Question

test the hypothesis that the true slope of the 
regression line is zero (as opposed to nonzero).  State the appropriate null 
and alternative hypotheses, give the value of the test statistic and give the 
appropriate P-value.  (Use a significance level of 0.05.)  Explain precisely 
what this means in terms of the relationship between the two variables.

Data

x y

77.5 45
80 73
78 43
78.5 61
77.5 52
83 56
83.5 70
81.5 70
75.5 53
69.5 51
70 39
73.5 55
77.5 55
79 57
80 68
79 73
76 57
76 51
75.5 55
79.5 56
78.5 72
82 73
71.5 69
70 38
68 50
66.5 37
69 43
70.5 42
63 25
64 31
64.5 31
65 32
66.5 35
67 32
66.5 34
67.5 35
75 41
75.5 51
71.5 34
63 19 
60 19
64 30
62.5 23 
63.5 35
73.5 29
68 55
77.5 56

Answer 1

Solution: We can use the excel regression data analysis tool to find the test statistic and the p-value. The excel output is given below:

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.842883361
R Square	0.71045236
Adjusted R Square	0.704017968
Standard Error	8.399174148
Observations	47
ANOVA
	df	SS	MS	F	Significance F
Regression	1	7789.339207	7789.339207	110.4148393	0.00000
Residual	45	3174.575687	70.54612638
Total	46	10963.91489
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	-98.24168878	13.88043618	-7.07770905	0.0000	-126.1983219	-70.28505564
x	2.005686407	0.190875114	10.50784656	0.0000	1.621244199	2.390128615

regression line is zero (as opposed to nonzero). State the appropriate null and alternative hypotheses, give the value of the test statistic and give the appropriate P-value. (Use a significance level of 0.05.) Explain precisely what this means in terms of the relationship between the two variables.

The null and alternative hypotheses are:

From the excel output, we have:

The test statistic is:

The p-value is:

Since the p-value is less than the significance level 0.05, we, therefore, reject the null hypothesis and conclude that there is a significant relationship between the two variables.