Question

1. now form a frequency distribution for the data. add more rows if needed. (3 points)

hours of sleep (x) frequency (f)
4 2
5 2
6 7
7 6
8 8
9 3
10 1
11 0
calculate probabilities

1. find the following probabilities. (3 points)

a. p(x < 8) =
b. p(x > 8) =

1. if three individuals are selected from the sample, without replacement, what is the probability that they all sleep less than 8 hours? (3 points)
2. 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)

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. $\frac{4}{29}$

1. $\frac{17\times16\times15}{29\times28\times27}\approx0.186$
2. $\frac{20604}{21924}\approx0.94$