Math / Data & Statistics

Question6. Metabolic rate, the rate at which the body consumes energy, is important in studies of weight gain, dieting, and exercise. We have data on lean body mass and resting metabolic rate for 12 women who are subjects in a dieting study. Lean body mass, given in kilograms, is a person's weight leaving out all fat. Metabolic rate is measured in calories burned per 24 hours. The scatterplot shows the relationship between metabolic rate and lean body mass. The correlation is $r=0.88$ a) What would be the value of the correlation if metabolic rate was plotted on the horizontal axis and lean body mass was plotted on the vertical axis. The direction woulo be regat b) What would be the value of the correlation if lean body mass was measured in pounds instead of kilograms? c) Howard claims that the correlation between metabolic rate and lean body mass is $r=0.88 \mathrm{cal} / \mathrm{kg}$ .
Is this correct?

Studdy Solution

STEP 1

What is this asking? We're exploring how the correlation coefficient, r, changes (or doesn't change!) when we switch the axes of a scatterplot or change the units of one of the variables.
We also need to check if Howard understands what the correlation coefficient represents. Watch out! Don't assume the correlation changes just because the units or axes do!
Think about what correlation really measures.

STEP 2

1. Flipping the Axes
2. Changing the Units
3. Correcting Howard

STEP 3

Imagine our scatterplot.
We've got lean body mass on the x-axis and metabolic rate on the y-axis.
The correlation, r, tells us how strongly these two variables are linearly related.

STEP 4

Now, flip the axes!
Metabolic rate is on the x-axis and lean body mass is on the y-axis.
Did the relationship between the two variables change?
Nope! The points are in different spots visually, but they still show the same relationship.

STEP 5

Since the relationship didn't change, the correlation, r, stays the same: $*r = 0.88$ .

STEP 6

We're told that lean body mass is measured in kilograms.
What if we measured it in pounds instead?
This is like multiplying every lean body mass value by a constant ( $2.20462$ to be exact).

STEP 7

Think about it this way: if someone has twice the lean body mass of someone else in kilograms, they'll also have twice the lean body mass in pounds!
The relative relationship between the data points stays the same, even though the numbers themselves change.

STEP 8

Since the relative relationship doesn't change, the correlation, r, also stays the same: $*r = 0.88$ .

STEP 9

Howard says the correlation is $r = 0.88$ cal/kg.
Correlation is a unitless* measurement!
It's just a number that tells us the strength and direction of a linear relationship.
It doesn't have units like calories or kilograms.

STEP 10

So, Howard is incorrect!
The correlation is simply $*r = 0.88$ .

STEP 11

a) The correlation would still be $r = 0.88$ . b) The correlation would still be $r = 0.88$ . c) Howard is incorrect.
The correlation is a unitless value, so it's simply $*r = 0.88$ .

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Studdy Solution

STEP 1

STEP 2

1. Flipping the Axes
2. Changing the Units
3. Correcting Howard

STEP 3

Imagine our scatterplot.
We've got lean body mass on the x-axis and metabolic rate on the y-axis.
The correlation, r, tells us how strongly these two variables are linearly related.

STEP 4

Now, flip the axes!
Metabolic rate is on the x-axis and lean body mass is on the y-axis.
Did the relationship between the two variables change?
Nope! The points are in different spots visually, but they still show the same relationship.

STEP 5

Since the relationship didn't change, the correlation, r, stays the same: $*r = 0.88$ .

STEP 6

We're told that lean body mass is measured in kilograms.
What if we measured it in pounds instead?
This is like multiplying every lean body mass value by a constant ( $2.20462$ to be exact).

STEP 7

Think about it this way: if someone has twice the lean body mass of someone else in kilograms, they'll also have twice the lean body mass in pounds!
The relative relationship between the data points stays the same, even though the numbers themselves change.

STEP 8

Since the relative relationship doesn't change, the correlation, r, also stays the same: $*r = 0.88$ .

