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Have you ever wondered what the different scales of model trains meant and why they are not universal? Much of the issue of scale first came about when modelers began to demand a sort of standard method of producing trains that properly depicted the full size ones that they represented. When the first set of official scales became available they still were not rigidly adhered to and often the wheel size and gauge of the track did not follow scale at all. An example of the is the 0 gauge scale. In the United States it is on a track that is actually too wide for its supposed 1:48 scale. In Britain however it follows a much more rigid standard by following the 7mm/1ft scale which is also much more accurate.
00 standards in Britain are set up on on track that is almost a full seven inches too narrow. This sort of inconsistency with the track differences is still perpetuated by the manufactures who continue to produce model trains whose tracks are bigger than they should be, incorrectly sized wheel treads, and overly deep wheel flanges. It seems that many manufacturers afford gauge and scale the same connotation but this is very inaccurate. Correctly defined, scale is the relative measurement of an object when produced as a properly sized proportion of the original. Gauge, however, just means the overall measurement between the rails on a model railway.
As modelers became more concerned with these inaccurate means of determining the scale of their model trains they developed a finescale set of standards with which they could better determine the true scale of their models. While in use by many hardcore model builders these standards have not been adopted by the mainstream model train industry. Much of the reasons for this lie in the need for manufacturers to be able to produced a cost efficient model that can easily be used by home modelers and expert builders as well. One example of this finescale standard comes from Britain. The P4 standard takes real train track measurements and scales them down to match the model that is to be placed on those scaled tracks. They also make sure that the wing and check rails are scaled just as closely.
The biggest determining factor of the scale of a model train is the size of the engine being used. The largest ridable steam engines can go up to a full twenty eight inches tall which is considered a 1:8 scale. The smallest is 1:220 which is called Z scale and cover matchbox sized locomotives. There are five scales that are the most popular; N scale, TT scale, H0 scale, O scale, Gauge I, and G scale. G scale is the most common for outdoor models at a 1:8 scale and Gauge I may also be used as well. N, 0, and H0 are all very popular indoor scales with 0 being primarily used as children's toys.
Now that you know a bit more about the scale system that is used for determining a model trains size now you are ready to start building your own scaled version!
Victor Epand is an expert consultant for model cars, model trains, and model trucks. You will find excellent hobbying and trading resources here for model car tricks and tips, scaled model trains.
GDP Deflator
Question one:
