Testing Hypotheses About Regression Coefficients If the coefficient has a nonzero value, then it is helpful in predicting the value of the response variable. If it is not helpful in predicting the value of the response variable and can be eliminated from the regression equation. To test the claim that use the test statistic Critical values or P-values can be found using the t distribution with degrees of freedom, where k is the number of predictor variables and n is the number of observations in the sample. The standard error is often provided by software. For example, see the accompanying StatCrunch display for Example 1, which shows that (found in the column with the heading of “Std. Err.” and the row corresponding to the first predictor variable of height). Use the sample data in Data Set 1 “Body Data” and the StatCrunch display to test the claim that Also test the claim that What do the results imply about the regression equation?

Verified step by step guidance

1. Step 1: Identify the null and alternative hypotheses for the regression coefficients. For each predictor variable (Height and Waist), the null hypothesis (H0) states that the coefficient is equal to zero, meaning the variable is not helpful in predicting the response variable. The alternative hypothesis (H1) states that the coefficient is not equal to zero, meaning the variable is helpful in predicting the response variable.
2. Step 2: Locate the test statistic (T-Stat) and the corresponding P-value for each predictor variable from the StatCrunch display. For Height, the T-Stat is 10.813917 and the P-value is <0.0001. For Waist, the T-Stat is 29.856261 and the P-value is <0.0001.
3. Step 3: Compare the P-values to the significance level (typically α = 0.05). If the P-value is less than α, reject the null hypothesis (H0) for that predictor variable. Both Height and Waist have P-values <0.0001, which are less than 0.05, so the null hypotheses for both variables are rejected.
4. Step 4: Interpret the results. Since the null hypotheses are rejected for both Height and Waist, this implies that both variables are statistically significant and helpful in predicting the response variable. Their coefficients should be retained in the regression equation.
5. Step 5: Evaluate the regression equation. The results suggest that the regression equation, which includes the coefficients for Height and Waist, is effective in predicting the response variable. The statistical significance of these predictors supports their inclusion in the model.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about a population based on sample data. It involves formulating a null hypothesis (H0) and an alternative hypothesis (H1), then using a test statistic to determine whether to reject H0. In regression analysis, this often involves testing whether a regression coefficient is significantly different from zero, indicating its predictive power.

P-value

The P-value is a measure that helps determine the significance of results in hypothesis testing. It represents the probability of observing the test results, or something more extreme, assuming the null hypothesis is true. A low P-value (typically less than 0.05) suggests that the null hypothesis can be rejected, indicating that the predictor variable has a significant effect on the response variable.

Standard Error

The standard error (SE) quantifies the amount of variability or dispersion of a sample statistic, such as a regression coefficient. It is calculated as the standard deviation of the sampling distribution of the statistic. A smaller standard error indicates more precise estimates of the regression coefficients, which is crucial for determining the reliability of the hypothesis tests conducted on these coefficients.