Module 4 Exam - Statistics: Correlation, Variance, and Data Analysis

This document provides a detailed solution to a statistics exam (Module 4) focusing on correlation, variance, and data analysis. The solution includes explanations of the relationship between variables (A and C), examples of positive correlations, and an analysis of how correlation might differ between different student groups (entire class vs. top students). The document calculates deviation scores, determines the nature of correlation (positive or negative), and calculates the correlation coefficient. It also addresses questions related to the variance of combined test scores and correlation and provides true/false answers with explanations related to correlation, bivariate data, and the interpretation of correlation coefficients. The student has provided the solutions to all the questions from the exam with step-by-step explanations.

Exam Module 4
1. Describe the relationship between variables A and C. Think of things these variables could
represent in real life.
Solution: The Variables A and C according to the above graph show a negative correlation
i.e the increase in the value of A shows a decrease in the value of C .
2. Make up a data set with 10 numbers that has a positive correlation.
Solution: I have created two Variables 1 & 2 as shown below and plotted the graph in Excel
which shows a positive correlation
Variable
1
Variable
2
2 3
4 5
7 8
8 11
11 13
14 15
17 18
19 21
21 23
Table: Variables 1 & 2

0 5 10 15 20 25
0
5
10
15
20
25
3
5
8
11
13
15
18
21
23
Variable 2
Variable 1
Fig: Positive Correlation between 1 & 2
3. Would you expect the correlation between High School GPA and College GPA to be higher
when taken from your entire high school class or when taken from only the top 20 students?
Why?
Solution: The Correlation will be higher only when we consider the top 20 Students as there
may not be higher as the Academic Performance of Topper i.e the top 20 may not Vary that
much.
4. For the following data:
A B
2
5
6
8
9
8
5
2
4
1
a. Find the deviation scores for Variable A that correspond to the raw scores of 2 and 8.
Solution: The Standard Deviation for the A variable is 2.07 hence the deviation for
score 2 is 4.07 as 2+ 2.07 =4.07
The Deviation for score 8 is 8 – 2.07 = 5.93.

b. Find the deviation scores for Variable B that correspond to the raw scores of 5 and 4.
Solution: The Standard Deviation for the B Variable is 2.73 hence the deviation for
score 5 is 5 – 2.73 = 2.27 and for 4 is 4 – 2.73 = 1.27
c. Just from looking at these scores, do you think these variable A and B are positively
or negatively correlated? Why?
Solution: The Values is the A columna are grdually increasing whereas the values in
the B column are decreasing. Hence there is a negative Correlation between both the
variables.
d. Now calculate the correlation. Were you right?
Solution: The Correlation between the variables is -0.79. So they are negatively
correlated.
5. Students took two parts of a test, each worth 50 points. Part A has a variance of 25, and
Part B has a variance of 36. The correlation between the test scores is 0.8. (a) If the teacher
adds the grades of the two parts together to form a final test grade, what would the
variance of the final test grades be? (b) What would the variance of Part A - Part B be?
Solution(A): The formula for Covariance between two variables is 24
Solution:
Covariance = Correlation * Standard deviation of first variable * Standard Deviation of
Second variable
Covariance = 0.8 * 5 (Square root of 25) * 6 (Square root of 36) = 24
6. True/False: Two variables with a correlation of 0.3 have a stronger linear relationship than
two variables with a correlation of -0.7.
Solution: True
Explanation: Variables having correlation of 0.3 are having positive correlation , but the
variables those have negative correlation i.e -0.7 wouldn’t have a better linear relationship
as compared to the positive ones.
7. True/False: After polling a certain group of people, researchers found a 0.5 correlation
between the number of car accidents per year and the driver's age. This means that older
people get in more accidents.
Solution: True

Explanation: Yes, as there is a positive correlation between the car accidents and the
driver’s age, it means increase in the age increases the possibility of accidents.
8. True/False: The correlation between R and T is the same as the correlation between T and
R.
Solution: True
Explanation: Yes , the correlation between the variables remains the same if vice versa
also
9. True/False: To examine bivariate data graphically, the best choice is two side by side
histograms.
Solution: False
Explanation: The best graph to analyze bivariate data is Scatter plot or a line graph
10. True/False: A correlation of r=1.2 is not possible.
Solution: True
Explanation: Correlation only varies between 0 to 1.