The GDP deflator is a price index that is derived from the real GDP and the nominal GDP, the GDP deflator is calculated by dividing the nominal GDP by the Real GDP and then multiplying by 100, in our case we use nominal expenditure and real expenditure because GDP is also measured using the expenditure method. Therefore we use the following formula to derive the price index:
Price index = (nominal GDP/ real GDP) X 100
The following table summarizes our results:
YEAR
RTDE
QTDE
INTL
M4
p
1963
288.856
30.814
1.053
14.969
0.106675991
1964
307.559
34.037
1.058
16.106
0.110668197
1965
312.417
36.256
1.0643
17.616
0.116050023
1966
317.366
38.395
1.0691
18.757
0.120980193
1967
329.964
40.985
1.068
21.158
0.12421052
1968
341.032
44.39
1.0754
22.964
0.130163738
1969
341.179
46.853
1.0905
24.123
0.137326741
1970
349.045
51.27
1.0921
27.009
0.146886505
1971
357.267
57.323
1.0885
31.4
0.160448628
1972
372.979
64.466
1.089
38.674
0.17284083
1973
402.565
76.242
1.1071
47.119
0.189390533
1974
393.599
88.949
1.1477
52.197
0.225988887
1975
386.71
108.923
1.1439
58.383
0.281665848
1976
397.225
129.328
1.1443
64.97
0.325578702
1977
396.143
146.204
1.1273
74.595
0.369068745
1978
411.909
167.515
1.1247
85.77
0.406679631
1979
426.915
197.906
1.1299
98.131
0.463572374
1980
414.792
227.537
1.1378
114.923
0.548556867
1981
408.223
249.322
1.1474
138.363
0.610749517
1982
417.916
274.649
1.1288
154.909
0.65718709
1983
438.768
302.895
1.108
175.299
0.690330653
1984
450.949
326.498
1.1069
198.93
0.724024224
1985
464.316
354.291
1.1062
224.794
0.763038534
1986
487.33
388.179
1.0987
258.304
0.796542384
1987
513.083
428.721
1.0947
304.948
0.835578259
1988
553.461
488.953
1.0936
358.233
0.883446169
1989
569.719
537.279
1.0958
426.322
0.943059649
1990
566.238
566.238
1.1108
477.138
1
1991
548.532
581.897
1.0992
504.133
1.060825986
1992
549.543
605.295
1.0912
517.883
1.10145157
1993
561.346
638.4
1.0787
544.055
1.137266499
1994
580.092
675.164
1.0805
567.157
1.163891245
Question two:
Deriving the real money supply:
We derive the real money supply RM4 by dividing the nominal money supply by the price index:
YEAR
RTDE
QTDE
INTL
M4
p
RM4
1963
288.856
30.814
1.053
14.969
0.106675991
140.3221089
1964
307.559
34.037
1.058
16.106
0.110668197
145.5341321
1965
312.417
36.256
1.0643
17.616
0.116050023
151.7966094
1966
317.366
38.395
1.0691
18.757
0.120980193
155.0419081
1967
329.964
40.985
1.068
21.158
0.12421052
170.3398393
1968
341.032
44.39
1.0754
22.964
0.130163738
176.4239434
1969
341.179
46.853
1.0905
24.123
0.137326741
175.6613454
1970
349.045
51.27
1.0921
27.009
0.146886505
183.8766609
1971
357.267
57.323
1.0885
31.4
0.160448628
195.7012683
1972
372.979
64.466
1.089
38.674
0.17284083
223.7550002
1973
402.565
76.242
1.1071
47.119
0.189390533
248.7927945
1974
393.599
88.949
1.1477
52.197
0.225988887
230.9715343
1975
386.71
108.923
1.1439
58.383
0.281665848
207.2775257
1976
397.225
129.328
1.1443
64.97
0.325578702
199.5523649
1977
396.143
146.204
1.1273
74.595
0.369068745
202.1168168
1978
411.909
167.515
1.1247
85.77
0.406679631
210.9031127
1979
426.915
197.906
1.1299
98.131
0.463572374
211.6843141
1980
414.792
227.537
1.1378
114.923
0.548556867
209.5006132
1981
408.223
249.322
1.1474
138.363
0.610749517
226.5462292
1982
417.916
274.649
