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Introduction to statistics - Essay Example

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Summary
The data below shows the percentage of goods moved by Railways and Waterways within a country for the years 2000 and 2009.
No; the data does not…
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Introduction to statistics
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INTRODUCTION TO STATISTICS By of the of the School Q1: It has been suggested that the use of Railways and Waterways for the internal transportation of goods in Europe is in decline. The data below shows the percentage of goods moved by Railways and Waterways within a country for the years 2000 and 2009.
Table 1: Data
Country
Rail and Waterways
2000
2009
Belgium
 
22.6
27.1
Denmark
 
7.9
9.2
Germany
 
34.7
33.0
Ireland
 
3.8
0.6
Spain
 
7.2
3.4
France
 
24.0
19.0
Italy
 
11.0
9.0
Netherlands
36.6
36.6
Portugal
 
7.5
5.7
United Kingdom
10
13.3
(a) Do the data support the suggestion?
SOLUTION
No; the data does not provide any evidence of decline in the use of Railways and Waterways for the internal transportation of goods in Europe. The above results can be confirmed from the linear graph shown below. We used a line graph because they are used to display data or information that changes continuously over time. Line graphs also allow us to see overall trends such as an increase or decrease in data over time. Clearly, looking at the graph we don’t observe any disparity in the data given
The intuition can further be confirmed from the t-test significance analysis conducted. From the table below, the percent of goods moved in the year 2000 was 16.53 while that moved in the year 2009 was 15.69, this shows no significant difference. Also from the same table (table 2), we observe that the p-value=0.206>0.05 (significance level), we thus fail to reject the null hypothesis and conclude that indeed there is no significant difference in the goods moved in the year 2000 and those moved in the year 2009.
Table 2: T-Test Analysis
t-Test: Paired Two Sample for Means
a
0.05
 
Year-2000
Year-2009
Mean
16.53
15.69
Variance
144.8823
161.0166
Observations
10
10
Pearson Correlation
0.970264
Hypothesized Mean Difference
0
df
9
t Stat
0.861
P(T 0.05 (Means are the same)
T Critical one-tail
1.833
P(T 0.05 (Means are the same)
T Critical Two-tail
2.262
 
(b) If the data DOES support the suggestion find the mean difference and its 95% confidence limits.
SOLUTION
N/A
(c) If the data DOES NOT support the suggestion produce a model to show the relationship between the data for 2009 and 2000 with 95% confidence limits and explain BRIEFLY what the model shows.
SOLUTION
In this part we regressed the 2009 data on the 2000 data since the data was found to follow a normal distribution we opted to use a parametric test (Bivariate regression analysis), we used bivariate because we only had two variables to test (simple linear regression) and the model produced is;
Where represents the year 2000
From table 3 below, we observe that the R-Squared (R2) is 0.941 which implies that 94.1% of the variation in 2009 data is affected (or explained) by the 2000 data (explanatory variable)
Table 3:Regression Statistics
Multiple R
0.970
R Square
0.941
Goodness of Fit >= 0.80
Adjusted R Square
0.934
Standard Error
3.258
Observations
10
From table 4 below, we observe that the coefficient for the 2000 data is 1.023 which implies that for a unit change in 2000 data, the 2009 data changes by a factor of 1.023
Table 4: ANOVA
 
df
SS
MS
F
P-value
Regression
1
1364.2
1364.25
128.55
0.000
Residual
8
84.902
10.6128
Total
9
1449.1
 
 
95%  Confidence Level
 
Coefficients
Standard Error
t Stat
P-value
Lower 95%
Upper 95%
Intercept
-1.22
1.8125
-0.672
0.521
-5.3976
2.9617
Year-2000
1.02
0.0902
11.3379
0.000
0.8148
1.2309
(d) Using the model, find the 95% confidence limits for the proportion expected to be moved in 2009 for a country where 20% of goods were moved in this way in 2000.
SOLUTION
If 20% of goods were moved in 2000, using our model we can compute the proportion goods moved in 2009. The model is;
Thus in 2009, the proportion of goods moved was 19.242.
We now find the 95% confidence limits as follows;
However, for our case we use excel to compute this;
The confidence as obtained from the excel is 7.460292. hence the corresponding 95% confidence limits are;
i.e. the upper limit is while the lower limit is Read More
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