FREC 608 Guided Exercise 9

Transcription

1 FREC 608 Guided Eercise 9 Problem. Model of Average Annual Precipitation An article in Geography (July 980) used regression to predict average annual rainfall levels in California. Data on the following variables were collected for 30 meteorological weather stations scattered throughout California. For the group work we will focus on a bivariate regression of Annual Percip on Latitude. You will have the option of eamining all the variables for this problem for the last assignment Annual Precip DEPENDENT VARIABLE: Annual Precipitation in inches Altitude The altitude of the station in feet Latitude The latitude of the station in degrees Distance Distance from the coast in miles Facing I made this into a dummy variable. Stations on the Westward facing slopes of the California mountains were coded as, whereas stations on the leeward side were coded as 0 a. The following are the descriptive statistics on each of the variable. Briefly describe Annual Precipitation using the mean, median, std deviation and so forth. Annual Percip Altitude Latitude Distance Facing The Mean Annual Precipitation is Mean The mean is larger than the Standard Error median indicating right skew. There Median Mode is an etreme range in the data Standard Deviation from.66 inches to 73.2 inches. Sample Variance The spread of the data is large relative Kurtosis to the mean: the CV is 83.8%. Skewness Range Minimum Maimum Sum Count b. The following is the Covariance matri on the variables. Using the formula to the right, generate the Correlation Matri for this data. Remember, the variances (and thus the standard deviations) for each variable is on the diagonal of the covariance matri. Annual Precip Altitude Latitude Distance Facing Annual Precip Altitude Latitude Distance Facing r = Cov σ σ 2 2 Annual Precip Altitude Latitude Distance Facing Annual Precip.000 Altitude Latitude Distance Facing For Annual Precip and Latitude: /( * ) =.577

2 c. Briefly describe the correlation between Annual Precip and Latitude. Does this correlation make sense? Remember, this data is from California weather stations. r =.577 As the Latitude increases the annual precipitation also increases. The correlation is moderately in strength. This makes sense since as you move further north in CA (higher Latitude) there tends to be more rainfall. Inches of rain CA Annual Precipitation by Latitude Latitude d. Facing is a dummy variable. Stations on the Westward facing slopes of the California mountains were coded as, whereas stations on the leeward side were coded as 0. Interpret the correlation between Annual Precip and Facing. r =.598. Since FACING is a dummy variable the correlation interpretation is a little different. Instead of as FACING increases, ANNUAL PRECIP increases, we will say that Westward facing stations tend to have more rainfall. Since the correlation is moderate in strength, there are moderate differences in average rainfall on west side and lee side stations. e. Now we will shift to the bivariate regression of Annual Precip on Latitude. The following are formulas for regression based on covariance β SS = SS XY X β 0 = β X Y Using the covariances, the variance for Latitude, and the means, calculate estimates for the regression coefficients. For b it is simply the covariance divided by the variance for Latitude. b = /6.873 = b 0 = *37.03 = -3.33

3 Confirm your results from the regression output from Ecel. SUMMARY OUTPUT Regression Statistics Multiple R R Square Adjusted R Square Standard Error 3.89 Observations 30 df SS MS F Sig F Regression Residual Total Coeff Std Error t Stat P-value Intercept Latitude f. Verify that R 2 in a bivariate regression is simply the correlation (r) squared. Interpret R 2 for this model. r 2 = =.333 R 2 =.333 One third (33.3%) of the variability in Annual Precipitation is eplained by knowing the Latitude of the station. g. What does the model predict for annual precipitation when the latitude is 33 degrees est. Annual Precip = (33) = degrees est. Annual Precip = (36) = degrees est. Annual Precip = (40) =

4 PROBLEM 2. This focuses on whether females mid-level managers have lower salaries than males. The data set contains the following variables for 220 mid-level managers of firms (we will only focus on these four variables): SALARY Dependent Variable Base annual salary in $,000s SEX POSITION = Female; 0 = Male An inde of the position of the employee in the firm, based on the number of employees supervised, size of budget and so forth. A higher number means higher level in the company YEARS EXP The number of years of eperience a. The following is the correlation matri for this data. Briefly describe the correlations between each of the independent variables and the dependent variable SALARY Salary Se Position YearsEper Salary.000 Se Position YearsEper Correlation between Se and Salary is -.38 Women earn slightly less salary than men Correlation between Position and Salary is.89 Strong correlation, those in higher positions earn more salary Correlation between YearsEper and Salary is.32 There is a weak positive relationship between eperience and salary b. For reference, I am including the results for this data. Then we will do the very same thing in regression. Note the means, variances, and conclusion from the results. Based on the result, could we conclude there is a difference in salary between men and women at alpha =.05? Briefly summarize the results. Anova: Single Factor SUMMARY Groups Count Sum Average Variance Females Males Source of Variation SS df MS F P-value F crit Between Groups Within Groups Total The provides a test of the difference in salary between men and women in the sample. The test confirms that there is a significant difference in salary between men and women women earn less. The F-test is significant at p =.04.

5 c. The following is the regression statistics for the multi-variate regression of SALARY on SEX. SEX is a dummy variable where = Females and 0 = Males. SUMMARY OUTPUT Regression Statistics Multiple R 0.38 R Square 0.09 Adjusted R Square 0.05 Standard Error Observations 220 df SS MS F Sig F Regression Residual Total Coef Std Error t Stat P-value Lower 95% Upper 95% Intercept Se d. Confirm for yourself the following: The R-square for both and Regression are the same. The Tables are identical - sums of squares, df, Mean squares and the F-test are the same. The pooled variances are the same (think about this one!) e. Solve the equation to get the estimated salary for males and females. To do this you need to use the estimated coefficients and realize that SEX can only take on two values: 0 and. Confirm that: The equation estimates the mean salary for males and females When Se = (Females) est Salary = () = the average salary for women When Se = 0 (Males) est Salary = (0) = 44.0 the average salary for men The slope coefficient represents the difference in salary between males and females The difference in mean salary is: = The intercept represents the reference group (in this case the group represented as zero for SEX) Since SEX = represents females, the reference group is males. The intercept is the mean for males