Question

The cost of a leading liquid laundry detergent in different sizes is given below.

Size (ounces)	Cost ($)	Cost per ounce
16	3.79
32	4.89
64	5.49
200	10.99

Calculate the least squares line. Put the equation in the form of:

ŷ = a + bx.

Find the correlation coefficient r

If the laundry detergent were sold in a 50-ounce size, find the estimated cost.

If the laundry detergent were sold in an 86-ounce size, find the estimated cost.

What is the slope of the least squares (best-fit) line?

Answer 1

Solution:

Here, we have to develop the regression model for the prediction of the dependent or response variable cost based on the independent variable size. The required regression model by using excel is given as below:

Regression Statistics
Multiple R	0.996490187
R Square	0.992992693
Adjusted R Square	0.989489039
Standard Error	0.329246172
Observations	4
ANOVA
	df	SS	MS	F	Significance F
Regression	1	30.72319392	30.72319392	283.4163451	0.003509813
Residual	2	0.216806084	0.108403042
Total	3	30.94
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	3.309391635	0.241758068	13.68885708	0.005294269	2.269190626	4.349592644
Size	0.038212928	0.002269854	16.83497387	0.003509813	0.028446536	0.04797932

From above regression output, the least squares regression line is given as below:

ŷ = 3.309391635 + 0.038212928*x

Y-intercept = a = 3.309391635

Slope = b = 0.038212928

The correlation coefficient between two variables is given as r = 0.996490187.

If the laundry detergent were sold in a 50-ounce size, find the estimated cost.

We are given x = 50

ŷ = 3.309391635 + 0.038212928*x

ŷ = 3.309391635 + 0.038212928*50

ŷ = 5.220038

Estimated cost = $5.22

If the laundry detergent were sold in an 86-ounce size, find the estimated cost.

We are given x = 86

ŷ = 3.309391635 + 0.038212928*x

ŷ = 3.309391635 + 0.038212928*86

ŷ = 6.595703443

Estimated cost = $6.60

What is the slope of the least squares (best-fit) line?

The slope of the least squares regression line is given as

Slope = b = 0.038212928

Which indicate approximately 0.038 increments in the cost as the size of liquid laundry detergent increases by 1 ounce.