Question

A soap manufacturer tested a bar of soap to see how long it would last. A test subject showered with the soap each day for 15 days and recorded the Weight​ (in grams) of the soap after the shower. Part of the regression output is shown below. Answer parts a through e.

Variable	Coefficient	​SE(Coeff)	​t-ratio	​P-value		Dependent variable​ is: Weight ​R-squared=94.0​% s=7.1
Intercept	126.392	3.39	37.24	≤0.0001
Day	−5.68865	0.386	−14.74	≤0.0001

​a) The standard deviation of the residuals

se=nothing

​(Round to one decimal place as​ needed.)

​b) The slope of the regression line

b1=nothing

​(Round to five decimal places as​ needed.)

​c) The standard error of b1

SEb1=nothing

​(Round to three decimal places as​ needed.)​d) The​ P-value appropriate for testing

H0: β1=0

versus HA: β1≠0The appropriate​ P-value is

nothing.

​(Round to four decimal places as​ needed.)

​e) Is the null hypothesis rejected at

α=0.05​?

Choose the correct answer below.

A. ​No, because the​ P-value is less than 0.05.

B. ​No, because the​ P-value is greater than 0.05.

C. ​Yes, because the​ P-value is less than 0.05.

D. Yes, because the​ P-value is greater than 0.05.

Answer 1

From the given output:

The fitted model can be expressed as:

Expected Weight = 126.392 - 5.68865 (Day)

where the estimated intercept and slope coefficients are 12.392 and -5.8865 respectively

​a) The standard deviation of the residuals

SE = 7.1

This is a measure of goodness of fit of the model, with smaller values indicating a better precision and hence a better fit for the data.

​b) The slope of the regression line

b1 = −5.68865

The slope is interpreted as the mean change in the response variable - Weight for a unit change in the predictor - Day. Here, the mean weight is estimated to decrease by 5.9991425 units as each day passes. (Here, change observed is an 'decrease', since, the slope for 'Day' is positive) .

​c) The standard error of b1

SEb1 = 0.386

d) The​ P-value appropriate for testing

H0: β1=0 versus HA: β1≠0

The appropriate​ P-value is

<0.0001

​e) Since, the p-value of the test 0.0001 < 0.05 is significant, we may reject the null hypothesis at 5% level. Hence, the correct option would be:

C. ​Yes, because the​ P-value is less than 0.05.