4. now form a frequency distribution for the data. add more rows if needed. (3 points)\nhours of sleep (x)…

4. now form a frequency distribution for the data. add more rows if needed. (3 points)\nhours of sleep (x) frequency (f)\n4 2\n5 3\n6 7\n7 5\n8 8\n9 3\n10 1\n11 0\ncalculate probabilities\n12. find the following probabilities. (3 points)\na. p(x < 8) = \nb. p(x > 8) =\n13. if three individuals are selected from the sample, without replacement, what is the probability that they all sleep less than 8 hours? (3 points)\n14. if three individuals are selected from the sample, without replacement, what is the probability that at least one of them sleeps less than 8 hours? (4 points)

4. now form a frequency distribution for the data. add more rows if needed. (3 points)\nhours of sleep (x) frequency (f)\n4 2\n5 3\n6 7\n7 5\n8 8\n9 3\n10 1\n11 0\ncalculate probabilities\n12. find the following probabilities. (3 points)\na. p(x < 8) = \nb. p(x > 8) =\n13. if three individuals are selected from the sample, without replacement, what is the probability that they all sleep less than 8 hours? (3 points)\n14. if three individuals are selected from the sample, without replacement, what is the probability that at least one of them sleeps less than 8 hours? (4 points)

Answer

Explanation:

Step1: Calculate total frequency

Sum all frequencies: $2 + 3+7 + 5+8+3 + 1+0=29$

Step2: Calculate $P(x < 8)$

Sum frequencies of $x = 4,5,6,7$: $2+3 + 7+5=17$. Then $P(x < 8)=\frac{17}{29}$

Step3: Calculate $P(x>8)$

Sum frequencies of $x = 9,10,11$: $3 + 1+0=4$. Then $P(x>8)=\frac{4}{29}$

Step4: Calculate probability for question 13

Use combination formula. Number of ways to choose 3 from total 29 is $C(29,3)=\frac{29!}{3!(29 - 3)!}=\frac{29\times28\times27}{3\times2\times1}=3654$. Number of ways to choose 3 from 17 (less - than - 8 group) is $C(17,3)=\frac{17!}{3!(17 - 3)!}=\frac{17\times16\times15}{3\times2\times1}=680$. Probability is $\frac{C(17,3)}{C(29,3)}=\frac{680}{3654}=\frac{340}{1827}$

Step5: Calculate probability for question 14

First, find probability that none sleeps less than 8 hours. Number of ways to choose 3 from $29 - 17=12$ (8 - or - more group) is $C(12,3)=\frac{12!}{3!(12 - 3)!}=\frac{12\times11\times10}{3\times2\times1}=220$. Probability that none sleeps less than 8 hours is $\frac{C(12,3)}{C(29,3)}=\frac{220}{3654}=\frac{110}{1827}$. Then probability that at least one sleeps less than 8 hours is $1-\frac{110}{1827}=\frac{1827-110}{1827}=\frac{1717}{1827}$

Answer:

a. $\frac{17}{29}$ b. $\frac{4}{29}$ 13. $\frac{340}{1827}$ 14. $\frac{1717}{1827}$