What is the null hypothesis to test the significance of the slope in a regression equation
Inferences About the Slope - The Regression t-TestYou may use this project freely under the Creative Commons Attribution-ShareAlike 4.0 International License. Please cite as follow: Hartmann, K., Krois, J., Waske, B. (2018): E-Learning Project SOGA: Statistics and Geospatial Data Analysis. Department of Earth Sciences, Freie Universitaet Berlin. Show
We now show how to test the value of the slope of the regression line. Observation: By Theorem 1 of One Sample Hypothesis
Testing for Correlation, under certain conditions, the test statistic t has the property But by Property 1 of
Method of Least Squares and by Definition 3 of
Regression Analysis and Property 4 of Regression Analysis Putting these elements together we get that where Since by the population version of Property 1 of Method of Least Squares
Example 1: Test whether the slope of the regression line in Example 1 of Method of Least Squares is zero. Figure 1 shows the worksheet for testing the null hypothesis that the slope of the regression line is 0. Figure 1 – t-test of the slope of the regression line Since p-value = .0028 < .05 = α (or |t| = 3.67 > 2.16 = tcrit) we reject the null hypothesis, and so we can’t conclude that the population slope is zero. Note that the 95% confidence interval for the population slope is b ± tcrit · sb= -628 ± 2.16(.171) = (-.998, -.259) Observation: We can also test whether the slopes of the regression lines arising from two independent populations are significantly different. This would be useful for example when testing whether the slope of the regression line for the population of men in Example 1 is significantly different from that of women. Click here for additional information and an example about
Hypothesis Testing for Comparing the Slopes of Two Independent Samples. Excel Functions: where R1 = the array of observed values and R2 = the array of observed values. STEYX(R1, R2) = standard error of the estimate sy∙x = SQRT(MSRes) LINEST(R1, R2, TRUE, TRUE) – an array function that generates a number of useful statistics. To use LINEST, begin by highlighting a blank 5 × 2 region, enter =LINEST( and then highlight the R1 array, enter a comma, highlight the R2 array and finally enter ,TRUE,TRUE) and press Ctrl-Shft-Enter. The LINEST function returns a number of values, but unfortunately no labels for these values. To make all of this clearer, Figure 2 displays the output from LINEST(A4:A18, B4:B18, TRUE, TRUE) using the data in Figure 1. I have added the appropriate labels manually for clarity. Figure 2 – LINEST(B4:B18,A4:A18,TRUE,TRUE) output R Square is the correlation of determination r2 (see Definition 2 of Basic Concepts of Correlation), while all the other values are as described above with the exception of the standard error of the y-intercept, which will be explained shortly. Excel also provides a Regression data analysis tool. The creation of a regression line and hypothesis testing of the type described in this section can be carried out using this tool. Figure 3 displays the principal output of this tool for the data in Example 1. Figure 3 – Output from Regression data analysis tool The following is a description of the fields in this report: Summary Output:
ANOVA:
Coefficients (third table): The third table gives key statistics for testing the y-intercept (Intercept in the table) and slope (Cig in the table). We will explain the intercept statistics in Confidence and Prediction Intervals for Forecasted Values. The slope statistics are as follows:
In addition to the principal results described in Figure 3, one can optionally generate a table of residuals and a table of percentiles as described in Figure 4. Figure 4 – Additional output from Regression data analysis tool Residual Output:
For example. for Observation 1 we have
Note that the mean of the residuals is approximately 0 (which is consistent with a key assumption of the regression model) and standard deviation 7.69. There is also the option to produce certain charts, which we will review when discussing Example 2 of Multiple Regression Analysis. What is the null hypothesis for the test of the slope of the regression model?The null hypothesis states that slope there is no linear relationship between the height and the weight of the individuals in the students data set.
How do you test the significance of the slope of a regression line?Conducting a Hypothesis Test for a Regression Slope. State the hypotheses. The null hypothesis (H0): B1 = 0. ... . Determine a significance level to use. What is this? ... . Find the test statistic and the corresponding p-value. ... . Reject or fail to reject the null hypothesis. ... . Interpret the results.. What is the null hypothesis for testing beta coefficients slope in linear regression?The null hypothesis states that the coefficient β1 is equal to zero. In other words, there is no statistically significant relationship between the predictor variable, x, and the response variable, y.
How do you find the value of the slope of a regression equation?To calculate slope for a regression line, you'll need to divide the standard deviation of y values by the standard deviation of x values and then multiply this by the correlation between x and y. The slope can be negative, which would show a line going downhill rather than upwards.
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