Question

Suppose you estimate a simple linear regression model and obtain a t-value for the slope coefficient of -3.1. Based on this, explain which of the following statements are correct or wrong:

a) A 95% confidence interval for the true slope would exclude 0.
b) It is possible that the point estimate for the slope is b_1=4.
c) At the 10% level of significance you fail to reject the null hypothesis that the true slope is equal to 0.
d) The probability that the true slope is negative is greater than the probability that the true slope is positive.

Answer 1

a) A 95% confidence interval for the true slope would exclude 0. TRUE

Explanation: Since the t statistic is large, the p-value will be lower than the 0.05 hence we can reject the null hypothesis (slope = 0) at a 95% confidence interval. (Let the degree of freedom = 30, the p-value for the t statistic = 0.002 which far less than 0.05.)

b) It is possible that the point estimate for the slope is b_1=4. FALSE

Explanation:

The slope coefficient is defined as,

Since the Standard Error is a positive value, the slope estimate will have the same sign as of t value. Hence the slope coefficient will be a negative number.

c) At the 10% level of significance, you fail to reject the null hypothesis that the true slope is equal to 0. FALSE

Explanation: Since the t statistic is large, the p-value will be lower than the 0.10 hence we can reject the null hypothesis (slope = 0) at a 90% confidence interval.

d) The probability that the true slope is negative is greater than the probability that the true slope is positive. TRUE

Explanation: Since the t statistic is negative and far below the 0 (slope coefficient is approximately 3.1 times of standard error below the 0 value), there is very little chance that the slope coefficient value will be positive.