Question 1: The following table gives measurements of the area ( x in km 2 ) and p H level ( y) of 13 lakes in Ontario, Canada. ( hand calculation in nothing else is stated)
Ar ea( x)
|
33
|
161
|
189
|
149
|
47
|
170
|
352
|
187
|
76
|
52
|
175
|
53
|
200
|
p H( y)
|
6 . 6
|
6 . 4
|
6 . 5
|
6 . 9
|
7 . 1
|
7 . 5
|
8 . 8
|
6 . 4
|
5 . 9
|
6 . 7
|
7 . 1
|
6 . 6
|
8 . 0
|
- ) Sketch the scatterplot of y vs x and comment on the plot. ( use Minitab or R/ Rstudio)
- ) Use the Principle of Least Squares to f it the simple linear regression model to the data. Superimpose this line of best f it on the scatterplot in part ( a).
- ) Perform an ANOVA test to deduce whether there is a l inear relationship between area and p H level.
- ) Perform all appropriate residual checks using R/ Rstudio or MINITAB and clearly explain if any of the model assumptions have been violated.
- ) Another lake in the same region was found to have an area of 2050 km 2. Predict its p H level and f ind a 99% confidence interval of this prediction.
( 3 + 8+ 5 + 8+ 3 = 27)
Question 2: ( Everything to be done by hand expect is otherwise is stated) The following data are provided:
a/a
|
1
|
2
|
3
|
4
|
5
|
6
|
7
|
8
|
9
|
10
|
11
|
12
|
13
|
X
|
35.3
|
29.7
|
30.8
|
58.8
|
61.4
|
71.3
|
74.4
|
76.7
|
70.7
|
57.5
|
46.4
|
28.9
|
28.1
|
Y
|
10.98
|
11.13
|
12.51
|
8.40
|
9.27
|
8.73
|
6.36
|
8.50
|
7.82
|
9.14
|
8.24
|
12.19
|
11.88
|
a/a
|
14
|
15
|
16
|
17
|
18
|
19
|
20
|
21
|
22
|
23
|
24
|
25
| |
X
|
39.1
|
46.8
|
48.5
|
59.3
|
70
|
70
|
74.5
|
72.1
|
58.1
|
44.6
|
33.4
|
28.6
| |
Y
|
9.57
|
10.94
|
9.58
|
10.09
|
8.11
|
6.83
|
8.88
|
7.68
|
8.47
|
8.86
|
10.36
|
11.08
|
Where X represent the steam in pounds per months and Y is the mean atmosphere temperature measured in Fahrenheit.
Calculate the followings:
Calculate the followings:
- Fit a l inear regression model and give the least square estimates for the constant and the slope.
- Calculate the residuals for each of the 25 observations.
- Make the ANOVA table and complete it by performing all the required calculations. W hat is the ANOVA table used for?
- Find the coefficient of determination and the correlation coefficient. Explain its value.
- Calculate the std for the error, the std for the slope and the std for the constant.
- Test whether the slope and the constant are significant. State the needed hypotheses and explain your results.
- Construct the confidence intervals for the slope and for the constant.
( 8 + 4+ 6 + 4+ 6 + 4+ 6 = 38 )
Question 3: For the data provided in the above question 2 do the following using a statistical software ( either Minitab or R/ Rstudio will do). Include the used code for R or describe the steps in details for Minitab.
- Generate a scatterplot of the data and comment on it .
- Answer all the queries a)- g) of Question 2 and comment on the derived outputs.
- Using the generated residuals, test the assumption behind the simple linear regression.
( 2 + 26+ 7= 35)
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