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Answer :
We are given the following data for the number of cars observed over 12 hours:
[tex]\[
\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
\text{Hour} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\
\hline
\text{Number of cars} & 52 & 76 & 90 & 75 & 91 & 104 & 89 & 105 & 119 & 103 & 121 & 135 \\
\hline
\end{array}
\][/tex]
A brief inspection shows that the number of cars is increasing overall, suggesting a trend. However, there is also a fluctuation about this trend that appears to repeat with a certain regularity.
### Step 1. Removing the Trend
Because there is a noticeable overall increase, imagine that the data consists of two parts:
- A linear trend (increasing over the hours).
- A cyclic (oscillatory) component.
To analyze the periodic component, one could imagine subtracting the linear trend from the data. This leaves us with what are called the “residuals.”
### Step 2. Identifying the Period
After the trend is removed, observe the pattern in the remaining fluctuations. By looking at the residuals or even directly at the hourly values, one may notice that the oscillatory behavior tends to repeat every 3 hours. For example, the pattern from hours 1–3 may resemble that from hours 4–6, 7–9, and 10–12.
Thus, based on this observation, we conclude that the periodic component has a period of:
[tex]\[
\text{Period} = 3 \text{ hours}
\][/tex]
### Step 3. Estimating the Amplitude
The amplitude of the periodic component is defined as half the difference between the maximum and minimum values of the oscillatory (detrended) fluctuations. When the oscillatory behavior is inspected, the typical variation (or “wiggle”) is estimated to be about 15 from peak to trough. Therefore, the amplitude is roughly:
[tex]\[
\text{Amplitude} \approx \frac{15}{2} = 7.5
\][/tex]
### Conclusion
Based on the analysis, the data set is approximately periodic. The periodic fluctuation repeats every 3 hours, and the amplitude of these fluctuations is about 7.5. Among the answer choices given, the correct answer is:
[tex]\[
\textbf{Periodic with period of 3 and amplitude of about 7.5.}
\][/tex]
[tex]\[
\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
\text{Hour} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\
\hline
\text{Number of cars} & 52 & 76 & 90 & 75 & 91 & 104 & 89 & 105 & 119 & 103 & 121 & 135 \\
\hline
\end{array}
\][/tex]
A brief inspection shows that the number of cars is increasing overall, suggesting a trend. However, there is also a fluctuation about this trend that appears to repeat with a certain regularity.
### Step 1. Removing the Trend
Because there is a noticeable overall increase, imagine that the data consists of two parts:
- A linear trend (increasing over the hours).
- A cyclic (oscillatory) component.
To analyze the periodic component, one could imagine subtracting the linear trend from the data. This leaves us with what are called the “residuals.”
### Step 2. Identifying the Period
After the trend is removed, observe the pattern in the remaining fluctuations. By looking at the residuals or even directly at the hourly values, one may notice that the oscillatory behavior tends to repeat every 3 hours. For example, the pattern from hours 1–3 may resemble that from hours 4–6, 7–9, and 10–12.
Thus, based on this observation, we conclude that the periodic component has a period of:
[tex]\[
\text{Period} = 3 \text{ hours}
\][/tex]
### Step 3. Estimating the Amplitude
The amplitude of the periodic component is defined as half the difference between the maximum and minimum values of the oscillatory (detrended) fluctuations. When the oscillatory behavior is inspected, the typical variation (or “wiggle”) is estimated to be about 15 from peak to trough. Therefore, the amplitude is roughly:
[tex]\[
\text{Amplitude} \approx \frac{15}{2} = 7.5
\][/tex]
### Conclusion
Based on the analysis, the data set is approximately periodic. The periodic fluctuation repeats every 3 hours, and the amplitude of these fluctuations is about 7.5. Among the answer choices given, the correct answer is:
[tex]\[
\textbf{Periodic with period of 3 and amplitude of about 7.5.}
\][/tex]
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