STEP 9

Howard says the correlation is $r = 0.88$ cal/kg.
Correlation is a unitless* measurement!
It's just a number that tells us the strength and direction of a linear relationship.
It doesn't have units like calories or kilograms.

STEP 10

So, Howard is incorrect!
The correlation is simply $*r = 0.88$ .

STEP 11

a) The correlation would still be $r = 0.88$ . b) The correlation would still be $r = 0.88$ . c) Howard is incorrect.
The correlation is a unitless value, so it's simply $*r = 0.88$ .

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Studdy Solution

STEP 1

STEP 2

1. Flipping the Axes2. Changing the Units3. Correcting Howard

STEP 3

Imagine our scatterplot.We've got lean body mass on the x-axis and metabolic rate on the y-axis.The correlation, *r*, tells us how strongly these two variables are *linearly* related.

STEP 4

Now, **flip the axes**!Metabolic rate is on the x-axis and lean body mass is on the y-axis.Did the *relationship* between the two variables change?Nope! The points are in different spots visually, but they still show the same relationship.

STEP 5

Since the relationship didn't change, the correlation, *r*, stays the same: ∗r=0.88*r = 0.88∗r=0.88.

STEP 6

We're told that lean body mass is measured in kilograms.What if we measured it in pounds instead?This is like multiplying every lean body mass value by a constant (2.204622.204622.20462 to be exact).

STEP 7

Think about it this way: if someone has twice the lean body mass of someone else in kilograms, they'll *also* have twice the lean body mass in pounds!The *relative* relationship between the data points stays the same, even though the numbers themselves change.

STEP 8

Since the relative relationship doesn't change, the correlation, *r*, also stays the same: ∗r=0.88*r = 0.88∗r=0.88.

STEP 9

Howard says the correlation is ∗r=0.88*r = 0.88∗r=0.88 cal/kg.Correlation is a *unitless* measurement!It's just a number that tells us the strength and direction of a linear relationship.It doesn't have units like calories or kilograms.

STEP 10

So, Howard is incorrect!The correlation is simply ∗r=0.88*r = 0.88∗r=0.88.

STEP 11

a) The correlation would still be ∗r=0.88*r = 0.88∗r=0.88. b) The correlation would still be ∗r=0.88*r = 0.88∗r=0.88. c) Howard is incorrect.The correlation is a unitless value, so it's simply ∗r=0.88*r = 0.88∗r=0.88.

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Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

1. Flipping the Axes
2. Changing the Units
3. Correcting Howard

Imagine our scatterplot.
We've got lean body mass on the x-axis and metabolic rate on the y-axis.
The correlation, r, tells us how strongly these two variables are linearly related.

Now, flip the axes!
Metabolic rate is on the x-axis and lean body mass is on the y-axis.
Did the relationship between the two variables change?
Nope! The points are in different spots visually, but they still show the same relationship.

Since the relationship didn't change, the correlation, r, stays the same: $*r = 0.88$ .

We're told that lean body mass is measured in kilograms.
What if we measured it in pounds instead?
This is like multiplying every lean body mass value by a constant ( $2.20462$ to be exact).

Think about it this way: if someone has twice the lean body mass of someone else in kilograms, they'll also have twice the lean body mass in pounds!
The relative relationship between the data points stays the same, even though the numbers themselves change.

Since the relative relationship doesn't change, the correlation, r, also stays the same: $*r = 0.88$ .

Howard says the correlation is $r = 0.88$ cal/kg.
Correlation is a unitless* measurement!
It's just a number that tells us the strength and direction of a linear relationship.
It doesn't have units like calories or kilograms.

So, Howard is incorrect!
The correlation is simply $*r = 0.88$ .

a) The correlation would still be $r = 0.88$ . b) The correlation would still be $r = 0.88$ . c) Howard is incorrect.
The correlation is a unitless value, so it's simply $*r = 0.88$ .