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Answer :
We begin by noting that the data appears to follow a repeating pattern every few days. To investigate this, we group the measurements by assuming the data repeats every 4 days. That is, we look at the days as belonging to one of four groups:
- Days 1, 5, and 9
- Days 2, 6, and 10
- Days 3, 7, and 11
- Days 4, 8, and 12
The given measurements are:
[tex]$$
\begin{array}{|c|cccccccccccc|}
\hline
\text{Day} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\
\hline
\text{Cups} & 16 & 30 & 8 & 14 & 28 & 10 & 15 & 31 & 11 & 14 & 29 & 9 \\
\hline
\end{array}
$$[/tex]
### Step 1. Grouping the Data
1. Group 1 (Days 1, 5, 9):
Values: 16, 28, 11
2. Group 2 (Days 2, 6, 10):
Values: 30, 10, 14
3. Group 3 (Days 3, 7, 11):
Values: 8, 15, 29
4. Group 4 (Days 4, 8, 12):
Values: 14, 31, 9
### Step 2. Determine the Amplitude for Each Group
The amplitude of a periodic function is defined as one-half the difference between its maximum and minimum values. For each group, we calculate:
1. Group 1:
Maximum = 28, Minimum = 11
[tex]$$
\text{Amplitude} = \frac{28 - 11}{2} = \frac{17}{2} = 8.5
$$[/tex]
2. Group 2:
Maximum = 30, Minimum = 10
[tex]$$
\text{Amplitude} = \frac{30 - 10}{2} = \frac{20}{2} = 10
$$[/tex]
3. Group 3:
Maximum = 29, Minimum = 8
[tex]$$
\text{Amplitude} = \frac{29 - 8}{2} = \frac{21}{2} = 10.5
$$[/tex]
4. Group 4:
Maximum = 31, Minimum = 9
[tex]$$
\text{Amplitude} = \frac{31 - 9}{2} = \frac{22}{2} = 11
$$[/tex]
### Step 3. Find the Overall Amplitude
To get an overall idea of the amplitude that describes the repeating pattern, we calculate the average of the amplitudes from all four groups:
[tex]$$
\text{Average Amplitude} = \frac{8.5 + 10 + 10.5 + 11}{4} = \frac{40}{4} = 10
$$[/tex]
### Final Conclusion
Since the same grouping into 4-day cycles yields similar amplitudes, we conclude that the data is approximately periodic with a period of [tex]$4$[/tex] days and a typical amplitude of about [tex]$10$[/tex] cups of coffee.
- Days 1, 5, and 9
- Days 2, 6, and 10
- Days 3, 7, and 11
- Days 4, 8, and 12
The given measurements are:
[tex]$$
\begin{array}{|c|cccccccccccc|}
\hline
\text{Day} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\
\hline
\text{Cups} & 16 & 30 & 8 & 14 & 28 & 10 & 15 & 31 & 11 & 14 & 29 & 9 \\
\hline
\end{array}
$$[/tex]
### Step 1. Grouping the Data
1. Group 1 (Days 1, 5, 9):
Values: 16, 28, 11
2. Group 2 (Days 2, 6, 10):
Values: 30, 10, 14
3. Group 3 (Days 3, 7, 11):
Values: 8, 15, 29
4. Group 4 (Days 4, 8, 12):
Values: 14, 31, 9
### Step 2. Determine the Amplitude for Each Group
The amplitude of a periodic function is defined as one-half the difference between its maximum and minimum values. For each group, we calculate:
1. Group 1:
Maximum = 28, Minimum = 11
[tex]$$
\text{Amplitude} = \frac{28 - 11}{2} = \frac{17}{2} = 8.5
$$[/tex]
2. Group 2:
Maximum = 30, Minimum = 10
[tex]$$
\text{Amplitude} = \frac{30 - 10}{2} = \frac{20}{2} = 10
$$[/tex]
3. Group 3:
Maximum = 29, Minimum = 8
[tex]$$
\text{Amplitude} = \frac{29 - 8}{2} = \frac{21}{2} = 10.5
$$[/tex]
4. Group 4:
Maximum = 31, Minimum = 9
[tex]$$
\text{Amplitude} = \frac{31 - 9}{2} = \frac{22}{2} = 11
$$[/tex]
### Step 3. Find the Overall Amplitude
To get an overall idea of the amplitude that describes the repeating pattern, we calculate the average of the amplitudes from all four groups:
[tex]$$
\text{Average Amplitude} = \frac{8.5 + 10 + 10.5 + 11}{4} = \frac{40}{4} = 10
$$[/tex]
### Final Conclusion
Since the same grouping into 4-day cycles yields similar amplitudes, we conclude that the data is approximately periodic with a period of [tex]$4$[/tex] days and a typical amplitude of about [tex]$10$[/tex] cups of coffee.
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