15, "Does not have an A, Slept 6 hours or less" is 6. So 15 + 6 = 21? Wait total students: 5 (Has A, >6) +15 (Has A, ≤6) +2 (No A, >6) +6 (No A, ≤6) = 5+15=20, 2+6=8, 20+8=28. Then number of students who slept 6 hours or less is 15 + 6 = 21? Wait but 15 + 6 is 21, total is 28. Wait but let's check again: 5 (Has A, >6) +15 (Has A, ≤6) =20 (Has A total), 2 (No A, >6) +6 (No A, ≤6)=8 (No A total). 20+8=28. Then students who slept ≤6: 15 (Has A) +6 (No A)=21. Then probability is 21/28. Simplify: divide numerator and denominator by 7: 21÷7=3, 28÷7=4. Wait 21/28=3/4? Wait no, 15+6 is 21? Wait wait, maybe I made a mistake. Wait the table:

• Has an A: Slept more than 6 hours: 5; Slept 6 hours or less:15
• Does not have an A: Slept more than 6 hours:2; Slept 6 hours or less:6

So total students: 5 + 15 + 2 + 6 = 28.

Students who slept 6 hours or less: 15 (Has A) + 6 (No A) = 21.

So probability is 21/28. Simplify: 21 ÷ 7 = 3, 28 ÷7=4. Wait 21/28=3/4? Wait no, 15+6 is 21? Wait 15+6=21, total is 28. So 21/28 reduces to 3/4? Wait no, 21 divided by 28: 21/28 = 3/4? Wait 37=21, 47=28. Yes. Wait but let me check again. Wait maybe I added wrong. Wait 5 (Has A, >6) +15 (Has A, ≤6) =20 (Has A), 2 (No A, >6) +6 (No A, ≤6)=8 (No A). 20+8=28. Students who slept ≤6: 15 (Has A) +6 (No A)=21. So 21/28=3/4? Wait no, 15+6 is 21? Wait 15+6=21, yes. Then 21/28=3/4. Wait but let's check the arithmetic again. 5+15=20, 2+6=8, 20+8=28. 15+6=21. So 21/28=3/4. Wait, but wait, maybe I misread the table. Wait the "Slept 6 hours or less" column: Has an A is 15, Does not have an A is 6. So 15+6=21. Total students 28. So probability is 21/28=3/4? Wait no, 21 divided by 28: 21 ÷ 28 = 0.75, which is 3/4. Wait but let me check again. Wait maybe the numbers are different? Wait the table:

• Has an A: Slept more than 6 hours: 5; Slept 6 hours or less:15

• Does not have an A: Slept more than 6 hours:2; Slept 6 hours or less:6

Yes. So total students: 5+15+2+6=28. Students who slept 6 hours or less:15+6=21. So probability is 21/28=3/4. Wait, but 21 and 28: GCD(21,28)=7. 21÷7=3, 28÷7=4. So 3/4.

Answer:

\(\frac{3}{4}\) (or 0.75)