Question

Test the slope of the regression line, ŷ = -46.292 + 15.240 x of the following data. Use α = .10 and s_e = 108.7.

Advertising	Sales
12.5	148
3.7	55
21.6	338
60.0	994
37.6	541
6.1	89
16.8	126
41.2	379

(Round the intermediate values to 3 decimal places. Round your answer to 2 decimal places.)

Observed t= ^______

The decision is to _______

A) Fail to reject the null hypothesis

B) Reject the null hypothesis.

Answer 1

The regression line is of the form:

Y=b0+b1*X

We need to test the slope b1 is significant or not.

i.e to test H0:b1=0 (slope not significant) vs H1:b1≠0 (slope is significant)

The test statistic is

t=b1^/SE(b1^)

where b1^ is the estimate of the slope=15.240 (From the given equation)

SE(b1^)=sqrt [ Σ(yi - ŷi)2 / (n - 2) ] / sqrt [ Σ(xi - x)2 ]

Table for calculation:

Advertising (xi)	Sales (yi)	predicted (yi^)	(yi-yi^)2	(xi-xbar)2
12.5	148	144.208	14.379264	154.6914063
3.7	55	10.096	2016.369216	451.0314063
21.6	338	282.892	3036.891664	11.13890625
60	994	868.108	15848.79566	1229.378906
37.6	541	526.732	203.575824	160.3389063
6.1	89	46.672	1791.659584	354.8514063
16.8	126	209.74	7012.3876	66.21890625
41.2	379	581.596	41045.13922	264.4689063
Total			70969.19803	2692.11875

*Predicted (yi^)=-46.292 + 15.240 *xi

SE(b1^)=sqrt [ Σ(yi - ŷi)2 / (n - 2) ] / sqrt [ Σ(xi - x)2 ] =√{(7069.198)/6}/√(2692.119) =108.758/51.886=2.096

So,t=15.240/2.096=7.27

Ans: observed t=7.27

The test statistic follows a t distribution with (n-2) degrees of freedom.

So, degrees of freedom =n-2=8-2=6

α=0.10, α/2=0.10/2=0.05

Hence the critical value for the two-tailed test is:

t(α/2, df) =t(0.05,6)=1.943

The p-value for two-tailed for t=7.27 for degrees of freedom=6 is =0.000345

Decision: observed t=7.27 is > critical value of t =1.943, and p-value=0.000345 is < α=0.10

So we reject the null hypothesis at α=0.10

Ans : Decision :  Reject the null hypothesis.