The Null Hypothesis H0 - Assignment Example

Q1a Step Identifying the null hypothesis H0 and the alternate hypothesis H1. The null hypothesis is that homes (referred to as population consume as much electricity as consumed by the units (referred to as population 2) whereas the alternative hypothesis is that homes consume more electricity. Step 2: Choosing α α = 0.05 Choosing α = 0.05 means that there is always a 5% chance that one rejects the null hypothesis when it was actually correct. Type I error Conversely there is a 5% chance that one fails to reject the null hypothesis when it is actually incorrect. Type II error Step 3: Selecting the test statistic and determining its value from the sample data. We choose a two-sample one-tailed t-test of the null hypothesis because both the populations are normally distributed, are independent and have equal but unknown variances. We first compute the standard error (SE), degrees of freedom (DF), and the t-score test statistic (t). = 40.5973 =48 where S1 is the standard deviation of sample 1, S2 is the standard deviation of sample 2, n1 is the size of sample 1, n2 is the size of sample 2,  is the mean of sample 1,  is the mean of sample 2, is the mean of first population and is the mean of the second population and SE is the standard error. Step 4: Comparing the observed value of the statistic to the critical value obtained for the chosen α. For this one-tailed test, the P-value is the probability of obtaining a t-score test statistic that is more extreme than (i.e., greater than 2.3827), assuming the null hypothesis is true. If the P-value is less than the significance level, we reject the null hypothesis. To find the probability that the t-score test statistic is greater than 2.3827, we use the Excel “TDIST(2.3827,48,1)” command. The P-value is: 0.0106. Step 5: Making a decision. Since the P-value (0.0106) is smaller than the significance level (0.05), we can reject the null hypothesis that the homes and units consume equal amount of electricity. Q1b Step 1: Identifying the null hypothesis H0 and the alternate hypothesis H1. The null hypothesis is that electricity consumption rate of homes (referred to as population 1) equals that of units (referred to as population 2) whereas the alternative hypothesis is that electricity consumption rate of homes is higher than that of units. Step 2: Choosing α α = 0.05 Choosing α = 0.05 means that there is always a 5% chance that one rejects the null hypothesis when it was actually correct. Type I error Conversely there is a 5% chance that one fails to reject the null hypothesis when it is actually incorrect. Type II error Step 3: Selecting the test statistic and determining its value from the sample data. We choose a two-sample one-tailed t-test of the null hypothesis because both the populations are normally distributed, are independent and have equal but unknown variances. We first compute the standard error (SE), degrees of freedom (DF), and the t-score test statistic (t). = 0.1218 =48 where S1 is the standard deviation of sample 1, S2 is the standard deviation of sample 2, n1 is the size of sample 1, n2 is the size of sample 2,  is the mean of sample 1,  is the mean of sample 2, is the mean of first population and is the mean of the second population and SE is the standard error. Step 4: Comparing the observed value of the statistic to the critical value obtained for the chosen α. For this one-tailed test, the P-value is the probability of obtaining a t-score test statistic that is more extreme than -1.1058 (i.e., greater than -1.1058), assuming the null hypothesis is true. If the P-value is less than the significance level, we reject the null hypothesis. To find the probability that the t-score test statistic is greater than -1.1058, we use the Excel “1-TDIST(1.1058,48,1)” command. The P-value is: 0.8628. Step 5: Making a decision. Since the P-value (0.8628) is greater than the significance level (0.05), we cannot reject the null hypothesis that the consumption rate of electricity for houses and units is the same. Q1c The homes consume more electricity than units only when their size is not kept under consideration as we have seen in the Q1a. If their size is held under consideration, we have reason to believe that both homes and units consume equal amount of electricity as is evident in Q1b. Q2 (a) Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept 0.534983 6.725214 0.07954 0.93692 -12.987 14.05693 Size (Squared Meters) 1.08177 0.050125 21.58151 2.35E-26 0.980987 1.182552 (b) Regression Statistics Multiple R 0.95214 R Square 0.906571 Adjusted R Square 0.904625 Standard Error 13.12236 Observations 50 R² is the coefficient of determination. In this