a.
Construct the probability distribution for the random variable X.
a.
Answer to Problem 4RE
The probability distribution for the random variable X is,
x | P(x) |
1 | 0.5632 |
2 | 0.2500 |
3 | 0.1147 |
4 | 0.0473 |
5 | 0.0171 |
6 | 0.0053 |
7 | 0.0018 |
8 | 0.0006 |
Explanation of Solution
Calculation:
The given table represents the frequency distribution of the number of AP tests taken by students who took one or more AP tests. The random variable X denotes the number of exams taken by a student who took one or more AP tests.
Here, the total number of people is 1,692. 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=1:
Similarly the remaining probabilities are obtained as follows:
x | P(x) | |
1 | 0.5632 | |
2 | 0.2500 | |
3 | 0.1147 | |
4 | 0.0473 | |
5 | 0.0171 | |
6 | 0.0053 | |
7 | 0.0018 | |
8 | 0.0006 | |
Total | 1.0000 |
Thus, the discrete probability distribution for the random variable X is obtained.
b.
Find the probability that a student took exactly one exam.
b.
Answer to Problem 4RE
The probability that a student took exactly one exam is 0.5632.
Explanation of Solution
Calculation:
The table represents the probability distribution of the random variable X, the exams taken by a student who took one or more AP tests.
The probability that a student took exactly one exam is the probability at the point
Thus, the probability that a student took exactly one exam is 0.5632.
c.
Find the
c.
Answer to Problem 4RE
The mean value is 1.7312.
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 | ||
1 | 0.5632 | 0.5632 |
2 | 0.2500 | 0.5000 |
3 | 0.1147 | 0.3441 |
4 | 0.0473 | 0.1892 |
5 | 0.0171 | 0.0855 |
6 | 0.0053 | 0.0318 |
7 | 0.0018 | 0.0126 |
8 | 0.0006 | 0.0048 |
Total | 1.0000 | 1.7312 |
Thus, the mean value is 1.7312.
d.
Find the standard deviation.
d.
Answer to Problem 4RE
The standard deviation is 1.049.
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 | ||||
1 | 0.5632 | –0.7312 | 0.535 | 0.3011 |
2 | 0.2500 | 0.2688 | 0.072 | 0.0181 |
3 | 0.1147 | 1.2688 | 1.610 | 0.1847 |
4 | 0.0473 | 2.2688 | 5.147 | 0.2435 |
5 | 0.0171 | 3.2688 | 10.685 | 0.1827 |
6 | 0.0053 | 4.2688 | 18.223 | 0.0966 |
7 | 0.0018 | 5.2688 | 27.760 | 0.0500 |
8 | 0.0006 | 6.2688 | 39.298 | 0.0236 |
Total | 1.0000 | 22.1504 | 103.330 | 1.1001 |
Therefore,
Thus, the variance is 1.1001.
The standard deviation is,
That is, the standard deviation is 1.049.
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Chapter 5 Solutions
Essential Statistics
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