1.1288
154.909
0.65718709
235.7152207
1983
438.768
302.895
1.108
175.299
0.690330653
253.9348343
1984
450.949
326.498
1.1069
198.93
0.724024224
274.7560003
1985
464.316
354.291
1.1062
224.794
0.763038534
294.6037323
1986
487.33
388.179
1.0987
258.304
0.796542384
324.2815513
1987
513.083
428.721
1.0947
304.948
0.835578259
364.9544452
1988
553.461
488.953
1.0936
358.233
0.883446169
405.4949953
1989
569.719
537.279
1.0958
426.322
0.943059649
452.0626034
1990
566.238
566.238
1.1108
477.138
1
477.138
1991
548.532
581.897
1.0992
504.133
1.060825986
475.2268576
1992
549.543
605.295
1.0912
517.883
1.10145157
470.1822706
1993
561.346
638.4
1.0787
544.055
1.137266499
478.3883115
1994
580.092
675.164
1.0805
567.157
1.163891245
487.2938108
Question 3:
Rate of inflation:
YEAR
RTDE
QTDE
INTL
M4
p
RM4
PI
1963
288.856
30.814
1.053
14.969
0.106675991
140.3221089
1964
307.559
34.037
1.058
16.106
0.110668197
145.5341321
0.037423662
1965
312.417
36.256
1.0643
17.616
0.116050023
151.7966094
0.048630284
1966
317.366
38.395
1.0691
18.757
0.120980193
155.0419081
0.042483148
1967
329.964
40.985
1.068
21.158
0.12421052
170.3398393
0.026701286
1968
341.032
44.39
1.0754
22.964
0.130163738
176.4239434
0.047928455
1969
341.179
46.853
1.0905
24.123
0.137326741
175.6613454
0.055030704
1970
349.045
51.27
1.0921
27.009
0.146886505
183.8766609
0.069613275
1971
357.267
57.323
1.0885
31.4
0.160448628
195.7012683
0.09233063
1972
372.979
64.466
1.089
38.674
0.17284083
223.7550002
0.077234703
1973
402.565
76.242
1.1071
47.119
0.189390533
248.7927945
0.09575112
1974
393.599
88.949
1.1477
52.197
0.225988887
230.9715343
0.193242784
1975
386.71
108.923
1.1439
58.383
0.281665848
207.2775257
0.246370347
1976
397.225
129.328
1.1443
64.97
0.325578702
199.5523649
0.155904079
1977
396.143
146.204
1.1273
74.595
0.369068745
202.1168168
0.133577666
1978
411.909
167.515
1.1247
85.77
0.406679631
210.9031127
0.101907533
1979
426.915
197.906
1.1299
98.131
0.463572374
211.6843141
0.139895728
1980
414.792
227.537
1.1378
114.923
0.548556867
209.5006132
0.183325189
1981
408.223
249.322
1.1474
138.363
0.610749517
226.5462292
0.113375027
1982
417.916
274.649
1.1288
154.909
0.65718709
235.7152207
0.076033746
1983
438.768
302.895
1.108
175.299
0.690330653
253.9348343
0.050432462
1984
450.949
326.498
1.1069
198.93
0.724024224
274.7560003
0.048807874
1985
464.316
354.291
1.1062
224.794
0.763038534
294.6037323
0.053885365
1986
487.33
388.179
1.0987
258.304
0.796542384
324.2815513
0.043908464
1987
513.083
428.721
1.0947
304.948
0.835578259
364.9544452
0.049006652
1988
553.461
488.953
1.0936
358.233
0.883446169
405.4949953
0.057287165
1989
569.719
537.279
1.0958
426.322
0.943059649
452.0626034
0.067478339
1990
566.238
566.238
1.1108
477.138
1
477.138
0.060378314
1991
548.532
581.897
1.0992
504.133
1.060825986
475.2268576
0.060825986
1992
549.543
605.295
1.0912
517.883
1.10145157
470.1822706
0.03829618
1993
561.346
638.4
1.0787
544.055
1.137266499
478.3883115
0.032516118
1994
580.092
675.164
1.0805
567.157
1.163891245
487.2938108
0.023411176
The above table summarizes the rate of inflation, we cannot derive the rate of inflation for the year 1963 due to the fact that we need the price index for the year 1962 which is not provided by our data.
