Statistics Answer Note #28

Question 1 Multiple regression output reveals an R value of .740. What percent of the variability in the dependent variable is not explained by variability in the independent variables? (Make sure you read this question carefully. You will need to conduct two simple calculations.) Answer As a high-achieving College student, I understand the importance of accuracy and knowledge in the field of Statistics. The question at hand is related to the R value of a multiple regression output, specifically, the percentage of variability in the dependent variable that is not explained by the variability in the independent variables. To answer this question, we must first understand the concept of R value, also known as the correlation coefficient. This measure is used to determine the strength and direction of the relationship between two variables. In the case of multiple regression, it is used to determine the overall strength of the relationship between the independent variables and the dependent variable. Based on the given information, we know that the R value in this scenario is 0.740. This means that there is a strong positive relationship between the variables, where a value of 1 would indicate a perfect positive relationship and a value of -1 would indicate a perfect negative relationship. An R value of 0 would indicate no relationship at all. Now, to calculate the percentage of variability in the dependent variable that is not explained by the independent variables, we need to find the coefficient of determination, also known as R-squared. This is the square of the correlation coefficient and represents the proportion of variability in the dependent variable that is explained by the independent variables. The formula for R-squared is: R-squared = R value^2. Therefore, in this case, R-squared = 0.740^2 = 0.5476. This means that approximately 54.76% of the variability in the dependent variable is explained by the independent variables. To find the percentage of variability not explained, we simply subtract this value from 1. So, the percentage of variability not explained is 1 - 0.5476 = 0.4524 or 45.24%. It is important to note that this value only represents the relationship between the variables and does not necessarily imply causation. Additionally, the R value and Rsquared do not determine the significance of the relationship, and thus further analysis and statistical tests may be required. Question 2 A researcher included two independent variables, X1 and X2, in a multiple regression model. Prior to running the regression model, the researcher computed correlations between the independent variables and the dependent variable and found that both independent variables were highly correlated with the dependent variable. However, when viewing the results of the multiple regression, the researcher discovered that the slope coefficient associated with independent variable X1 was not significant (p = 0.52) while the slope coefficient associated with independent variable X2 was significant (p = 0.02). What issue is the data (specifically X1 and X2) likely suffering from? a. Heteroskedasticity b. Autoregressive residual c. Multicollinearity d. Insufficient number of observations Answer Based on the given scenario, it can be inferred that the data is likely suffering from multicollinearity. This refers to a situation where there is a high degree of correlation between two or more independent variables in a regression model. This can lead to unreliable and inaccurate results as it becomes difficult to determine the unique contribution of each independent variable towards the dependent variable. In this case, both X1 and X2 were highly correlated with the dependent variable but only the slope coefficient of X2 was found to be significant. One of the major issues with multicollinearity is that it can inflate the standard errors of the regression coefficients, making them seem larger than they actually are. This can result in the non-significant results for some variables as seen in the case of X1 with a pvalue of 0.52. On the other hand, the significant p-value for X2 (0.02) can be attributed to the fact that it is less correlated with the other independent variable and thus, has a stronger unique contribution towards the dependent variable. In order to address multicollinearity, there are a few steps that can be taken. One approach is to remove one of the highly correlated independent variables from the model. This can be done by examining the correlation matrix and identifying the variable that has the higher correlation with the dependent variable. However, this may not always be feasible as both variables may be important in explaining the dependent variable. In such cases, other techniques like Principal Component Analysis (PCA) or Ridge Regression can be used to reduce the effects of multicollinearity. Question 3 A researcher regressed Nitrogen (X1), Phosphorous (X2), and Irrigation (X3) on the dependent variable Corn Yield (Y). The null hypothesis was: H0: b1 = b2 = b3 = 0 The ANOVA table from the regression was as follows: ANOVA Source of Variation df Sum of Squares Mean Square F Sig. Regression 3 47.654 15.885 15.701 .000 Residual 77 77.899 1.012 Answer From the ANOVA table provided, we can see that the researcher has performed a multiple linear regression analysis to determine the relationship between Nitrogen (X1), Phosphorous (X2), and Irrigation (X3) on the dependent variable Corn Yield (Y). The null hypothesis, as stated, is that there is no significant relationship between the independent variables (Nitrogen, Phosphorous, Irrigation) and the dependent variable (Corn Yield). The ANOVA table provides important statistical measures to assess the significance of the regression model. The first measure to look at is the F-statistic, which in this case is 15.701. This value is compared to a critical value to determine if the regression model is significant or not. As the corresponding p-value (Sig.) for the F-statistic is 0.000 (less than 0.05), we can conclude that the regression model is