Question

Paragraph A soap manufacturer tested a standard bar of soap to see how long it would last. A test subject showered with the soap each day for 15 days and recorded Weight (in grams) of the soap after the shower. The resulting regression computer output looks, in part, like this: 3) Dependent Variable: Weight R squared = 99.5% s 2.949 Variable Intercept Day Coefficient SE (Coeff) t-ratio 123.141 -5.57476 P-value s 0.0001 s0.0001 1.382 89.1 0.1068 -52.2 a) Write the estimated regression equation b) Use a t test to determine whether the number of days affects the weight of the soap at α = 0.05 c) Predict the weight of the soap at 10 days, 15 days and 30 days.

Answer 1

a) the estimated regression equation is

Weight -Y

Days - X

Y-123.141 5.57476X

b) Null Hypothesis $H0: \beta _{1} =0$

Alternative Hypothesis $H1: \beta _{1} \neq 0$

Under H0, the test statistic is

$t = \frac{b_{1} -\beta _{1}}{S_{b_{1}}} = \frac{-5.57476-0}{0.1068} =-52.2$

The calculated value of t is =-52.2

The critical value of t for 14 df, at 5% level of significance is +/-2.145

The P-Value is < .00001

Since p value is less than level of significance, we reject H0 and conclude that the number of days affects the weight of the soap .

c) When X =10

$\widehat{Y} =123.141-5.57476X = 123.141-5.57476 * 10 = 67.3934$

When X = 15

$\widehat{Y} =123.141-5.57476X = 123.141-5.57476 * 15 = 39.5196$

When X =30

$\widehat{Y} =123.141-5.57476X = 123.141-5.57476 * 30 = -44.1018$

Since weight cannot be negative , When X = 30, $\widehat{Y} =0$