Question

For all calculations, please keep two decimals for the final answer.

Use the data set below for Questions 1, & 2:

ID	Sit-ups (Test A)	Softball Throw / ft (Test B)	40-Yard Dash (Test C)
1	23	25	9
2	34	40	7.7
3	40	38	7.1
4	31	38	7.5
5	35	39	7.4
6	20	24	8.3
7	44	53	7.2
8	22	34	7.6
9	39	42	6.9
10	37	45	7.2
11	32	45	7.3
12	33	39	7.6

Make three scatter plots:

the first one is for the data sets of Test A (Sit-Ups) & Test C (40-Yard Dash),

the second one is for the data sets of Test A (Sit-Ups) & Test B (Softball Throw), and

the last one is for the data sets of Test B (Softball Throw) & Test C (40-Yard Dash).

For each scatter plot, briefly describe your observations (e.g., does the plot show you a positive or negative relationship between the two variables? Is the relationship between the two variables strong, weak or moderate? Does a straight line fit the data better than a curve line?)

show all work. Calculate correlation coefficients for:

            Test A (Sit-Ups) & Test C (40 Yard Dash)             = __________

            Test A (Sit-Ups) & Test B (Softball Throw)              = __________

            Test B (Softball Throw) & Test C (40-Yard Dash) = __________

Based on your calculation, please answer the following question for each correlation coefficient: what does the correlation coefficient indicate (hint: direction, degree and form)?

Answer 1

Solution :

Question 1) Construction of the 3 Scatter Plots :

Scatter Plot for the data sets of Test A (Sit-Ups) & Test C (40-Yard Dash)

We construct the Scatter Plot of Test A (Sit-Ups) & Test C (40-Yard Dash) using R Software.

Observations : From the above Scatter Plot , we can clearly see that there is a Negative Relationship between Test A (Sit-Ups) and Test C (40-Yard Dash). That is , from the Scatter Plot , we can say that as the Test A scores (Number of Sit-Ups) increase , the Test C scores (Time taken for the 40-Yard Dash) decrease. From the above Scatter Plot , we can also observe that the relation between the two variables , Test A (Sit-Ups) and Test C (40-Yard Dash) , is Moderate in nature. A straight line representation is enough to fit the data , but a curve line always has better Coefficient of Determination (r²) value compared to a Straight Line. The value of the Coefficient of Determination (r²) increases as the degree of polynomial increases.

Scatter Plot for the data sets of Test A (Sit-Ups) & Test B (Softball Throw)

We construct the Scatter Plot of Test A (Sit-Ups) & Test B (Softball Throw) using R Software.

Observations : From the above Scatter Plot , we can clearly see that there is a Positive Relationship between Test A (Sit-Ups) and Test B (Softball Throw). That is , from the Scatter Plot , we can say that as the Test A scores (Number of Sit-Ups) increase , the Test B scores (Total feet for the Softball Throw) also increase. From the above Scatter Plot , we can also observe that the relation between the two variables , Test A (Sit-Ups) and Test B (Softball Throw) , is Strong in nature. A straight line representation is enough to fit the data , but a curve line always has better Coefficient of Determination (r²) value compared to a Straight Line. The value of the Coefficient of Determination (r²) increases as the degree of polynomial increases.

Scatter Plot for the data sets of Test B (Softball Throw / ft) & Test C (40-Yard Dash)

We construct the Scatter Plot of Test B (Softball Throw / ft) & Test C (40-Yard Dash) using R.

Observations : From the above Scatter Plot , we can clearly see that there is a Negative Relationship between Test B (Softball Throw / ft) and Test C (40-Yard Dash). That is , from the Scatter Plot , we can say that as the Test B scores (Total feet for the Softball Throw) increase , the Test C scores (Time taken for the 40-Yard Dash) decrease. From the above Scatter Plot , we can also observe that the relation between the two variables , Test B (Softball Throw / ft) and Test C (40-Yard Dash) , is Moderate in nature. A straight line representation is enough to fit the data , but a curve line always has better Coefficient of Determination (r²) value compared to a Straight Line. The value of the Coefficient of Determination (r²) increases as the degree of polynomial of the regression equation increases.

