Passengers (thousands) 150000 145000 140000 135000 130000 125000 120000 115000 110000 Time Series plot of quarterly light rail usage (first quarter 2015 through third quarter 2024) wwww 0 8 12 16 20 Time (Quarters) 24 28 32 36 40 40 Year Quarter t y(t) 2015 1 1 110569 2 2 113433 3 3 118183 4 4 114932 2016 1 5 112337 2 6 117224 3 7 118863 4 8 116554 2017 1 9 116287 2 10 124077 3 11 126540 4 12 123559 2018 1 13 122607.4 2 14 129549.7 3 15 128105.9 4 16 126818.9 2019 1 17 123020.3 2 18 130158.9 3 19 133584.3 4 20 132147.9 2020 1 21 126932.4 2 22 123051.92] 3 23 134041.56 4 24 135031.20 2021 1 25 130020.840 2 26 128010.480 3 27 138000.12 4 28 139989.76 2022 1 29 140979.40! 2 30 135969.04 3 31 130958.68! 4 32 142948.32! 2023 1 33 144321.12! 2 34 139782.059 3 35 135372.74 4 36 147611.86 2024 123 37 149334.83 38 145124.004 39 141022.359
Figure 1 below displays the quarterly number of U.S. passengers (in thousands) using light rail as a mode of transportation. The series begins in the first quarter of 2015 and ends with the third quarter of 2024.
We can see a regularity to the series: the first quarter’s ridership tends to be lowest; then there is a progressive rise in ridership going into the second and third quarters, followed by a decline in the fourth quarter. Superimposed on the series are the moving-average forecasts based on a span of k = 4.
Notice that the seasonal pattern in the time series is not present in the moving averages. The moving averages are a smoothed-out version of the original time series, reflecting only the general trend in the series, which is upward.
Question:
1. Compute a
= [a + b (t)] × SR =?
where SR is the seasonality ratio for the appropriate quarter corresponding to the value of t.
![Year
Quarter
t
y(t)
2015
1
1
110569
2
2
113433
3
3
118183
4
4
114932
2016
1
5
112337
2
6
117224
3
7
118863
4
8
116554
2017
1
9
116287
2
10
124077
3
11
126540
4
12
123559
2018
1
13
122607.4
2
14
129549.7
3
15
128105.9
4
16 126818.9
2019
1
17
123020.3
2
18
130158.9
3
19 133584.3
4
20
132147.9
2020
1
21 126932.4
2
22 123051.92]
3
23 134041.56
4
24 135031.20
2021
1
25 130020.840
2
26 128010.480
3
27 138000.12
4
28 139989.76
2022
1
29 140979.40!
2
30 135969.04
3
31 130958.68!
4
32 142948.32!
2023
1
33 144321.12!
2
34 139782.059
3
35 135372.74
4
36 147611.86
2024
123
37 149334.83
38 145124.004
39 141022.359](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe3c20e04-668d-4e7f-ab67-18ddfb8e3211%2Faf0c5251-8727-4a0b-87a2-c56fd8373f94%2Fb144k9a_processed.jpeg&w=3840&q=75)
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