
Concept explainers
(a)
Section 1:
The mean of the number of heads.
(a)
Section 1:

Answer to Problem 87E
Solution: The mean of the number of heads (μX) is 0.5.
Explanation of Solution
Mean(μX)=2∑i=1xi×p(xi)=0×0.5+1×0.5=0.5
Section 2:
The standard deviation of number of heads.
Section 2:

Answer to Problem 87E
Solution: The standard deviation (σX) is 0.5.
Explanation of Solution
Variance (σ2X)=(xi−μX)2×pi
And it is calculated as
Variance(σ2X)=(xi−μX)2×pi=(0−0.5)2×12+(1−0.5)2×12=0.25
So, the standard deviation is calculated as
Standard deviation (σ)=√Variance =√0.25=0.5
(b)
Section 1:
The mean of the number of heads.
(b)
Section 1:

Answer to Problem 87E
Solution: The mean is 2.
Explanation of Solution
For 0 head, there will be 4C0=1 arrangement that has no heads.
For 1 head, there will be 4C1=4 arrangements that have 1 head.
For 2 heads, there will be 4C2=6 arrangements that have 2 heads.
For 3 heads, there will be 4C3=4 arrangements that have 3 heads.
For 4 heads, there will be 4C4=1 arrangement that has 4 heads.
So, there will be total 16(1+4+6+4+1) arrangements. Thus, the corresponding probabilities are as follows:
Number of heads (xi) |
Favorable arrangements |
Probability (pi) |
0 |
1 |
1/16 |
1 |
4 |
4/16 |
2 |
6 |
6/16 |
3 |
4 |
4/16 |
4 |
1 |
1/16 |
Now, the formula of mean is as follows:
Mean (μX)=5∑i=1xi×pi
And it is calculated as
Mean (μX)=5∑i=1xi×pi=(0×116)+(1×416)+(2×616)+(3×416)+(4×116)=2
Section 2:
The standard deviation.
Section 2:

Answer to Problem 87E
Solution: The standard deviation is 1.
Explanation of Solution
Standard deviation (σX)=√5∑i=1(xi−μX)2×pi
And it is calculated as
Standard deviation (σX)=√5∑i=1(xi−μX)2×pi=√((0−2)2×116+(1−2)2×416+(2−2)2×616+(3−2)2×416+(4−2)2×116)=1
(c)
Section 1:
The mean using the distribution provided in Example 4.23.
(c)
Section 1:

Answer to Problem 87E
Solution: The mean is 2.
Explanation of Solution
Mean μX=5∑i=1xi×pi=0×0.0625+1×0.25+2×0.375+3×0.25+4×0.0625=2
Section 2:
The standard deviation using the distribution provided in Example 4.23.
Section 2:

Answer to Problem 87E
Solution: The standard deviation is 1.
Explanation of Solution
Standard deviation (σX)=√5∑i=1(xi−μX)2×pi
And it is calculated as
Standard deviation (σX)=√5∑i=1(xi−μX)2×pi=√((0−2)2×0.0625+(1−2)2×0.25+(2−2)2×0.375+(3−2)2×0.25+(4−2)2×0.0625)=1
Section 3:
Whether the results obtained in part (b) and part (c) are the same.
Section 3:

Answer to Problem 87E
Solution: Yes, the results obtained in part (b) and parts (c) are same.
Explanation of Solution
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Chapter 4 Solutions
Introduction to the Practice of Statistics: w/CrunchIt/EESEE Access Card
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