#### Solved Examples and Worksheet for Correlation Coefficient, Causation and Correlation

Q1In Colorado, after repeated experiments, doctors proved that excess intake of junk food results in obesity. But when the correlation was calculated between the two, it was found to be negligible. Which of the following would you conclude?

A. there is no causal factor present between the two
B. causal factor is present between the two
C. if correlation exists then causation exists
D. correlation doesn′t imply causation

Step: 1
Correlation is a measure of the strength of the relationship between two variables. It is used to predict the value of one variable given the value of the other.
Step: 2
Causation is the act of causing one variable to happen due to the other variable's effect.
Step: 3
Eating junk food makes you obese therefore there is a causal factor present.
Correct Answer is :   causal factor is present between the two
Q2A research institute found that a strong correlation between coffee consumption and the intake of water. What would you conclude?

A. I dont know
B. correlation doesn′t imply causation
C. if correlation exists then causation exists
D. only correlation exists

Step: 1
Correlation is a measure of the strength of the relationship between two variables. It is used to predict the value of one variable given the value of the other.
Step: 2
Causation is the act of causing one variable to happen due to the other variable's effect
Step: 3
Eating junk food makes you obese therefore there is a causal factor present.
Correct Answer is :   correlation doesn′t imply causation
Q3Is there any relationship between student's scores on an examination and student's cumulative grade- point average (GPA) upon graduation?
A. There is both correlation and causation.
B. There is causation but no correlation.
C. There is a correlation but no causation.
D. There is neither correlation nor causation.

Step: 1
Correlation is a measure of the strength of the relationship between two variables. It is used to predict the value of one variable given the value of the other.
Step: 2
Causation is the act of causing one variable to happen due to the other variable's effect.
Step: 3
There is direct relation of correlation and causation as student's scores increases his cumulative GPA also increases.
Correct Answer is :   There is both correlation and causation.
Q4A company is interested in seeing if a relationship between the age of a truck and the cost to repair the truck are related. They do find that the older truck, the more costly the repair bills. Which statement can be concluded about the data?

A. There is causation but no correlation.
B. There is neither correlation nor causation.
C. There is both correlation and causation.
D. There is a correlation but no causation.

Step: 1
Correlation is a measure of the strength of the relationship between two variables. It is used to predict the value of one variable given the value of the other.
Step: 2
Causation is the act of causing one variable to happen due to the other variable's effect.
Step: 3
In the problem, the age of truck and the cost of repairs are the two variables.
Step: 4
If the age of truck is more, then the cost of repairs is also more. So, there exists a causal factor between the two.
Step: 5
Therefore, both correlation and causation exists.
Correct Answer is :   There is both correlation and causation.
Q5A chemical firm spends $10 million for R & D in 2002 and expects to earn$ 45 million in profits that year. This has been the trend over 6 years. Is there any relationship between the money spent on research and development and the firm's annual profits?

A. There is both correlation and causation.
B. There is a correlation but no causation.
C. There is neither correlation nor causation.
D. There is causation but no correlation.

Step: 1
Correlation is a measure of the strength of the relationship between two variables. It is used to predict the value of one variable given the value of the other.
Step: 2
Causation is the act of causing one variable to happen due to the other variable's effect.
Step: 3
In the problem, the money spent on R & D and the firm's annual profits are the two variables.
Step: 4
If the money spent on R & D is more, then the firm's annual profit is also more. So, there exists a causal factor between the two.
Step: 5
Therefore, both correlation and causation exists.
Correct Answer is :   There is both correlation and causation.
Q6The table shows the relationship between the hours of study per day and the test scores of 10 students.Find the correlation coefficient and describe the relationship.
 Student Hours of study(x) Test scores (y) A 6 48 B 12 72 C 8 65 D 10 70 E 12 78 F 7 65 G 12 82 H 6 50 I 14 85 J 15 90

A. - 0.953; strong negative relationship
B. 0.953; strong positive relationship
C. - 0.173; weak negative relationship
D. 0.173; weak positive relationship

Step: 1
Prepare a table for values x, y, x2, y2, xy. Step: 2
Correlation coefficient,r = n(Σxy)-(Σx)(Σy)[n(Σx2)-(Σx)2][n(Σy2)-(Σy)2]
[Formula]
Step: 3
r = 10(7587)-(102)(705)[10(1138)-(102)2][10(51471)-(705)2]
[Substitute the values from the table and simplify.]
Step: 4
r = 0.953
Step: 5
Therefore, there is a strong positive relationship between the hours of study and test scores .
[When r is close to ± 1, the relationship between the variables is strong and when r is away from ± 1, it is a weak relationship.]
Correct Answer is :   0.953; strong positive relationship
Q7State whether true or false: "Two uncorrelated variables may not be independent."
 x -2 -1 0 1 2 y 64 1 0 1 64
Choose the correct statement for the data given in the table.

A. false; there does not exist a relationship
B. false; two uncorrelated variables are always independent
C. true; there exists a relation between x and y i.e., y = x6
D. true; two uncorrelated variables are always dependent

Step: 1
Make a table with values for x, y, x2, y2, xy. Step: 2
Correlation coefficient, r = n(Σxy)-(Σx)(Σy)[n(Σx2)-(Σx)2][n(Σy2)-(Σy)2] = 0
[Substitute the values from the table and simplify.]
Step: 3
Therefore, x and y are uncorrelated.
Step: 4
But, we can see that x and y are connected and are such that y = x6
Step: 5
Therefore, two uncorrelated variables need not be independent, i.e., they can be dependent also.
Correct Answer is :   true; there exists a relation between x and y i.e., y = x6
Q8A sample of average heights and the shoe sizes of 5 people from a population are given below.
 Height(in inches) 60 62 64 66 66 Shoe size(in inches) 7 8 9 10 11
Calculate the correlation coefficient and describe the relation.

