Calculate the correlation coefficient in the given problem

In summary, the conversation discusses the calculation of the correlation coefficient and the equation of least-squares for determining the relationship between waistline measurements and percentage of body fat. There is a question about whether the variables can be switched, and a discussion about rounding off numbers in the calculations. The conversation ends with a clarification that the independent variable is waistline measurements and the dependent variable is percentage of body fat.
  • #1
chwala
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Homework Statement
see attached
Relevant Equations
stats
Unless there is another alternative method, i would appreciate...ms did not indicate working...thought i should share my working though...

1680380454844.png


Let Waistline= ##X## and Percentage body fat =##Y## and we know that ##n=11##

##\sum X=992, \sum XY=13,772## and ## \sum Y=150##

Then it follows that,

Correlation coefficient

= ##\dfrac{(11×13,772)-992×150}{\sqrt {(11×89,950)-992^2)(11×2,202)-150^2)}}=\dfrac{151,492-148,800}{3045.4379}=\dfrac{2,692}{3045.4379}=0.8839=0.88## (to two decimal places).

switching ##x## and ##y## would that be appropriate? considered wrong with correct working? ...just asking. By letting ##X## be the Percentage body fat, that is...

...next i would want to determine the equation of least-squares...

Cheers!
 
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  • #2
chwala said:
switching x and y would that be appropriate? considered wrong with correct working?
I don't believe that switching x and y would be appropriate.

"Estimates for percentage of body fat can be determined by ... waistline measurements."

This statement implies that the independent variable X is the set of waistline measurements, and the dependent variable Y is the set of percentages of body fat.
 
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  • #3
yap...i can from my calculations that the equation of least squares would be given by;

##y=β_1x+β_0##

where,

##β_1=\dfrac{13,772-\frac{992×150}{11}}{89,950-\frac{992×992}{11}}=\dfrac{244.73}{489.64}=0.4998=0.5## to one decimal place.

##β_0=13.636-(0.4998×90.18)=13.636-45.07=31.43##

thus,##y=0.4998x+31.43=0.5x+31.4##

I noted that if we input,

##0.5## instead of ##0.4998## in the equation, ##β_0=13.636-(0.5×90.18)=13.636-45.09=31.454##
which rounds to ##-31.5##(to one decimal place) which is not as is indicated on ms below. At what point does one round off? or rather what ##β_1## value should one use?

Mark scheme solution

1680385631930.png
cheers!
 
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  • #4
chwala said:
yap...i can from my calculations that the equation of least squares would be given by;
##y=β_1x+β_0##
where,
##β_1=\dfrac{13,772-\frac{992×150}{11}}{89,950-\frac{992×992}{11}}=\dfrac{244.73}{489.64}=0.4998=0.5## to one decimal place.
##β_0=13.636-(0.4998×90.18)=13.636-45.07=31.43##
Sign error above. That last number should be -31.43.
chwala said:
thus,
##y=0.4998x+31.43=0.5x+31.4##

I noted that if we input,
##0.5## instead of ##0.4998## in the equation, ##β_0=13.636-(0.5×90.18)=13.636-45.09=31.454##
which rounds to ##-31.5##(to one decimal place) which is not as is indicated on ms below. At what point does one round off? or rather what ##β_1## value should one use?
There are different rules about rounding when the digit following the digit to round is 5. One rule says that if the digit to be rounded, round towards an even digit in the digit in front of that one. So, using this rule, -31.45 would round to -31.4 while -31.35 would round to -31.4.
 
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  • #5
Yes, correlation is,symmetric; Corr(X,Y)=Corr( Y,X).
But , regarding the line of best fit Y^=m^x ×b^
you can't just solve for X to get the best fit between Y and X. For one, if Y depends on X, it doesn't follow that X depends on Y; consider for one Y= height, X = age.
 
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  • #6
WWGD said:
Yes, correlation is,symmetric; Corr(X,Y)=Corr( Y,X).
But , regarding the line of best fit Y^=m^x ×b^
you can't just solve for X to get the best fit between Y and X. For one, if Y depends on X, it doesn't follow that X depends on Y; consider for one Y= height, X = age.
Meaning that we can indeed switch ##x## and ##y## in determining the correlation coefficient. I will check on this ... cheers @WWGD
 
  • #7
I believe correlation is the inner-product in a space of Random Variables. Will double check on that.
 
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  • #8
Mark44 said:
I don't believe that switching x and y would be appropriate.

"Estimates for percentage of body fat can be determined by ... waistline measurements."

This statement implies that the independent variable X is the set of waistline measurements, and the dependent variable Y is the set of percentages of body fat.
Your statement seems to be correct.
 

What is the correlation coefficient?

The correlation coefficient is a statistical measure that indicates the strength and direction of the relationship between two variables. It ranges from -1 to 1, where a value of 1 indicates a perfect positive correlation, 0 indicates no correlation, and -1 indicates a perfect negative correlation.

How do you calculate the correlation coefficient?

The correlation coefficient can be calculated using the formula r = (n∑xy - (∑x)(∑y)) / √((n∑x^2 - (∑x)^2)(n∑y^2 - (∑y)^2)), where n is the number of data points, ∑xy is the sum of the product of x and y values, ∑x and ∑y are the sum of x and y values respectively, and ∑x^2 and ∑y^2 are the sum of squared x and y values respectively.

What does a positive correlation coefficient mean?

A positive correlation coefficient indicates that as one variable increases, the other variable also tends to increase. This means that there is a positive relationship between the two variables, and they move in the same direction.

What does a negative correlation coefficient mean?

A negative correlation coefficient indicates that as one variable increases, the other variable tends to decrease. This means that there is a negative relationship between the two variables, and they move in opposite directions.

What is the significance of the correlation coefficient?

The correlation coefficient helps to determine the strength and direction of the relationship between two variables. It can also be used to make predictions about one variable based on the other variable. However, it does not necessarily imply causation, and other factors may influence the relationship between the variables.

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