Question

Which of the following statements is true? I. In the computer output for regression, s is the estimator of the standard deviation of the response variable. II. A null hypothesis that if true, implies that there's no correlation between the x and y variables. III. The t test for the slope of a regression line is always two-sided.

Answer 1

Concepts and reason

Statistical hypotheses testing: Hypotheses testing is used to make inferences about the population based on the sample data. The hypotheses test consists of null hypothesis and alternative hypothesis.

Null hypothesis: The null hypothesis states that there is no difference in the test, which is denoted by ${H_0}$ . Moreover, the sign of null hypothesis is equal $\left( = \right)$ , greater than or equal $\left( \ge \right)$ and less than or equal $\left( \le \right)$ .

Alternative hypothesis: The hypothesis that differs from the ${H_0}$ is called alternative hypothesis. This signifies that there is a significant difference in the test. The sign of alternative hypothesis is less than $\left( < \right)$ , greater than $\left( > \right)$ , or not equal $\left( \ne \right)$ .

Dependent variable: The variable is change or affected by other variables is known as dependent variable.

Independent variable: The variable that does not change or affected by other variables is known as independent variable.

Regression: Regression is a technique that is used to determine relationship between two or more variables. That is, the change in the predictor variable influences the change in the dependent variable is determined. Moreover, in regression analysis which involves more than one independent variable, the change in the dependent is analyzed when the one independent variable is varied by keeping all other independent variables as constant.

If the data set is bivariate, then linear regression best suits the data. The straight line known as least squares regression line is obtained which best represents the data with two variables.

Multiple Regression: When there are two or more independent variables then multiple regression is used for predicting the dependent variable. This helps in determining the relationship between dependent variable and more number of independent variables.

If the data set is bivariate, then linear regression best suits the data. The straight line known as least squares regression line is obtained which best represents the data with two variables.

Slope: The slope of a least squares regression line is interpreted as the predicted change in the average response variable for a one-unit change in the explanatory variable.

Intercept: The y-intercept of a regression line is interpreted as the predicted value of the response variable when the explanatory variable has a value of zero (though be wary of extrapolation in interpreting the intercept or other values outside the original data range).

T distribution is a continuous probability distribution that is similar to the normal distribution. The T distribution is used for the normally distributed populations when the sample size is small (less than 30) and when the population standard deviation is not known. The main difference between the T distributions and Normal distribution is that the degrees of freedom are included in the T distribution.

A hypothesis test that is useful to determine the significance related to mean in the given situations from the data collected from two random samples is known as two sample t test.

Assumptions:

• The selected samples are simple random sample.

• The sample size is smaller.

• Population variance is unknown.

• Population is approximately normal.

Fundamentals

If the data set is bivariate, then linear regression best suits the data. The straight line known as least squares regression line is obtained which best represents the data with two variables. The equation of the line is given by,

$\begin{array}{l}\\\hat y = a + bx\\\\{\rm{where,}}\\\\a{\rm{ - Intercept}}\\\\b{\rm{ - Regression coefficient}}\\\\\hat y{\rm{ - Predicted value of the dependent variable}}\\\\x{\rm{ - Independent or predictor variable}}\\\end{array}$

${\rm{Degrees}}\,{\rm{of}}\,{\rm{freedom}} = n - 1$

Rejection rule based on p-value:

If $p{\rm{ - value}} \le \alpha \left( { = 0.05} \right)$ then reject null hypothesis.

In the regression output, the value of s represents the standard deviation for the total model but not for only response variable. Hence, the statement I is false. In regression analysis, the t test is used for testing the slope coefficient. Here, the alternative hypothesis may be one-sided or two-sided. Hence, the statement III is false.

In the regression analysis, the null hypothesis represents there is no relation between two variables. That is, there is no correlation between two variables. If the null hypothesis is true, it may be concluded that there is no correlation between x and y variables. Hence, the statement II is true.

Ans:

Thus, the statement “A null hypothesis that if true, implies that there's no correlation between the x and y variables” is true.