A. - 0.98; strong negative relationship
B. 1; strong positive relationship
C. -1; strong positive relationship
D. 0.98; strong negative relationship

Step: 1
Make a table with values for x, y, x2, y2, xy. Step: 2
Correlation coefficient, r = n(Σxy)-(Σx)(Σy)[n(Σx2)-(Σx)2][n(Σy2)-(Σy)2]
[Substitute the values from the table.]
Step: 3
r = 5(2900)-(320)(45)[5(20520)-(320)2][5(415)-(45)2] = 100100 = 1
[Simplify.]
Step: 4
r = 1
Step: 5
Therefore, there is a strong positive relationship between the heights and their shoe sizes of 5 people .
[When r is close to ± 1, the relationship between the variables is strong and when r is away from ± 1, it is a weak relationship.]
Correct Answer is :   1; strong positive relationship
Q9The average temperature (in °F) and precipitation (in cm) in 7 places of a state in the month of May-2010 are shown.
 Average temperature(in °F) 42.3 40.2 41.8 42.9 43.1 42.6 40.7 Average precipitation(in cm) 0.85 1.72 0.69 2.77 2.46 1.89 0.64
Find the correlation coefficient.

A. 0.76
B. - 0.54
C. 0.54
D. - 0.76

Step: 1
Make a table with values for x, y, x2, y2, xy. Step: 2
Correlation coefficient, r = n(Σxy)-(Σx)(Σy)[n(Σx2)-(Σx)2][n(Σy2)-(Σy)2]
[Substitute the values from the table.]
Step: 3
r = 0.54
[Simplify.]
Step: 4
The correlation coefficient, r is 0.54.
Q10Calculate the correlation coefficient for the data shown and describe the relationship.
 x 34 31 35 30 32 35 32 33 y 35 32 30 31 34 30 33 35

A. - 0.134; weak negative relationship
B. 0.1506; weak positive relationship
C. - 0.1506; weak negative relationship
D. 0.134; weak positive relationship

Step: 1
Make a table with values for x, y, x2, y2, xy. Step: 2
Correlation coefficient, r = n(Σxy)-(Σx)(Σy)[n(Σx2)-(Σx)2][n(Σy2)-(Σy)2]
[Substitute the values from the table.]
Step: 3
r = 8(8511)-(262)(260)[8(8604)-(262)2][8(8480)-(260)2] = -3245120 = -0.1506
[Simplify.]
Step: 4
r = -0.1506
Step: 5
Therefore, there is weak negative relationship between the variables.
[When r is close to ± 1, the relationship between the variables is strong and when r is away from ± 1, it is a weak relationship.]
Correct Answer is :   - 0.1506; weak negative relationship
Q11The relation between midterm scores and final scores of 10 students in a class are as shown. Calculate the correlation coefficient and describe the relationship.
 Student A B C D E F G H I J Midterm scores(x) 76 68 66 83 90 72 79 92 96 87 Final scores(y) 71 75 79 77 84 68 82 98 88 93

A. 2.105
B. 0.752
C. 1.243
D. 0.141

Step: 1
Prepare a table for values x, y, x2, y2, xy. Step: 2
Correlation coefficient, r = n(Σxy)-(Σx)(Σy)[n(Σx2)-(Σx)2][n(Σy2)-(Σy)2]
[Substitute the values from the table.]
Step: 3
r = 10(66524)-(809)(814)[10(66419)-(809)2][10(67080)-(814)2] = 0.752
[Simplify.]
Step: 4
The correlation coefficient, r = 0.752
Q12The local ice cream shop keeps track of ice cream sales in a day versus the temperature on that day, here are their figures for the last 7 days:
 Temperature °C(x) Ice Cream Sales (y) 14.2° $210 16.4°$320 11.9° $185 15.2°$328 18/5° $410 22.1°$518 19.4° \$416
Calculate the coefficient of correlation between the temperature and sales and describe the relationship.

A. -0.975
B. -0.645
C. 0.975
D. 0.645

Step: 1
Prepare a table for values x, y, x2, y2, xy. Step: 2
Correlation coefficient, r = n(Σxy)-(Σx)(Σy)[n(Σx2)-(Σx)2][n(Σy2)-(Σy)2]
[Substitute the values from the table.]
Step: 3
r = 7(42520.3)-(117.7)(2387)[7(2050.27)-(117.7)2][7(897789)-(2387)2] = 0.975
[Simplify.]
Step: 4
r = 0.975
Step: 5
Therefore, there is a strong positive relationship between the temperature and ice cream sales.
[When r is close to ± 1, the relationship between the variables is strong and when r is away from ± 1, it is a weak relationship.]
Q13At different speeds of a car, the time for a particular journey are shown in the table. Find the correlation coefficient between the speed (in miles per hour) of the car and the time taken(in hours) by the car.
 Speed(x) Time(y) 50 50 55 40 60 42 65 35 70 36

A. 1.321
B. -1.321
C. -1.098
D. 1.098

Step: 1
Prepare a table for values x, y, x2, y2, xy. Step: 2
Correlation coefficient, r = n(Σxy)-(Σx)(Σy)[n(Σx2)-(Σx)2][n(Σy2)-(Σy)2]
[Formula.]
Step: 3
r = 5(12015)-(300)(203)[5(19250)-(300)2][5(8385)-(203)2]
[Substitute the values from the table and simplify.]
Step: 4
r = 1.321
Step: 5
The correlation coefficient, r is 1.321.