Question

An economist was interested in studying the relationship between the way families
spend tax refunds and the size of their incomes. He took a sample of 6 families
and looked at their annual family income and the percentage of the tax refund
spent within three months of receipt. The results are below.
Family Income ($1000s)	Percentage of Refund Spent (%)
23	45
52	55
16	100
45	50
64	15
34	20
The sample covariance between the annual family income and the percentage of
refund spent is:		The sample covariance indicates (a positive /
no / a negative) linear relationship between these two variables. BOLD one.
Fill in the blank with the number.
The correlation coefficient between the annual family income and the percentage of
refund spent is:		This correlation coefficient indicates
(perfect positive / strong positive /moderate positive / weak positive /
perfect negative / strong negative / moderate negative / weak negative / no linear)
correlation.

Answer 1

a) To calculate the sample covariance for the given sample we do the table calculation as:

Based on the table calculations the sum of squares is calculated as:

and the sample covariance is calculated as:

b) To calculate the correlation coefficient of the data set we do the following table calculation as:

based on the table calculation the sum of squares are calculated as:

Thus

Now the correlation is calculated as:

The correlation is divided into three categories, strong, moderate, and weak. Where if the correlation is above 0.70 hence it is considered as strong and if it is between 0.50 to 0.70 hence it is generally stated as moderate relationship, and if it is below 0.50 then there will be a weak correlation.

Since the correlation coefficient is -0.627 which is almost moderate negative.