Problem 1: The Mean Corporation has been commissioned to conduct a study is into the relationship between the population of a city and the number of motor vehicle accidents in the city per year. A linear regression model is to be constructed. In the proposed regression model, number of motor vehicle accidents per year is the response variable and population of the city is the explanatory variable.
A random sample of 20 cities is selected and measurements are observed.
Population No. accidents per year
('000s)
1,030 4,155
2,890 6,658
740 2,900
3,440 7,381
730 3,273
1,010 3,784
3,190 6,676
1,620 5,053
1,900 5,250
2,310 6,288
2,960 7,137
1,840 5,033
2,380 6,276
2,940 7,375
2,390 5,540
1,790 5,217
1,980 5,608
2,010 5,751
2,620 5,939
2,120 6,076
a) Calculate the point prediction for the value x = 2,500. Give your answer as a whole number.
y^ =
b) Give the 95% prediction interval for the value x = 2,500. Give your answers as whole numbers.
≤ y^ ≤
Problem 2: An investigation has been conducted to determine whether there is a relationship between the salary paid to a Chief Executive Officer (CEO) and the productivity of that CEO as measured by the change in profits from the time the CEO was employed.
The scatter plot plots the salary paid (x) against the change in profits (y) of a sample of CEOs. Without doing any calculations and according only to this scatter diagram, a reasonable coefficient of correlation (r) between x and y would be:
- r = -1.25
- r = -0.59
- r = 0.68
- r = 1.35
Proble 3: Two variables (A and B) are hypothesised to have a linear relationship with one another, as represented by the following equation: A = β0 + β1B + ε
Data was gathered for these two variables and the correlation coefficient (r) was calculated to be -0.96. Select all from the following statements that are true:
- The graph of the linear relationship between A and B slopes up.
- Variable A is known as the explanatory variable, and variable B is known as the response variable.
- The linear regression line provides a strong fit to the observed data.
- The proportion of variability in A that is explained by the regression model is equal to 92.16%.
- Causation between A and B cannot be implied from the correlation that exists between them.
- β0 is known as the sample intercept.
- Variable B causes variable A since B is the independent variable.
Problem 4: A study is conducted to determine the simple linear regression relation (if any) between average temperature over a week and ice-cream sales in a particular city. The following table provides the regression data required to construct the model:
Average Temperature and Ice-cream Sales:
Average Temperature Weekly Ice-cream Sales
over a Week (°F) (000s)
85 202.238
37 118.724
65 167.318
54 148.53
31 108.186
59 156.77
75 185.198
33 111.082
64 166.63
53 146.902
48 138.302
88 208.182
Calculate the slope (b1) and intercept (b0) of the simple regression equation using the data provided. Give your answers to 2 decimal places.
(a)Slope = b1 =
(b)Intercept = b0 =
Problem 5: A biologist is studying the levels of heavy metal contaminants among a population of the South Nakaratuan Chubby Bat. The biologist is interested in constructing a simple linear regression model to investigate the relationship between weight of an animal and the level of heavy metal contamination. In the proposed regression model the level of contaminant is the response variable and weight is the explanatory variable. The contaminant level is measured in parts per billion (ppb) and weight in grams.
A random sample of 20 individuals is selected and measurements are taken.
Contaminant study in
South Nakaratuan Chubby Bat
Contaminant level
(ppb) Weight
(g)
210 144
149 102
157 105
193 132
179 129
177 131
165 114
145 106
218 143
170 121
176 128
157 116
208 147
183 120
155 110
152 112
186 139
203 139
135 101
251 110
Plotting the data, the researcher notices an obvious outlier. They decide to do the regression analysis with and without the outlier and compare the results.
Calculate the slope (b1) and intercept (b0) of the simple regression equation using the data provided. Give your answers to 2 decimal places.
a) Slope = b1 =
b)Intercept = b0 =
Find the proportion of variation in the values of contaminant level that is explained by the regression model. Give your answer as a decimal to 2 decimal places.
c) R2 =
Repeat this process omitting the outlier:
d) Slope = b1 =
e) Intercept = b0 =
Find the proportion of variation in the values of contaminant level that is explained by the regression model. Give your answer as a decimal to 2 decimal places.
Problem 6: A study was conducted into the relationship between the age in years of a person and their average weekly after tax income. You have been supplied with the following information regarding a regression model that had been developed as part of the study.
Regression analysis
sample intercept 509.91
sample slope 3.08
A particular observed value from the data is:
x = 20, y = 550.44
Calculate the residual of this observed value of the response variable. Give your answer to 2 decimal places.
Problem 7: You have constructed a simple linear regression model and are testing whether the assumption of independence of the residuals is reasonably satisfied. Select the scatter plot that indicates independence of the residuals:
Problem 8: The following table shows the average petrol price and the number of online shopping orders over a given month:
Petrol Price and Online Shopping
Average Petrol Price
per gallon over a month ($) Number of Online
Shopping Orders
3.94 3,920
1.995 2,278
1.37 1,575
3.9225 3,405
3.37 3,500
3.39 3,626
2.6525 2,202
2.5925 2,193
3.34 2,731
2.415 2,473
The relationship between the average petrol price and the number of online shopping orders in a given month is proposed to follow the simple regression equation below:show variables
y^ = b0 + b1x
Calculate the proportion of variability in the number of online shopping orders that is not explained by the average petrol price. Give your answer as a percentage to 1 decimal place.
Proportion = %
Problem 9: A cinema manager wishes to investigate the pattern of ticket sales over the different days of the week. Starting on a Monday, she records the number of tickets sold each day for one week. The results are recorded below
Day Tickets sold
1 321
2 545
3 412
4 901
5 1290
6 1215
7 1063
In the time-series plot below, only four of the values have been marked. Drag the three red markers onto the plot to complete it and represent the data for all seven days.
Problem 10: It has been hypothesised that there is a linear relationship between the average annual interest rate and the volume of car sales, and that relationship can be represented as follows: show variables
y^ = b0 + b1x
The following table lists the average interest rates recorded in a particular year and the corresponding number of car sales:
Interest Rates and Car Sales
Interest Rate (% per annum) Car Sales Volume (millions)
8.4 2.508
4.9 3.983
3.7 4.109
5.6 4.002
8.4 2.158
5 4.32
3.4 4.418
7.4 2.348
7.3 2.581
3.8 4.356
Using the data provided, and at α = 0.05, the null hypothesis that a significant linear relationship between the average annual interest rate and the volume of car sales does not exist is . You may find this Student's t distribution table useful.
Problem 11: The regression equation: y^ = 5 + 1.2x was calculated from a sample. It is part of a regression model that has been developed in order to predict the score in an end-of-year exam based on the score in a mid-year exam for a particular university course. In the sample, mid-year scores ranged from 50 to 80.
Select whether or not each of the following conclusions are correct from the regression analysis:
Correct Not correct
a) For an increase by one in the mid-year score, the predicted increase in end-of-year score is 1.2.
b) If a student achieves a score of 45 in the mid-year exam then we know that the student will achieve a score of 59 in the end-of-year exam.
c) If a student achieves a score of 60 in the mid-year exam then we know that the student will achieve a score of 77 in the end-of-year exam.