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 6\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 6\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

First, find the sum of all frequencies. $2 + 3+7 + 6+8+3 + 1+0=30$.

Step2: Calculate $P(x < 8)$

Sum the frequencies of $x = 4,5,6,7$. $2+3 + 7+6=18$. Then $P(x < 8)=\frac{18}{30}=\frac{3}{5}$.

Step3: Calculate $P(x>8)$

Sum the frequencies of $x = 9,10,11$. $3 + 1+0=4$. Then $P(x>8)=\frac{4}{30}=\frac{2}{15}$.

Step4: Calculate probability for question 13

The number of individuals who sleep less than 8 hours is 18. The total number of individuals is 30. The probability that the first - selected individual sleeps less than 8 hours is $\frac{18}{30}$. For the second - selected individual (without replacement), the probability is $\frac{17}{29}$. For the third - selected individual (without replacement), the probability is $\frac{16}{28}$. The probability that all three sleep less than 8 hours is $\frac{18}{30}\times\frac{17}{29}\times\frac{16}{28}=\frac{2448}{24360}=\frac{204}{2030}\approx0.101$.

Step5: Calculate probability for question 14

The probability that none of them sleeps less than 8 hours is the probability that all three sleep 8 hours or more. The number of individuals who sleep 8 hours or more is $8 + 3+1+0 = 12$. The probability that the first - selected individual sleeps 8 hours or more is $\frac{12}{30}$. For the second - selected individual (without replacement), the probability is $\frac{11}{29}$. For the third - selected individual (without replacement), the probability is $\frac{10}{28}$. The probability that none of them sleeps less than 8 hours is $\frac{12}{30}\times\frac{11}{29}\times\frac{10}{28}=\frac{1320}{24360}=\frac{11}{203}$. The probability that at least one of them sleeps less than 8 hours is $1-\frac{11}{203}=\frac{203 - 11}{203}=\frac{192}{203}\approx0.946$.

Answer:

a. $\frac{3}{5}$ b. $\frac{2}{15}$ 13. $\frac{204}{2030}$ 14. $\frac{192}{203}$