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Question: Discussion Question 3: The Correlation Coefficient There are two ways we may calculate the correl...
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Discussion Question 3: The Correlation Coefficient
There are two ways we may calculate the correlation coefficient r (also called Pearson’s r,) the hard way, calculating each step in the formula, and the easy way, plugging the data into a calculator or computer that has the formula for the correlation coefficient programmed into it. In this Discussion Question, we will compute the correlation between two variables in a data set the hard way as a group. Each person who contributes to the thread will perform one more step toward finding the value of r. But each person will also compute r the easy way using a programmable calculator or computer.
Here is our scenario: Eight restaurants belonging to the same franchise operate in different cities. The restaurants have different levels of daily sales. You suspect that part of the variation in sales may come from competition with other restaurants. You collect two pieces of data for each restaurant in the franchise, the number of competitors within one mile and the average daily sales of the restaurant. The data are summarized in the table below:
ACTIVITIES AND ASSIGNMENTS FOR WEEK 3 3
Number of Competitors within 1 mile, (x) Average daily sale (in thousands), (y)
1 4.0
1 3.3
2 3.1
3 2.5
3 2.9
4 2.5
5 2.8
5 2
Question: (3) Calculate the mean value of y, ¯y.
Please show all work in how you got the answer to this question.
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mean of Y= sum of Y/ Total no of obsns
= (4+3.3+....+2)/8
= 23.1/8
= 2.8875
Mean of X = sum of X/ Total no of obsns
=24/8
=3
X.Y={ 4.0 , 3.3 , 6.2 , 7.5 , 8.7 ,10.0, 14.0 ,10.0}
mean(X.Y) = 933.7/8 = 7.9625
Cov(XY) = mean(X.Y) - mean(X)mean(Y)
= 7.9625 - 3(2.8875)
= -0.7
var(X) = mean(X^2) - [mean(X)]^2
= 8.6562 - 8.3376
=0.3186
SD(X) = 0.5644
Var(Y) = 2.25
SD(Y)= 1.5
corr(X,Y) = cov(X,Y)/(SD(X)*SD(Y))
= -0.7/(0.5644*1.5)
= -0.8268
corr(X,Y)= -0.8268
Comment
The question that I am asking is how did the person who did this question get the answer? I have to explain it and I am a little confused as how to do so? It is different from how our textbook does it and I have to explain how and why I used this method. Please help.
Explanation / Answer
Remember
Mean = Total of values/N
So, mean of Y= sum of Y/ Total no of obsns
= (4+3.3+....+2)/8
= 23.1/8
= 2.8875
Similarly, Mean of X = sum of X/ Total no of obsns
=24/8
=3
Now we multiply values of X and Y
X.Y={ 4.0 , 3.3 , 6.2 , 7.5 , 8.7 ,10.0, 14.0 ,10.0}
Finding mean of (X*Y)
mean(X.Y) = 933.7/8 = 7.9625
Now, Cov(XY) = mean(X*Y) - mean(X) * mean(Y)
= 7.9625 - 3 * (2.8875)
= -0.7
This is the formula: var(X) = mean(X^2) - [mean(X)]^2
= 8.6562 - 8.3376
=0.3186
SD is square root of variance: So,
SD(X) = 0.5644
Var(Y) = 2.25
SD(Y)= 1.5
corr(X,Y) = cov(X,Y)/(SD(X)*SD(Y))
= -0.7/(0.5644*1.5)
= -0.8268
corr(X,Y)= -0.8268
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