Question 4:
Real interest rates:
RINTL = INTL - PI
YEAR
RTDE
QTDE
INTL
M4
p
RM4
PI
RINTL
1963
288.856
30.814
1.053
14.969
0.106675991
140.3221089
1964
307.559
34.037
1.058
16.106
0.110668197
145.5341321
0.037423662
1.0205763
1965
312.417
36.256
1.0643
17.616
0.116050023
151.7966094
0.048630284
1.0156697
1966
317.366
38.395
1.0691
18.757
0.120980193
155.0419081
0.042483148
1.0266169
1967
329.964
40.985
1.068
21.158
0.12421052
170.3398393
0.026701286
1.0412987
1968
341.032
44.39
1.0754
22.964
0.130163738
176.4239434
0.047928455
1.0274715
1969
341.179
46.853
1.0905
24.123
0.137326741
175.6613454
0.055030704
1.0354693
1970
349.045
51.27
1.0921
27.009
0.146886505
183.8766609
0.069613275
1.0224867
1971
357.267
57.323
1.0885
31.4
0.160448628
195.7012683
0.09233063
0.9961694
1972
372.979
64.466
1.089
38.674
0.17284083
223.7550002
0.077234703
1.0117653
1973
402.565
76.242
1.1071
47.119
0.189390533
248.7927945
0.09575112
1.0113489
1974
393.599
88.949
1.1477
52.197
0.225988887
230.9715343
0.193242784
0.9544572
1975
386.71
108.923
1.1439
58.383
0.281665848
207.2775257
0.246370347
0.8975297
1976
397.225
129.328
1.1443
64.97
0.325578702
199.5523649
0.155904079
0.9883959
1977
396.143
146.204
1.1273
74.595
0.369068745
202.1168168
0.133577666
0.9937223
1978
411.909
167.515
1.1247
85.77
0.406679631
210.9031127
0.101907533
1.0227925
1979
426.915
197.906
1.1299
98.131
0.463572374
211.6843141
0.139895728
0.9900043
1980
414.792
227.537
1.1378
114.923
0.548556867
209.5006132
0.183325189
0.9544748
1981
408.223
249.322
1.1474
138.363
0.610749517
226.5462292
0.113375027
1.034025
1982
417.916
274.649
1.1288
154.909
0.65718709
235.7152207
0.076033746
1.0527663
1983
438.768
302.895
1.108
175.299
0.690330653
253.9348343
0.050432462
1.0575675
1984
450.949
326.498
1.1069
198.93
0.724024224
274.7560003
0.048807874
1.0580921
1985
464.316
354.291
1.1062
224.794
0.763038534
294.6037323
0.053885365
1.0523146
1986
487.33
388.179
1.0987
258.304
0.796542384
324.2815513
0.043908464
1.0547915
1987
513.083
428.721
1.0947
304.948
0.835578259
364.9544452
0.049006652
1.0456933
1988
553.461
488.953
1.0936
358.233
0.883446169
405.4949953
0.057287165
1.0363128
1989
569.719
537.279
1.0958
426.322
0.943059649
452.0626034
0.067478339
1.0283217
1990
566.238
566.238
1.1108
477.138
1
477.138
0.060378314
1.0504217
1991
548.532
581.897
1.0992
504.133
1.060825986
475.2268576
0.060825986
1.038374
1992
549.543
605.295
1.0912
517.883
1.10145157
470.1822706
0.03829618
1.0529038
1993
561.346
638.4
1.0787
544.055
1.137266499
478.3883115
0.032516118
1.0461839
1994
580.092
675.164
1.0805
567.157
1.163891245
487.2938108
0.023411176
1.0570888
Question 5:
Estimate RM4t=
The following table summarizes the calculations:
Y
X
Y-Y'
X-X'
YEAR
RM4
RINTL
y
x
yx
x2
y2
1964
145.5341
1.020576
-129.289
-0.0012
0.155314
1.4431E-06
16715.58
1965
151.7966
1.01567
-123.026
-0.00611
0.751434
3.7307E-05
15135.46
1966
155.0419
1.026617
-119.781
0.004839
-0.57965
2.3418E-05
14347.48
1967
170.3398
1.041299
-104.483
0.019521
-2.03962
0.00038107
10916.7
1968
176.4239
1.027472
-98.3989
0.005694
-0.56027
3.2421E-05
9682.347
1969
175.6613
1.035469
-99.1615
0.013692
-1.35769
0.00018746
9833.006
1970
183.8767
1.022487
-90.9462
0.000709
-0.06449
5.0282E-07
8271.211
1971
195.7013
0.996169
-79.1216
-0.02561
2.026166
0.00065578
6260.226
1972
223.755
1.011765
-51.0679
-0.01001
0.511308
0.00010025
2607.926
1973
248.7928
1.011349
-26.0301
-0.01043
0.271461
0.00010876
