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 2\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) =\n b. 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 2\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) =\n b. 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 total number of data - points. Add up all the frequencies: $2 + 2+7 + 6+8 + 3+1+0=29$.

Step2: Calculate $P(x < 8)$

The frequencies for $x = 4,5,6,7$ are $2,2,7,6$ respectively. The sum of these frequencies is $2 + 2+7 + 6=17$. So, $P(x < 8)=\frac{17}{29}$.

Step3: Calculate $P(x>8)$

The frequencies for $x = 9,10,11$ are $3,1,0$ respectively. The sum of these frequencies is $3 + 1+0 = 4$. So, $P(x>8)=\frac{4}{29}$.

Step4: Calculate probability for three individuals sleeping less than 8 hours (without replacement)

The number of individuals sleeping less than 8 hours is 17. The total number of individuals is 29. The probability that the first - selected individual sleeps less than 8 hours is $\frac{17}{29}$. The probability that the second - selected individual sleeps less than 8 hours (given the first one slept less than 8 hours) is $\frac{16}{28}$. The probability that the third - selected individual sleeps less than 8 hours (given the first two slept less than 8 hours) is $\frac{15}{27}$. The combined probability is $\frac{17}{29}\times\frac{16}{28}\times\frac{15}{27}=\frac{17\times16\times15}{29\times28\times27}=\frac{4080}{21924}\approx0.186$.

Step5: Calculate probability that at least one of three individuals sleeps less than 8 hours (without replacement)

First, find the probability that none of them sleeps less than 8 hours. The number of individuals sleeping 8 or more hours is $29 - 17=12$. The probability that the first - selected individual sleeps 8 or more hours is $\frac{12}{29}$. The probability that the second - selected individual sleeps 8 or more hours (given the first one slept 8 or more hours) is $\frac{11}{28}$. The probability that the third - selected individual sleeps 8 or more hours (given the first two slept 8 or more hours) is $\frac{10}{27}$. The probability that none of them sleeps less than 8 hours is $\frac{12}{29}\times\frac{11}{28}\times\frac{10}{27}=\frac{1320}{21924}\approx0.06$. The probability that at least one of them sleeps less than 8 hours is $1-\frac{1320}{21924}=\frac{21924 - 1320}{21924}=\frac{20604}{21924}\approx0.94$.

Answer:

a. $\frac{17}{29}$ b. $\frac{4}{29}$ 13. $\frac{17\times16\times15}{29\times28\times27}\approx0.186$ 14. $\frac{20604}{21924}\approx0.94$