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Question 2) Calculation of the three Correlation Coefficients :

Let "X" and "Y" be two random variables havng "n" observations each. That is , we can say ,

Then , the formula for calculating Correlation Coefficient between X and Y is given as ,

Correlation Coefficient for the data sets of Test A (Sit-Ups) & Test C (40-Yard Dash)

We calculate the Correlation Coefficient between Test A (Sit-Ups) & Test C (40-Yard Dash).

Now , Let X represent Test A (Sit-Ups) scores and Y represent Test C (40 Yard Dash) scores.

Thus , the Correlation Coefficient between Test A (Sit-Ups) & Test C (40-Yard Dash) is,

Interpretation : We clearly see that the Correlation Coefficient between Test A (Sit-Ups) & Test C (40-Yard Dash) is r_AC = -0.77. Thus , we can clearly state that the direction of the relationship between Test A (Sit-Ups) & Test C (40-Yard Dash) is in the Negative Direction. From the value of the Correlation Coefficient , we can say that a Moderate Negative Relationsip of a Linear Form exists between Test A (Sit-Ups) & Test C (40-Yard Dash).

Correlation Coefficient for the data sets of Test A (Sit-Ups) & Test B (Softball Throw)

We calculate the Correlation Coefficient between Test A (Sit-Ups) & Test B (Softball Throw).

Now , Let X represent Test A (Sit-Ups) scores and Y represent Test B (Softball Throw) scores.

Thus , the Correlation Coefficient between Test A (Sit-Ups) & Test B (Softball Throw) is,

Interpretation : We clearly see that the Correlation Coefficient between Test A (Sit-Ups) & Test B (Softball Throw) is r_AB = 0.86. Thus , we can clearly state that the direction of the relationship between Test A (Sit-Ups) & Test B (Softball Throw) is in the Positive Direction. From the value of the Correlation Coefficient , we can say that a Strong Positive Relationsip of a Linear Form exists between Test A (Sit-Ups) & Test B (Softball Throw).

Correlation Coefficient for the data sets of Test B (Softball Throw) & Test C (40-Yard Dash)

We compute Correlation Coefficient between Test B (Softball Throw) & Test C (40-Yard Dash).

Let X represent Test B (Softball Throw) scores and Y represent Test C (40 Yard Dash) scores.

Thus , the Correlation Coefficient between Test B (Softball Throw) & Test C (40-Yard Dash) is,

Interpretation : We clearly see that the Correlation Coefficient between Test B (Softball Throw) & Test C (40-Yard Dash) is r_BC = -0.81. Thus , we can clearly state that the direction of the relationship between Test B (Softball Throw) & Test C (40-Yard Dash) is in the Negative Direction. From the value of the Correlation Coefficient , we can say that a Strong Negative Relationsip of a Linear Form exists between Test B (Softball Throw) & Test C (40-Yard Dash).

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The R - codes for constructing the Scatter Plots and calculating the Correlation Coeffs :

## Data Entry ##

a=c(23,34,40,31,35,20,44,22,39,37,32,33);a
b=c(25,40,38,38,39,24,53,34,42,45,45,39);b
c=c(9,7.7,7.1,7.5,7.4,8.3,7.2,7.6,6.9,7.2,7.3,7.6);c

## Scatter Plots ##

plot(a,c,xlab="Test A (Sit-Ups)",ylab="Test C (40 Yard Dash)",main="Scatter Plot for Sit-Ups (A) & 40 Yard Dash (C)",text(35,8.5,"Correlation Coefficient (r) = -0.77"))
plot(a,b,xlab="Test A (Sit-Ups)",ylab="Test B (Softball Throw)",main="Scatter Plot for Sit-Ups (A) & Softball Throw (B)",text(25,50,"Correlation Coefficient (r) = +0.86"))
plot(b,c,xlab="Test B (Softball Throw)",ylab="Test C (40 Yard Dash)",main="Scatter Plot for Softball Throw (B) & 40 Yard Dash (C)",text(45,8.5,"Correlation Coefficient (r) = -0.81"))

## Correlation Coefficients ##

cor_ac=round(cor(a,c),2);cor_ac
cor_ab=round(cor(a,b),2);cor_ab
cor_bc=round(cor(b,c),2);cor_bc

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