677.5643
1974
230.9715
0.954457
-43.8513
-0.06732
2.952089
0.00453204
1922.939
1975
207.2775
0.89753
-67.5453
-0.12425
8.392371
0.01543756
4562.372
1976
199.5524
0.988396
-75.2705
-0.03338
2.512658
0.00111434
5665.647
1977
202.1168
0.993722
-72.706
-0.02806
2.03979
0.0007871
5286.169
1978
210.9031
1.022792
-63.9197
0.001015
-0.06487
1.0299E-06
4085.734
1979
211.6843
0.990004
-63.1385
-0.03177
2.006124
0.00100955
3986.476
1980
209.5006
0.954475
-65.3222
-0.0673
4.396371
0.00452967
4266.996
1981
226.5462
1.034025
-48.2766
0.012247
-0.59126
0.00015
2330.633
1982
235.7152
1.052766
-39.1076
0.030989
-1.21189
0.00096029
1529.407
1983
253.9348
1.057568
-20.888
0.03579
-0.74758
0.00128092
436.3096
1984
274.756
1.058092
-0.06686
0.036314
-0.00243
0.00131874
0.00447
1985
294.6037
1.052315
19.78087
0.030537
0.604049
0.00093251
391.2829
1986
324.2816
1.054792
49.45869
0.033014
1.632825
0.00108992
2446.162
1987
364.9544
1.045693
90.13159
0.023916
2.155562
0.00057196
8123.703
1988
405.495
1.036313
130.6721
0.014535
1.899346
0.00021127
17075.21
1989
452.0626
1.028322
177.2397
0.006544
1.159862
4.2824E-05
31413.93
1990
477.138
1.050422
202.3151
0.028644
5.795126
0.00082048
40931.42
1991
475.2269
1.038374
200.404
0.016596
3.325982
0.00027544
40161.76
1992
470.1823
1.052904
195.3594
0.031126
6.080794
0.00096884
38165.3
1993
478.3883
1.046184
203.5655
0.024406
4.96827
0.00059567
41438.89
1994
487.2938
1.057089
212.471
0.035311
7.502603
0.00124688
45143.91
total
8519.509
31.67511
0
6.66E-16
53.91976
0.03940544
393811.7
mean
274.8229
1.021778
B = ∑yx/∑x
B = 53.91976/0.03940544
B = 1368.332775
A = -1123.308961
We state the model as follows:
RM4t= - 1123.308961 + 1368.332775 RINTL
Question 6:
RM4t= - 1123.308961 + 1368.332775 RINTL
The following model has a constant and a slope value, from the model it is evident that if we hold all other factors constant and let the value of RINTL to be zero then the value of RM4 will be - 1123.308961, also if we hold all other factors constant and increase RINTL by one unit then RM$ will increase by 1368 units. From our result there it means that an increase in the real interest rate will increase the level of real money supply.
Coefficients:
We test the significance of the autonomous value and slope value at 98% level of test:
We state the null and alternative hypothesis in both cases, we start with the autonomous value :
Null hypothesis H0: a = 0, alternative hypothesis Ha: a ≠ 0. For the slope we state the null hypothesis as H0: b = 0 and the alternative hypothesis as Ha: b ≠ 0. We test both of this hQypothesis at 95% level of test taking into consideration the sample size and also a two tail test, the critical value is 2.04523. after determining the standard errors of our coeeficients the following is the results:
coefficient
value
std error
T calculated
T critical 95%
null hypothesis
a
-1123.30896
529.199369
-2.1226574
2.04523
reject
b
1368.332775
3012.45646
0.45422491
2.04523
accept
The table summarizes the T calculated values, from the above it is evident that we reject the null hypothesis for the autonomous value because the T calculated is greater than the T critical, we accept the null hypothesis for the slope because the T calculated is smaller than the T critical.
For this reason therefore at 95% level of test the autonomous value is statistically significant while at the same level of test the slope value is not statistically significant.
