Question

Estimating r For each of the following, estimate the value of the linear correlation coefficient r for the given paired data obtained from 50 randomly selected adults.

a. Their heights are measured in inches (x) and those same heights are recorded in centimetres (y).

b. Their IQ scores (x) are measured and their heights (y) are measured in centimetres.

c. Their pulse rates (x) are measured and their IQ scores are measured (y).

d. Their heights (x) are measured in centimetres and those same heights are listed again, but with negative signs (y) preceding each of these second listings.

Short answer

a. The estimated value of the linear correlation coefficient (r) between heights measured in inches and heights measured in centimeters is approximately 1.

b. The estimated value of the linear correlation coefficient (r) between the IQ scores and heights measured in centimeters is approximately 0.

c. The estimated value of the linear correlation coefficient (r) between the pulse rates and the IQ scores is approximately 0.

d. The estimated value of the linear correlation coefficient (r) between the heights measured in centimeters and the same heights with a negative sign is approximately -1.

Step-by-step solution

Given information

Few variables are listed for a set of 50 randomly selected adults.

Define a linear correlation coefficient

An estimate of the linear correlation coefficient for any set of variables is a value in the range of -1 to 1, which can be guessed (or computed if observations are known) using the prior knowledge of the relationship between the two variables.

For example:

• Correlation 0 implies there is no linear relationship.
• Correlation -1 implies there is a negative linear relationship.
• Correlation 1 implies there is a positive linear relationship.

Estimate the correlation coefficient for heights in two units

a.

Let x denote the heights of the 50 adults measured in inches.

Let y denote the heights of the 50 adults measured in centimeters.

It is known that one inch of measurement is equivalent to 2.54 centimeters.

Thus, each observation for the x variable increases by a multiple of 2.54 units.

Since the variables on both the axes represent the same set of observations measured in different units, the scatterplot between x and y will be a near-perfect straight line sloping upward (since when x increases, y will also increase).

A straight-line pattern sloping upward on a scatterplot implies a correlation coefficient equal to 1.

Estimate the correlation coefficient for heights and IQ

b.

Let x denote the IQ scores of the 50 adults.

Let y denote the heights of the 50 adults measured in centimeters.

Intuitively, there is no relationship between the height of an individual and IQ scores. Thus, it is expected that the scatterplot constructed between the variables will show the points randomly scattered over the graph. It is unlikely that the variables form a close to linear pattern.

Since randomly scattered points (no pattern) on a scatterplot imply a correlation coefficient equal to 0, the estimated correlation coefficient value is 0.

Estimate the correlation coefficient for pulse rates and IQ

c.

Let x denote the pulse rates of the 50 adults.

Let y denote the IQ scores of the 50 adults.

Intuitively, there is no relationship between the pulse rate of an individual and IQ scores. Thus, it is expected that the scatterplot constructed between the variables will be a randomly scattered set of observations with no specific pattern. It is unlikely that variables form a linear pattern as there is no association between the pulse rate of an adult and his/her IQ score

Since randomly scattered points (no pattern) on a scatterplot imply a correlation coefficient equal to 0, the estimated correlation coefficient value is 0.

Estimate the correlation coefficient for heights with opposite sign 

d.

Let x denote the heights of the 50 adults measured in centimeters.

Let y denote the heights of the 50 adults measured in centimeters with a negative sign.

From the definition of variables, each observation of y corresponds to a negative measure of the corresponding x observation.

Since the variables on both axes represent the same thing, the scatterplot between x and y will be a near-perfect straight line. However, here, the values have an opposite sign, which indicates that the straight line between x and y will be sloping downward (since when x increases, y will decrease).

Astraight-line pattern sloping downward on a scatterplot implies a correlation coefficient equal to -1.