Statistical inference: Hypothesis tests and confidence intervals

One of the main purposes for determining a regression line is to find the true value of the slope of the population regression line. However, in almost all cases, the regression line is estimated using sample data. Then based on the sample regression line, inferences are made about the population regression line. The slope b of a sample regression line is a point estimator of the slope of the population regression line. The different sample regression lines estimated for different samples taken from the same population will give different values of b. If only one sample is selected, then the value of b will depend on which elements are included in the sample. Thus, b is a random variable and it possesses a probability distribution called a sampling distribution.

Assume that assumptions 3.5.4 are hold. Then b is an unbiased estimator of and has a population variance

and unbiased estimator of is provided by

In applied regression analysis we first would like to know if there is a relationship. We see that if is zero then there is no relationship- y would not continuously increase or decrease with increase in x.

3.7.1. Hypothesis testing about

Let be a population regression slope and b its least square estimate based on n pairs of sample observations. Assume that assumptions 3.5.4 hold and also assume that the errors are normally distributed. Then the random variable

is distributed as Student’s t distribution with degree of freedom.

If we use notation

for the test statistic then the following tests have a significance level

1. To test either null hypothesis

or

against the alternative

the decision rule is

Reject if

2. To test either null hypothesis

or

against the alternative

the decision rule is

Reject if

3. To test null hypothesis

against the two sided alternative

the decision rule is

Reject if or

Remark1: To test the hypothesis that x does not determine y linearly and there is no linear relationship, we will test the null hypothesis that the slope of the regression line is zero, that is ; the alternative hypothesis that means x determines y linearly; means x determines y positively; means x determines y negatively.

Remark2: The null hypothesis does not always have to be .We may test the null hypothesis that is equal to a value different from zero.

Example:

Test at the 5% significance level if the slope of the population regression line for the example on incomes and food expenditure of seven households is positive.

Solution:

From earlier calculations we have

; and

; and .

We are to test whether or not slope of the population regression line is positive. The two hypotheses are

(Slope is zero)

(Slope is positive)

The decision rule is

reject if .

The value of the test statistic is

The significance level is 0.05. Therefore,

The value of the test statistic is greater than the critical value of

and it falls in the rejection region. Hence, we reject the null hypothesis and conclude that x (income) determines y (food expenditure) positively. That is, food expenditure increases with an increase in income and it decreases with a decrease in income.

3.7.2. Confidence intervals for the population regression slope

We can derive confidence intervals for the slope of the population regression line by using coefficient b and variance estimators we have developed.

If the assumptions 3.5.4 hold, and if the regression errors, , are normally distributed, then confidence interval for the population regression slope is given by

where is the number for which and the random variable follows Student’s t distribution with degrees of freedom.

Example:

Construct a 95% confidence interval for for the data on incomes and food expenditures of seven households.

Solution:

From earlier calculations we have

; ; and

The confidence level is 95%. So

The 95% confidence interval for is

Thus, we are 95% confident that slope for the population regression line is

between 0.17 and 0.35.

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