Concept explainers
a.
Construct the
a.
Answer to Problem 47E
The probability distribution for the random variable X is,
x | P(x) |
0 | 0.0680 |
1 | 0.1110 |
2 | 0.2005 |
3 | 0.1498 |
4 | 0.1885 |
5 | 0.1378 |
6 | 0.0889 |
7 | 0.0197 |
8 | 0.0358 |
Explanation of Solution
Calculation:
The table represents the results of the survey in which 1,676 people are asked how many hours per day they were able to relax. The random variable X denotes the number of hours of relaxation for a person sampled at random from the population.
Here, the total number of people is 1,676. The corresponding probabilities of the random variable X are obtained by dividing the frequency of each hour (f) by total frequency (N).
That is,
For x=0:
Similarly the remaining probabilities are obtained as follows:
x | P(x) | |
0 | 0.0680 | |
1 | 0.1110 | |
2 | 0.2005 | |
3 | 0.1498 | |
4 | 0.1885 | |
5 | 0.1378 | |
6 | 0.0889 | |
7 | 0.0197 | |
8 | 0.0358 | |
Total | 1.0000 |
Thus, the discrete probability distribution for the random variable X is obtained.
b.
Find the probability that a person relaxes more than 4 hours per day.
b.
Answer to Problem 47E
The probability that a person relaxes more than 4 hours per day is 0.282.
Explanation of Solution
Calculation:
The table represents the probability distribution of the random variable X, the number of hours of relaxation for a person.
The probability that a person relaxes more than 4 hours per day is the sum of the probabilities at the points
Substituting the values from the probability distribution table obtained in part (a),
Thus, the probability that a person relaxes more than 4 hours per day is 0.282.
c.
Find the probability that the person doesn’t relax at all.
c.
Answer to Problem 47E
The probability that the person doesn’t relax at all is 0.068.
Explanation of Solution
Calculation:
The probability that the person doesn’t relax at all is the probability at
From the probability distribution table, the probability at
Thus, the probability that the person doesn’t relax at all is 0.068.
d.
Find the
d.
Answer to Problem 47E
The mean value is 3.36.
Explanation of Solution
Calculation:
The formula for the mean of a discrete random variable is,
The mean of the random variable is obtained as given below:
x | P(x) | |
0 | 0.0680 | 0.00 |
1 | 0.1110 | 0.11 |
2 | 0.2005 | 0.40 |
3 | 0.1498 | 0.45 |
4 | 0.1885 | 0.75 |
5 | 0.1378 | 0.69 |
6 | 0.0889 | 0.53 |
7 | 0.0197 | 0.14 |
8 | 0.0358 | 0.29 |
Total | 1.0000 | 3.36 |
Thus, the mean value is 3.36.
f.
Find the standard deviation.
f.
Answer to Problem 47E
The standard deviation is 1.97.
Explanation of Solution
Calculation:
The standard deviation of the random variable X is obtained by taking the square root of variance.
The formula for the variance of the discrete random variable X is,
Where
The variance of the random variable X is obtained using the following table:
x | P(x) | |||
0 | 0.0680 | –3.3621 | 11.3037 | 0.7687 |
1 | 0.1110 | –2.3621 | 5.5795 | 0.6193 |
2 | 0.2005 | –1.3621 | 1.8553 | 0.3720 |
3 | 0.1498 | –0.3621 | 0.1311 | 0.0196 |
4 | 0.1885 | 0.6379 | 0.4069 | 0.0767 |
5 | 0.1378 | 1.6379 | 2.6827 | 0.3697 |
6 | 0.0889 | 2.6379 | 6.9585 | 0.6186 |
7 | 0.0197 | 3.6379 | 13.2343 | 0.2607 |
8 | 0.0358 | 4.6379 | 21.5101 | 0.7701 |
Total | 1.0000 | 5.7411 | 63.6622 | 3.8754 |
Therefore,
Thus, the variance is 3.8754.
The standard deviation is,
That is, the standard deviation is 1.97.
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Chapter 5 Solutions
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