# The test statistic from the table will be f the

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Chapter 8 / Exercise 2
Finite Mathematics for the Managerial, Life, and Social Sciences: An Applied Approach
Tan
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The test statistic from the table will be F= 𝑀𝑆𝑅𝑒𝑔𝑟𝑒𝑠𝑠𝑖𝑠𝑜𝑛 𝑀𝑆𝐸𝑟𝑟𝑜𝑟 . The degrees of freedom for the numerator will be 1, and the degrees of freedom for denominator will be 5-2=3. Critical Value for F at α of 5% with df num =1 and df den =3 is 10.13. Reject Ho if F >10.13. We will also run this test using the p-value method with statistical software, such as Minitab. Data/Results 341.422 12.859 26.551 F = = which is more than the critical value of 10.13, so Reject Ho. Also, the p-value = 0.0142 < 0.05 which also supports rejecting Ho. Conclusion Sales of Sunglasses and Rainfall are negatively correlated.
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Chapter 8 / Exercise 2
Finite Mathematics for the Managerial, Life, and Social Sciences: An Applied Approach
Tan
Expert Verified
P a g e | 200 2 e SSE s MSE n = = 13.5 Estimating σ , the standard error of the residuals The simple linear regression model ( ) 0 1 Y X β β ε = + + includes a random variable ε representing the residual which follows a Normal Distribution with an expected value of 0 and a standard deviation σ, which is independent of the value of X. The estimate of σ is called the sample standard error of the residuals and is represented by the symbol s e . We can use the fact that the Mean Square Error (MSE) from the ANOVA table represents the estimated variance of the residuals errors: Example – rainfall and sales of sunglasses For the rainfall data, the standard error of the residuals is determined as: 12.859 3.586 e s = Keep in mind that this is the standard deviation of the residual errors and should not be confused with the standard deviation of Y. 13.6 r 2 , the Correlation of determination The Regression ANOVA hypothesis test can be used to determine if there is a significant correlation between the independent variable (X) and the dependent variable (Y). We now want to investigate the strength of correlation. In the earlier chapter on descriptive statistics, we introduced the correlation coefficient (r), a value between -1 and 1. Values of r close to 0 meant there was little correlation between the variables, while values closer to 1 or -1 represented stronger correlations. In practice, most statisticians and researchers prefer to use r 2 , the coefficient of determination as a measure of strength as it represents the proportion or percentage of the variability of Y that is explained by the variability of X. 87
P a g e | 201 2 1 2 1 0, 0, If b r r If b r r > = < = % 100 % 0 2 2 = r SS SS r Total regression r 2 represents the percentage of the variability of Y that is explained by the variability of X. We can also calculate the correlation coefficient (r) by taking the appropriate square root of r 2 , depending on whether the estimate of the slope (b 1 ) is positive or negative: Example – rainfall and sales of sunglasses For the rainfall data, the coefficient of determination is: 2 341.422 89.85% 380 r = = 89.85% of the variability of sales of sunglasses is explained by rainfall.