case, R² = 0.9066 means that 90.66% of the variation of y can be explained by the relationship between x and y which is quite high whereas the remaining variation is unexplained and is due to extraneous factors. (c) Residual graph Classical assumptions for the residuals of the regression analysis include normality, homoscedasticity and no autocorrelation. Normality A simple test of normality can be provided by the ‘moments’ of the random variable, i.e. average, standard deviation, skewness and kurtosis. Using Monte Carlo analysis, 10000 iterations of the test statistics computed from the randomly generated standard normal variable given in the table below give us a range of values. A comparison of the test statistics of the residuals of the regression model with the simulated values indicates that we cannot reject the assumption of normality of errors. Predicted model Standard Normal Variable Residuals Mean Variance Lower CL Upper CL Average 0.0000 0.000216 0.019843 -0.27655 0.276908 SD 1.0000 0.995414 0.010199 0.803951 1.19899 Skewness 0.2689 0.000763 0.112775 -0.67591 0.665531 Kurtosis -0.8920 -0.00382 0.433187 -0.91402 1.659935 Homoscedasticity The White (1980) test is frequently used to test the homoscedasticity of the residuals. This test involves an auxiliary regression of {ût2} on the original regressors ( xit) and all their squares (xit2). The null is that the residuals are homoscedastic, and the alternative is that the variance of the {ut} process depends on xt and on the xit2. Below is the Excel output of the auxiliary regression. SUMMARY OUTPUT Regression Statistics Multiple R 0.205013 R Square 0.04203 Adjusted R Square 0.001266 Standard Error 173.1917 Observations 50 ANOVA   df SS MS F Significance F Regression 2 61853.1 30926.55 1.031044 0.364559 Residual 47 1409782 29995.36 Total 49 1471635         Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept 25.06269 282.481 0.088723 0.929679 -543.216 593.3411 Size 3.32481 4.659924 0.71349 0.479071 -6.04975 12.69937 Size2 -0.01603 0.018148 -0.88318 0.381632 -0.05254 0.020481 The t-stats of the independent variable Size and its square are insignificant individually and so is F-value which indicates that both the independent variables are also not jointly significant. So we cannot reject the null of homoscedasticity of errors in our model. Autocorrelation Another assumption of a valid regression model is that the residuals should not be auto-correlated. The Durbin-Watson (1951) statistic provides a test for significant residual autocorrelation at lag 1: the DW should be close to 2.0--say, between 1.4 and 2.6 for a sample size of 50. However, the estimated DW test statistic for our regression model is only 2.3 indicating the absence of autocorrelation in the residuals. The matter of the fact however is that we are dealing with cross-sectional data, so autocorrelation is not the relevant issue of this model. In the graph chart of the residuals below, the residuals are sorted randomly showing no autocorrelation. However in the chart further below, the residuals were first sorted in ascending order giving the indication of the presence of strong autocorrelation. This suggests that in the cross-section data, the randomizing of the variables solves the problem of autocorrelation in the residuals. (d) A new home with 180 square meters is completed. Construct a 90% prediction interval of the electricity consumption per week. The fitted regression model is: The estimated value based on the fitted regression model for the observation at 180 is: The 90% prediction interval on  is: = (191.998, 198.501) We can conclude with 90% confidence that when the size of the home is 180 square meters, the electricity consumption will be between 191.998 and 198.501 kw per week. Q3 In absolute terms, units consume more electricity than homes. However, homes consume more electricity than units only when their size is kept under consideration. The significant t-statistic of the independent variable Size in the estimated regression equation indicates that the size of the homes does affect the consumption of electricity. A significantly high R2 equal to 0.9065 shows that nearly 91% of the variation in the dependent variable is explained by the explanatory variable. The failure to reject the normality, independence and homoscedasticity assumption of the residuals points to the validity of the model and its predictive power. Bibliography White, H. (1980). "A heteroskedastic-consistent covariance matrix estimator and a direct test for heteroskedasticity" Econometrica, 48, 817--838. Durbin, J., and Watson, G. S. (1951). "Testing for serial correlation in least squares regression II" Biometrika, 38, 159--178. Read More