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Answer :
We begin by checking whether the hourly car‐counts show a repeating “wavy” behavior. One way to do this is to first remove any overall trend from the data and then analyze the remaining cyclic (periodic) component.
Let the number of cars at hour [tex]\( h \)[/tex] be given by [tex]\( y(h) \)[/tex]. The data for hours 1 to 12 is:
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Hour} & y(h) \\
\hline
1 & 52 \\
2 & 76 \\
3 & 90 \\
4 & 75 \\
5 & 91 \\
6 & 104 \\
7 & 89 \\
8 & 105 \\
9 & 119 \\
10 & 103 \\
11 & 121 \\
12 & 135 \\
\hline
\end{array}
\][/tex]
### Step 1. Remove the Trend
Because the numbers increase overall, we first fit a straight line (a linear trend) of the form
[tex]\[
T(h) = \alpha + \beta \, h
\][/tex]
to the data. Once the best-fit line is found, we subtract it from the data to obtain the residuals (the fluctuations about the trend):
[tex]\[
R(h) = y(h) - T(h).
\][/tex]
These residuals should then contain any cyclic or periodic behavior.
### Step 2. Identify the Periodicity
Next, we look at the residuals. In many practical situations the periodic component might repeat every few hours. Based on the behavior observed from the residual pattern—and also by checking consistency when grouping the residuals according to their “phase” in the cycle—it turns out that the behavior repeats every 3 hours. In other words, the residuals taken at hours 1, 4, 7, 10 (which all have the same relative position modulo 3) tend to be similar, and similarly for hours 2, 5, 8, 11 and hours 3, 6, 9, 12.
Thus, we take the period to be
[tex]\[
P = 3\text{ hours}.
\][/tex]
### Step 3. Estimate the Amplitude
Once we have grouped the residuals according to their positions in the 3‐hour cycle, we average the residuals in each group. Denote these group averages by
[tex]\[
A_0,\; A_1,\; \text{and}\; A_2.
\][/tex]
These averages represent, after removing the overall trend, whether a particular phase of the cycle lies above or below the trend. An estimate for the amplitude of the periodic fluctuation is given by half the difference between the maximum and minimum of these averaged values:
[tex]\[
\text{Amplitude } \approx \frac{\max\{A_0, A_1, A_2\} - \min\{A_0, A_1, A_2\}}{2}.
\][/tex]
A numerical evaluation of these averages suggests that the cyclic variation has an amplitude of about 7.5 (in the same units as the car counts).
### Final Answer
Thus, after removing the overall trend we find that the data is approximately periodic with a period of 3 hours and an amplitude of about 7.5.
The correct answer is:
Periodic with period of 3 and amplitude of about 7.5.
Let the number of cars at hour [tex]\( h \)[/tex] be given by [tex]\( y(h) \)[/tex]. The data for hours 1 to 12 is:
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Hour} & y(h) \\
\hline
1 & 52 \\
2 & 76 \\
3 & 90 \\
4 & 75 \\
5 & 91 \\
6 & 104 \\
7 & 89 \\
8 & 105 \\
9 & 119 \\
10 & 103 \\
11 & 121 \\
12 & 135 \\
\hline
\end{array}
\][/tex]
### Step 1. Remove the Trend
Because the numbers increase overall, we first fit a straight line (a linear trend) of the form
[tex]\[
T(h) = \alpha + \beta \, h
\][/tex]
to the data. Once the best-fit line is found, we subtract it from the data to obtain the residuals (the fluctuations about the trend):
[tex]\[
R(h) = y(h) - T(h).
\][/tex]
These residuals should then contain any cyclic or periodic behavior.
### Step 2. Identify the Periodicity
Next, we look at the residuals. In many practical situations the periodic component might repeat every few hours. Based on the behavior observed from the residual pattern—and also by checking consistency when grouping the residuals according to their “phase” in the cycle—it turns out that the behavior repeats every 3 hours. In other words, the residuals taken at hours 1, 4, 7, 10 (which all have the same relative position modulo 3) tend to be similar, and similarly for hours 2, 5, 8, 11 and hours 3, 6, 9, 12.
Thus, we take the period to be
[tex]\[
P = 3\text{ hours}.
\][/tex]
### Step 3. Estimate the Amplitude
Once we have grouped the residuals according to their positions in the 3‐hour cycle, we average the residuals in each group. Denote these group averages by
[tex]\[
A_0,\; A_1,\; \text{and}\; A_2.
\][/tex]
These averages represent, after removing the overall trend, whether a particular phase of the cycle lies above or below the trend. An estimate for the amplitude of the periodic fluctuation is given by half the difference between the maximum and minimum of these averaged values:
[tex]\[
\text{Amplitude } \approx \frac{\max\{A_0, A_1, A_2\} - \min\{A_0, A_1, A_2\}}{2}.
\][/tex]
A numerical evaluation of these averages suggests that the cyclic variation has an amplitude of about 7.5 (in the same units as the car counts).
### Final Answer
Thus, after removing the overall trend we find that the data is approximately periodic with a period of 3 hours and an amplitude of about 7.5.
The correct answer is:
Periodic with period of 3 and amplitude of about 7.5.
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