Question

Some research has indicated that as people age they need less sleep. A random sample of adults was selected and the number of hours that they slept last night was recorded. Simple regression analysis was performed on this sample and partial results are shown below.

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.0810376
R Square	0.0528784
Adjusted R Square	0.018598
Standard Error	0.8385297
Observations	50

	Coefficients	Standard Error	t Stat	P-value
Intercept	20.67619	3.196349
X Variable 1	0.385714	0.331149

Which one of the following statements is true using α = 0.10?

		We can reject the null hypothesis and conclude that there is a positive relationship between the hours of sleep and the age of the adult.
		We can reject the null hypothesis and conclude that there is a negative relationship between the hours of sleep and the age of the adult.
		We cannot reject the null hypothesis and conclude that there is a positive relationship between the hours of sleep and the age of the adult.
		We cannot reject the null hypothesis and conclude that there is a negative relationship between the hours of sleep and the age of the adult.
		We cannot reject the null hypothesis and conclude that there is no relationship between the hours of sleep and the age of the adult.

Answer 1

To test whether there is a relationship between the hours of sleep and the age of the adult,

i.e. to test against

The test statistic can be written as

which under H₀ follows a t distribution with n-2 df.

We reject H₀ at 10% level of significance if P-value < 0.1

Now,

The value of test statistic

P-value

Since P-value > 0.05, so we fail to reject the null hypothesis.

ans-> iv) We cannot reject the null hypothesis and conclude that there is no relationship between the hours of sleep and the age of the adult.