Math1312 Regression Analysis Assignment Brief
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)
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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:
- 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)