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Answer :
We start with the given data of cups of coffee consumed over 12 days:
[tex]$$
\begin{array}{|c|c|}
\hline
\text{Day} & \text{Cups of Coffee} \\
\hline
1 & 16 \\
2 & 30 \\
3 & 8 \\
4 & 14 \\
5 & 28 \\
6 & 10 \\
7 & 15 \\
8 & 31 \\
9 & 11 \\
10 & 14 \\
11 & 29 \\
12 & 9 \\
\hline
\end{array}
$$[/tex]
### Step 1. Check for Periodicity
By examining the daily values, there seems to be a suggestion that the pattern repeats after a fixed number of days. To explore this potential repetition, we divide the data into three groups corresponding to the positions within the period assuming a period of [tex]$3$[/tex] days.
Define the groups as follows:
- Group 1: Days [tex]$1, 4, 7, 10$[/tex]
Values: [tex]$16,\; 14,\; 15,\; 14$[/tex]
- Group 2: Days [tex]$2, 5, 8, 11$[/tex]
Values: [tex]$30,\; 28,\; 31,\; 29$[/tex]
- Group 3: Days [tex]$3, 6, 9, 12$[/tex]
Values: [tex]$8,\; 10,\; 11,\; 9$[/tex]
Since the pattern in each group seems consistent, we determine that the data is approximately periodic with a period of [tex]$3$[/tex] days.
### Step 2. Calculate Group Averages
Find the average (mean) consumption for each group:
- Group 1 Average:
[tex]$$
\bar{x}_1 = \frac{16+14+15+14}{4} = \frac{59}{4} = 14.75
$$[/tex]
- Group 2 Average:
[tex]$$
\bar{x}_2 = \frac{30+28+31+29}{4} = \frac{118}{4} = 29.5
$$[/tex]
- Group 3 Average:
[tex]$$
\bar{x}_3 = \frac{8+10+11+9}{4} = \frac{38}{4} = 9.5
$$[/tex]
### Step 3. Determine the Amplitude
The amplitude is measured as half the difference between the maximum and minimum group averages. Here, the maximum average is [tex]$29.5$[/tex] (from Group 2) and the minimum average is [tex]$9.5$[/tex] (from Group 3). Therefore, the amplitude is calculated as:
[tex]$$
\text{Amplitude} = \frac{29.5 - 9.5}{2} = \frac{20}{2} = 10
$$[/tex]
### Final Conclusion
Based on the analysis:
- The data is approximately periodic with a period of [tex]$3$[/tex] days.
- The amplitude of the oscillation is about [tex]$10$[/tex] cups of coffee.
Thus, the answer is: periodic with a period of [tex]$3$[/tex] and an amplitude of about [tex]$10$[/tex].
[tex]$$
\begin{array}{|c|c|}
\hline
\text{Day} & \text{Cups of Coffee} \\
\hline
1 & 16 \\
2 & 30 \\
3 & 8 \\
4 & 14 \\
5 & 28 \\
6 & 10 \\
7 & 15 \\
8 & 31 \\
9 & 11 \\
10 & 14 \\
11 & 29 \\
12 & 9 \\
\hline
\end{array}
$$[/tex]
### Step 1. Check for Periodicity
By examining the daily values, there seems to be a suggestion that the pattern repeats after a fixed number of days. To explore this potential repetition, we divide the data into three groups corresponding to the positions within the period assuming a period of [tex]$3$[/tex] days.
Define the groups as follows:
- Group 1: Days [tex]$1, 4, 7, 10$[/tex]
Values: [tex]$16,\; 14,\; 15,\; 14$[/tex]
- Group 2: Days [tex]$2, 5, 8, 11$[/tex]
Values: [tex]$30,\; 28,\; 31,\; 29$[/tex]
- Group 3: Days [tex]$3, 6, 9, 12$[/tex]
Values: [tex]$8,\; 10,\; 11,\; 9$[/tex]
Since the pattern in each group seems consistent, we determine that the data is approximately periodic with a period of [tex]$3$[/tex] days.
### Step 2. Calculate Group Averages
Find the average (mean) consumption for each group:
- Group 1 Average:
[tex]$$
\bar{x}_1 = \frac{16+14+15+14}{4} = \frac{59}{4} = 14.75
$$[/tex]
- Group 2 Average:
[tex]$$
\bar{x}_2 = \frac{30+28+31+29}{4} = \frac{118}{4} = 29.5
$$[/tex]
- Group 3 Average:
[tex]$$
\bar{x}_3 = \frac{8+10+11+9}{4} = \frac{38}{4} = 9.5
$$[/tex]
### Step 3. Determine the Amplitude
The amplitude is measured as half the difference between the maximum and minimum group averages. Here, the maximum average is [tex]$29.5$[/tex] (from Group 2) and the minimum average is [tex]$9.5$[/tex] (from Group 3). Therefore, the amplitude is calculated as:
[tex]$$
\text{Amplitude} = \frac{29.5 - 9.5}{2} = \frac{20}{2} = 10
$$[/tex]
### Final Conclusion
Based on the analysis:
- The data is approximately periodic with a period of [tex]$3$[/tex] days.
- The amplitude of the oscillation is about [tex]$10$[/tex] cups of coffee.
Thus, the answer is: periodic with a period of [tex]$3$[/tex] and an amplitude of about [tex]$10$[/tex].
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