Question

Rejecting the null hypothesis that the population slope is equal to zero or no relationship and concluding that the relationship between x and y is significant does not enable one to conclude that a cause-and-effect relationship is present between x and y. Explain why.

Answer 1

The test focuses on the slope of the regression line

Y = Β₀ + Β₁X

where Β₀ is a constant, Β₁ is the slope (also called the regression coefficient), X is the value of the independent variable, and Y is the value of the dependent variable.

If we find that the slope of the regression line is significantly different from zero, we will conclude that there is a significant relationship between the independent and dependent variables.

Test Requirements

The approach described is valid whenever the standard requirements for simple linear regression are met.

• The dependent variable Y has a linear relationship to the independent variable X.
• For each value of X, the probability distribution of Y has the same standard deviation σ.
• For any given value of X,

• The Y values are independent.
• The Y values are roughly normally distributed (i.e., symmetric and unimodal). A little skewness is ok if the sample size is large.

The test procedure consists of four steps:

(1) state the hypotheses,

(2) formulate an analysis plan,

(3) analyze sample data, and

(4) interpret results.

State the Hypotheses

If there is a significant linear relationship between the independent variable X and the dependent variable Y, the slope will not equal zero.

H_o: Β₁ = 0

H_a: Β₁ ≠ 0

The null hypothesis states that the slope is equal to zero, and the alternative hypothesis states that the slope is not equal to zero.

Formulate an Analysis Plan

The analysis plan describes how to use sample data to accept or reject the null hypothesis. The plan should specify the following elements.

• Significance level. Often, researchers choose significance levels equal to 0.01, 0.05, or 0.10; but any value between 0 and 1 can be used.
• Test method. Use a linear regression t-test (described in the next section) to determine whether the slope of the regression line differs significantly from zero.

Analyze Sample Data

Using sample data, find the standard error of the slope, the slope of the regression line, the degrees of freedom, the test statistic, and the P-value associated with the test statistic. The approach described in this section is illustrated in the sample problem at the end of this lesson.

• Standard error. Many statistical software packages and some graphing calculators provide the standard error of the slope as a regression analysis output. The table below shows hypothetical output for the following regression equation: y = 76 + 35x .

Predictor	Coef	SE Coef	T	P
Constant	76	30	2.53	0.01
X	35	20	1.75	0.04

• In the output above, the standard error of the slope (shaded in gray) is equal to 20. In this example, the standard error is referred to as "SE Coeff". However, other software packages might use a different label for the standard error. It might be "StDev", "SE", "Std Dev", or something else. If you need to calculate the standard error of the slope (SE) by hand, use the following formula:
• SE = s_b1 = sqrt [ Σ(y_i - ŷ_i)² / (n - 2) ] / sqrt [ Σ(x_i - x)² ]
• where y_i is the value of the dependent variable for observation i, ŷ_i is estimated value of the dependent variable for observation i, x_i is the observed value of the independent variable for observation i, x is the mean of the independent variable, and n is the number of observations.
• Slope. Like the standard error, the slope of the regression line will be provided by most statistics software packages. In the hypothetical output above, the slope is equal to 35.
• Degrees of freedom. For simple linear regression (one independent and one dependent variable), the degrees of freedom (DF) is equal to:

DF = n - 2

where n is the number of observations in the sample.

• Test statistic. The test statistic is a t statistic (t) defined by the following equation.

t = b₁ / SE

where b₁ is the slope of the sample regression line, and SE is the standard error of the slope.

• P-value. The P-value is the probability of observing a sample statistic as extreme as the test statistic. Since the test statistic is a t statistic, use the t Distribution Calculator to assess the probability associated with the test statistic. Use the degrees of freedom computed above.

Interpret Results

If the sample findings are unlikely, given the null hypothesis, the researcher rejects the null hypothesis. Typically, this involves comparing the P-value to the significance level, and rejecting the null hypothesis when the P-value is less than the significance level.

Test Your Understanding

Problem

The local utility company surveys 101 randomly selected customers. For each survey participant, the company collects the following: annual electric bill (in dollars) and home size (in square feet). Output from a regression analysis appears below.

Regression equation: Annual bill = 0.55 * Home size + 15
Predictor	Coef	SE Coef	T	P
Constant	15	3	5.0	0.00
Home size	0.55	0.24	2.29	0.01

Is there a significant linear relationship between annual bill and home size? Use a 0.05 level of significance.

Solution

The solution to this problem takes four steps: (1) state the hypotheses, (2) formulate an analysis plan, (3) analyze sample data, and (4) interpret results. We work through those steps below:

• State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

H_o: The slope of the regression line is equal to zero.

H_a: The slope of the regression line is not equal to zero.

If the relationship between home size and electric bill is significant, the slope will not equal zero.

• Formulate an analysis plan. For this analysis, the significance level is 0.05. Using sample data, we will conduct a linear regression t-test to determine whether the slope of the regression line differs significantly from zero.
• Analyze sample data. To apply the linear regression t-test to sample data, we require the standard error of the slope, the slope of the regression line, the degrees of freedom, the t statistic test statistic, and the P-value of the test statistic.

We get the slope (b₁) and the standard error (SE) from the regression output.

b₁ = 0.55       SE = 0.24

We compute the degrees of freedom and the t statistic test statistic, using the following equations.

DF = n - 2 = 101 - 2 = 99

t = b₁/SE = 0.55/0.24 = 2.29

where DF is the degrees of freedom, n is the number of observations in the sample, b₁ is the slope of the regression line, and SE is the standard error of the slope.

Based on the t statistic test statistic and the degrees of freedom, we determine the P-value. The P-value is the probability that a t statistic having 99 degrees of freedom is more extreme than 2.29. Since this is a two-tailed test, "more extreme" means greater than 2.29 or less than -2.29. We use the t Distribution Calculator to find P(t > 2.29) = 0.0121 and P(t < -2.29) = 0.0121. Therefore, the P-value is 0.0121 + 0.0121 or 0.0242.

• Interpret results. Since the P-value (0.0242) is less than the significance level (0.05), we cannot accept the null hypothesis.