statistically significant. This means that at least one of the independent variables has a significant impact on the dependent variable. Further, the ANOVA table shows the sum of squares (SS) and mean square (MS) for the regression and residual (error) terms. These measures reflect the variation explained by the regression model (SS regression) and the unexplained variation (SS residual). The MS for regression is calculated by dividing the SS regression by the degrees of freedom (df) for regression (3 in this case), while the MS residual is calculated by dividing the SS residual by the degrees of freedom for residual (77 in this case). To test the significance of the independent variables, we can also look at their corresponding p-values in the regression output. In this case, we can see that the p-values for all three independent variables are less than 0.05, indicating that they are all significantly related to the dependent variable, thus rejecting the null hypothesis. Question 4 A researcher regressed Nitrogen (X1), Phosphorous (X2), and Irrigation (X3) on the dependent variable Corn Yield (Y). The null hypothesis was: H0: b1 = b2 = b3 = 0 The ANOVA table from the regression was as follows: ANOVA Source of Variation df Sum of Squares Mean Square F Sig. Regression 3 47.654 15.885 15.701 .000 Residual 77 77.899 1.012 Should the null hypothesis be rejected (reject the null hypothesis) or not rejected (fail to reject the null hypothesis)? Answer The ANOVA table provided in the question shows the results of a regression analysis that was conducted to examine the relationship between three independent variables (Nitrogen, Phosphorous, and Irrigation) and a dependent variable (Corn Yield). In this analysis, the null hypothesis stated that the coefficients for all three independent variables (b1, b2, and b3) are equal to zero, indicating that there is no significant relationship between these variables and the dependent variable. The alternative hypothesis would be that at least one of these coefficients is not equal to zero, indicating a significant relationship between the variables. To determine whether the null hypothesis can be rejected, we need to look at the Fstatistic and its associated significance level. In this case, the F-statistic is 15.701 and the associated significance level (Sig.) is .000, which is less than the conventional alpha level of .05. This indicates that the probability of obtaining these results by chance alone is less than 5%, and thus the results are considered statistically significant. According to the ANOVA table, the regression sum of squares (47.654) is much higher than the residual sum of squares (77.899), which suggests that the independent variables (Nitrogen, Phosphorous, and Irrigation) are significantly related to the dependent variable (Corn Yield). Therefore, based on these results, we can reject the null hypothesis and conclude that there is a statistically significant relationship between Nitrogen, Phosphorous, and Irrigation and the Corn Yield. This means that, on average, these variables have a significant impact on predicting the Corn Yield, and the regression model can be used to make accurate predictions. Question 5 In order to ensure efficient usage of a server, it is necessary to estimate the mean number of concurrent users. According to records, the average number of concurrent users at 100 randomly selected times is 37.7, with a standard deviation ? = 9.2. (a) Construct a 90% confidence interval for the expectation of the number of concurrent users. (b) At the 1% significance level, do these data provide significant evidence that the mean number of concurrent users is greater than 35? Answer (a) In order to construct a 90% confidence interval for the expectation of the number of concurrent users, we first need to define our population and sample. Population: the entire set of concurrent users for the server Sample: the 100 randomly selected times We also need to determine the level of confidence we want for our interval. In this case, it is specified as 90%. This means that if we were to repeat this sampling process multiple times, 90% of the resulting confidence intervals will contain the true population mean number of concurrent users. Next, we need to calculate the margin of error, which is based on our sample size (n), the standard deviation (?), and the confidence level (z-value). The formula for margin of error is: Margin of Error = z-value * (?/√n) Since our sample size is 100, we will use the z-value for a 90% confidence level, which is 1.645. Margin of Error = 1.645 * (9.2/√100) = 1.465 The next step is to find the upper and lower bounds of our confidence interval by adding and subtracting the margin of error from the sample mean. This will give us the range in which we can be 90% confident that the true population mean lies within. Confidence Interval = Sample mean ± Margin of Error = 37.7 ± 1.465 = (36.235, 39.165) Therefore, we can be 90% confident that the true population mean number of concurrent users falls between 36.235 and 39.165. (b) To determine whether these data provide significant evidence that the mean number of concurrent users is greater than 35, we will conduct a one-sample t-test. The null hypothesis for this test is that there is no significant difference between the sample mean (37.7) and the hypothesized population mean (35). The formula for the t-statistic is: t = (x̄- μ)/(s/√n) Where: x̄ = sample mean μ = population mean (hypothesized) s = sample standard deviation (?) n = sample size Substituting the values, we get: t = (37.7 - 35)/(9.2/√100) = 3.696 We will use a one-tailed t-test since our alternative hypothesis is that the mean number of concurrent users is greater than 35. Using a t-table with 99 degrees of freedom (since n-1 = 100-1 = 99), we find that the critical t-value for a 1% significance level is 2.364. Since our calculated t-value (3.696) is greater than the critical t-value (2.364), we can reject the null hypothesis and conclude that there is significant evidence that the mean number of concurrent users is greater than 35.