Question

Structurally deficient highway bridges. Data on structurally deficient highway bridges is compiled by the Federal Highway Administration (FHWA) and reported in the National Bridge Inventory (NBI). For each state, the NBI lists the number of structurally deficient bridges and the total area (thousands of square feet) of the deficient bridges. The data for the 50 states (plus the District of Columbia and Puerto Rico). For future planning and budgeting, the FHWA wants to estimate the total area of structurally deficient bridges in a state based on the number of deficient bridges

NumberSD	SDArea
1899	432.71
155	60.92
181	110.57
997	347.35
3140	5177.97
580	316.92
358	387.78
20	9.05
24	59.34
302	412.92
1028	344.86
142	39.8
349	135.43
2501	1192.43
2030	688.19
5153	1069.71
2991	527.47
1362	458.37
1780	1453.26
349	131.13
388	236.18
585	521.83
1584	804.15
1156	325.9
3002	692.75
4433	1187.42
473	90.94
2382	335.75
47	20.08
383	127.66
750	752.43
404	196.67
2128	1427.73
2272	1034.61
743	101.42
2862	965.16
5793	1423.25
514	393.96
5802	2404.61
164	237.96
1260	626.38
1216	209.33
1325	481.31
2186	1031.45
233	102.56
500	153.8
1208	483.68
400	502.03
1058	331.59
1302	399.8
389	143.46
241	195.43

e) Build a 90% CI, confidence interval, for coefficient of NumberSD ( ).

f) Repeat (e) with a 95% CI. What is the difference between your answer in (e) and (f)?

Answer 1

We will use R-software to make scatterplot ,and to fit a regression model.

Given data is

NumberSD	SDArea
1899	432.71
155	60.92
181	110.57
997	347.35
3140	5177.97
580	316.92
358	387.78
20	9.05
24	59.34
302	412.92
1028	344.86
142	39.8
349	135.43
2501	1192.43
2030	688.19
5153	1069.71
2991	527.47
1362	458.37
1780	1453.26
349	131.13
388	236.18
585	521.83
1584	804.15
1156	325.9
3002	692.75
4433	1187.42
473	90.94
2382	335.75
47	20.08
383	127.66
750	752.43
404	196.67
2128	1427.73
2272	1034.61
743	101.42
2862	965.16
5793	1423.25
514	393.96
5802	2404.61
164	237.96
1260	626.38
1216	209.33
1325	481.31
2186	1031.45
233	102.56
500	153.8
1208	483.68
400	502.03
1058	331.59
1302	399.8
389	143.46
241	195.43

First we will import data into R

> NumberSD=scan("clipboard")
Read 52 items
> SDArea=scan("clipboard")
Read 52 items

> head(data.frame(NumberSD,SDArea),10)          # to print first 10 observations
   NumberSD SDArea
1      1899       432.71
2       155 60.92
3       181       110.57
4       997 347.35
5      3140      5177.97
6       580        316.92
7       358        387.78
8        20          9.05
9        24         59.34
10      302         412.92

e)

Build a 90% CI, confidence interval, for coefficient of NumberSD ( b1).

90% CI, confidence interval, for ( b1). is given by

CI = { - * SE() , + * SE() }

Here = 0.34560    and    SE() = 0.06158

is t-distributed with n-2 = 52-2 = 50 degree of freedom and =0.10, { for 90% CI, }

It can be computed from statistical book or more accurately from any software like R,Excel

From R

> qt(1-.1/2,50)
[1] 1.675905

Thus = 1.675905

Hence 90% CI, confidence interval, for ( b1) is given by

CI = { - * SE() , + * SE() }

    = { 0.34560 - 1.675905 * 0.06158 , 0.34560 + 1.675905 * 0.06158 }

    = { 0.2423978 , 0.4488022 }

90% CI, confidence interval, for coefficient of NumberSD ( b1) is { 0.24240 , 0.44880 }

f)

Repeat (e) with a 95% CI. What is the difference between your answer in (e) and (f)?

90% CI, confidence interval, for ( b1). is given by

CI = { - * SE() , + * SE() }

is t-distributed with 50 degree of freedom but =0.05, { for 95% CI, }

From R

> qt(1-.05/2,50)
[1] 2.008559

Thus = 2.008559

Hence 95% CI, confidence interval, for ( b1) is given by

CI = { - * SE() , + * SE() }

    = { 0.34560 - 2.008559 * 0.06158 , 0.34560 + 2.008559 * 0.06158 }

    = { 0.2219129 , 0.4692871 }

95% CI, confidence interval, for coefficient of NumberSD ( b1) is { 0.22191 , 0.46929 }

Difference between part in (e) and (f) is that 95% confidence interval which is { 0.22191 , 0.46929 } is greater than that of 90% confidence interval { 0.24240 , 0.44880 } . ie A 90 % confidence interval for ( b1) is narrower .

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