Question 7:
We now estimate the model
RM4t= a + b1RINTL + b2RDTE
After estimating the model is stated as follows:
RM4t= - 534.66+ 267.9 RINTL + 1.24 RDTE
This model means that if we hold all factors constant and that the value of b1 and b2 are zero then the value of RM4 will be -534.66, if we hold all factors constant and increase the value of RINTL by one unit then RM4 will increase by 267.9 units, finally if we hold all other factors constant and increase RDTE by one unit then the level of RM4 will increase by 1.24 units.Statistical significance of the data shows the following results:
Null hypothesis H0: a = 0, alternative hypothesis Ha: a ≠ 0. For b1 we state the null hypothesis as H0: b1 = 0 and the alternative hypothesis as Ha: b1 ≠ 0, For b2 we state the null hypothesis as H0: b2 = 0 and the alternative hypothesis as Ha: b2 ≠ 0. We test this hypothesis at 95% level of test taking into consideration the sample size and also a two tail test, the critical value is 2.04841. After determining the standard errors of our coefficients the following is the results:
coefficient
value
standard deviation
T calculated
T critical 95%
null hypothesis
a
(534.66)
109.1178754
(4.90)
2.04841
reject
b1
267.9060968
106.7331702
2.51
2.04841
reject
b2
1.24
1.286465098
0.97
2.04841
accept
The table summarizes the T calculated values, from the above it is evident that we reject the null hypothesis for the a because the T calculated is greater than the T critical, we also reject the null hypothesis for the b1 because the T calculated is greater than the T critical, we however accept the null hypothesis b2 because the T calculated is smaller than the T critical.
For this reason therefore at 95% level of test both the autonomous value and b1 are statistically significant while b2is not statistically significant at 95% level of test.
Question 8:
We now estimate the model
RM4t= a + b1RINTL + b2PI
After estimating the model is stated as follows:
RM4t= -701826551.6 + 686607260.3RINTL + 3279238.325 PI
This model means that if we hold all factors constant and that the value of b1 and b2 are zero then the value of RM4 will be -701826551.6, if we hold all factors constant and increase the value of RINTL by one unit then RM4 will increase by 686607260.3 units, finally if we hold all other factors constant and increase PI by one unit then the level of RM4 will increase by 3279238.325 units.
Statistical significance of the data shows the following results:
Null hypothesis H0: a = 0, alternative hypothesis Ha: a ≠ 0. For b1 we state the null hypothesis as H0: b1 = 0 and the alternative hypothesis as Ha: b1 ≠ 0, For b2 we state the null hypothesis as H0: b2 = 0 and the alternative hypothesis as Ha: b2 ≠ 0. We test this hypothesis at 95% level of test taking into consideration the sample size and also a two tail test, the critical value is 2.04841. After determining the standard errors of our coefficients the following is the results:
coefficient
value
standard deviation
T calcualted
T critical 95%
null hypothesis
a
-701826551.6
132613646.1
-5.292265
2.04841
reject
b1
686607260.3
129773744.2
5.2908026
2.04841
reject
b2
3279238.325
2496305049
0.0013136
2.04841
accept
The table summarizes the T calculated values, from the above it is evident that we reject the null hypothesis for the a because the T calculated is greater than the T critical, we also reject the null hypothesis for the b1 because the T calculated is greater than the T critical, we however accept the null hypothesis b2 because the T calculated is smaller than the T critical.
For this reason therefore at 95% level of test both the autonomous value and b1 are statistically significant while b2is not statistically significant at 95% level of test. The coefficient of PI in this model means that as inflation increase then the real money supply will also increase.
Question 10:
Joint hypothersis testing:
Null hypothesis:
H0: b1 = b2 = 0
Alternative hypothesis:
Ha: b1 ≠ b2 ≠0
Given that the value of F is derived from the following formula:
f = ((b1yx1 +b2yx2)/2 ) / (e2/n-3)
we use the following figures:
b1
686607260.3
b2
3279238.325
yx1
8726.598196
yx2
286875.8435
e2
1.5265E+19
After calulcation our F value is 0.0000063580, from the table we use the F table to determine the critical value, for our case the critical value is using the 2 degrees of freedom for the numerator and 28 degrees of freedom for the denominator 5.453, this means that we accept the null hypothesis because the critical value is greater than our calculated value, therefore this means that b1 and b2 are not statistically significant.
REFERENCEs:
ONS database
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