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Answer :
To find the mean (or expected value) of the random variable [tex]$G$[/tex], we use the formula
[tex]$$
E[G] = \sum_{g} g \cdot P(g),
$$[/tex]
where [tex]$g$[/tex] represents the number of days and [tex]$P(g)$[/tex] is the probability that a member worked out [tex]$g$[/tex] days.
The table provides:
[tex]\[
\begin{array}{c|cccccccc}
\text{Number of Days } g & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\
\hline
P(g) & 0.49 & 0.12 & 0.13 & 0.15 & 0.06 & 0.02 & 0.02 & 0.01
\end{array}
\][/tex]
We calculate the contribution of each day as follows:
1. For [tex]$g = 0$[/tex]:
[tex]$$
0 \cdot 0.49 = 0.00.
$$[/tex]
2. For [tex]$g = 1$[/tex]:
[tex]$$
1 \cdot 0.12 = 0.12.
$$[/tex]
3. For [tex]$g = 2$[/tex]:
[tex]$$
2 \cdot 0.13 = 0.26.
$$[/tex]
4. For [tex]$g = 3$[/tex]:
[tex]$$
3 \cdot 0.15 = 0.45.
$$[/tex]
5. For [tex]$g = 4$[/tex]:
[tex]$$
4 \cdot 0.06 = 0.24.
$$[/tex]
6. For [tex]$g = 5$[/tex]:
[tex]$$
5 \cdot 0.02 = 0.10.
$$[/tex]
7. For [tex]$g = 6$[/tex]:
[tex]$$
6 \cdot 0.02 = 0.12.
$$[/tex]
8. For [tex]$g = 7$[/tex]:
[tex]$$
7 \cdot 0.01 = 0.07.
$$[/tex]
Now, add all these contributions together:
[tex]$$
E[G] = 0.00 + 0.12 + 0.26 + 0.45 + 0.24 + 0.10 + 0.12 + 0.07 \approx 1.36.
$$[/tex]
Interpretation:
The computed mean of [tex]$G$[/tex] is approximately [tex]$1.36$[/tex]. This means that if many members of the gym were randomly selected, the average number of days per week a member worked out would be about [tex]$1.36$[/tex] days.
[tex]$$
E[G] = \sum_{g} g \cdot P(g),
$$[/tex]
where [tex]$g$[/tex] represents the number of days and [tex]$P(g)$[/tex] is the probability that a member worked out [tex]$g$[/tex] days.
The table provides:
[tex]\[
\begin{array}{c|cccccccc}
\text{Number of Days } g & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\
\hline
P(g) & 0.49 & 0.12 & 0.13 & 0.15 & 0.06 & 0.02 & 0.02 & 0.01
\end{array}
\][/tex]
We calculate the contribution of each day as follows:
1. For [tex]$g = 0$[/tex]:
[tex]$$
0 \cdot 0.49 = 0.00.
$$[/tex]
2. For [tex]$g = 1$[/tex]:
[tex]$$
1 \cdot 0.12 = 0.12.
$$[/tex]
3. For [tex]$g = 2$[/tex]:
[tex]$$
2 \cdot 0.13 = 0.26.
$$[/tex]
4. For [tex]$g = 3$[/tex]:
[tex]$$
3 \cdot 0.15 = 0.45.
$$[/tex]
5. For [tex]$g = 4$[/tex]:
[tex]$$
4 \cdot 0.06 = 0.24.
$$[/tex]
6. For [tex]$g = 5$[/tex]:
[tex]$$
5 \cdot 0.02 = 0.10.
$$[/tex]
7. For [tex]$g = 6$[/tex]:
[tex]$$
6 \cdot 0.02 = 0.12.
$$[/tex]
8. For [tex]$g = 7$[/tex]:
[tex]$$
7 \cdot 0.01 = 0.07.
$$[/tex]
Now, add all these contributions together:
[tex]$$
E[G] = 0.00 + 0.12 + 0.26 + 0.45 + 0.24 + 0.10 + 0.12 + 0.07 \approx 1.36.
$$[/tex]
Interpretation:
The computed mean of [tex]$G$[/tex] is approximately [tex]$1.36$[/tex]. This means that if many members of the gym were randomly selected, the average number of days per week a member worked out would be about [tex]$1.36$[/tex] days.
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