Statistical Inference Business Estimates

Question: Discuss about the Statistical Inference for Business Estimates. Answer: 1. The below table shows the results of an OLS regression of US real GDP growth rates (REALGDP) on changes of oil prices (OIL), interest rate (INTERESTRATE) and inflation rates (INFLATION) (monthly data from 1990 to 2013): Discuss the statistical significance of the parameters, interpret the sign and magnitude of the estimates, and overall fit of the model. Solution: The p-value for the independent variable or predictor oil is given as 0.003 which is less than level of significance or alpha value 0.01, so we reject the null hypothesis that the predictor oil is not significant. This means we conclude that the independent variable or the predictor oil for the given regression model is statistically significant. The p-value for the independent variable or predictor interest rate is given as 0.032 which is greater than the significance level or alpha 0.01, so we do not reject the null hypothesis that the predictor interest rate is not significant at 1% level of significance. For the independent variable or predictor inflation, the p-value is given as 0.145 which is greater than significance level or alpha value 0.01, so at the 1% level of significance we do not reject the null hypothesis that the predictor inflation is not statistically significant. So, overall we conclude that the predictor oil is significant while the predictors interest rate and i nflation are not statistically significant for the given regression model. The constant for the regression equation or the y-intercept is statistically significant as the p-value is given as 0.00 which is less than the significance level 0.01. The signs for the given three predictors are negative; this means there is negative impact of these variables on the real GDP growth rates. The coefficient of determination or the value of the R square plays an important role in determining the explained variation in the dependent variable due to the independent variables in the given model. The value of the adjusted R-square or the coefficient of determination is given as 0.58 which means about 58% of the variation in the response variable real GDP is explained by the predictors oil, interest rates and inflation. Are the results in line with the predictions of the theory and why? Solution: For answering this question regarding with the predictions of the theory, we need to use the decision rule. We reject the null hypothesis if the p-value is less than the given level of significance or alpha value and we do not reject the null hypothesis if the p-value is greater than the given level of significance or alpha value. From the given information for the regression model, it is observed that predictor oil is statistically significant and other two predictors interest rates and inflation are not statistically significant. From the given information it is observed that the predictor oil is statistically significant because for this predictor the p-value is less than the given level of significance. So, we reject the null hypothesis that the predictor oil is not a statistically significant. This means that the predictor oil is significant. For the other two predictors interest rates and inflation, the p-values for both predictors are greater than the given level of significan ce, so we do not reject the null hypothesis that the given predictors are not statistically significant. So, these results are not in line with the predictions of the theory. For the purpose of the unbiased prediction of the response variable real GDP, it is important to have all predictors statistically significant. 3. A company wants to produce three different mobile phones, with low-range, mid-range and high-range specifications, respectively. A survey with 100 respondents has been used to reveal the choices of potential customers. The company wants to review the figures to see if the three mobile phones would be equally popular. The results of the Chi-Square test are given in the following tables: Describe the null hypothesis for the Chi-Square test. Solution: For the given scenario, company wants to check whether the three different mobile phones are equally popular or not. For checking this claim we have to use the chi square test. The null and alternative hypothesis for this chi square test is given as below: Null hypothesis: H0: The popularity of low-range, mid-range and high-range specification mobile phones is same. Alternative hypothesis: Ha: The popularity of low-range, mid-range and high-range specification mobile phones is not same. Discuss the results and explain whether there are statistically significant differences in the preference for the three devices. Solution: We know that the decision rule for rejecting or do not rejecting the null hypothesis. We reject the null hypothesis if the p-value is less than the given level of significance or alpha value and we do not reject the null hypothesis if the p-value is greater than the given level of significance or alpha value. The p-value for the chi square test is given as 0.032 which is greater than the level of significance or alpha value 0.01, so we do not reject the null hypothesis that the popularity of low-range, mid-range and high-range specification mobile phones is same. This means at the 1% level of significance there is no any statistically significant differences in the preferences for the three devices. What are the underlying assumptions of the Chi-Square test? Explain if, in your opinion, those are met in the above examples. Solution: The assumptions for the chi square test are given as below: One variable should be categorical variable. The observations should be independent. The group of categorical variables should be mutually exclusive. There must be at least 5 expected frequencies. For the given chi square test, it is observed that the variable specification of mobile phone is a categorical variable. Also all observations are independent and there are 0 cells have expected frequencies less than 5. So, all assumptions are followed by this test. References: Casella, G, and Berger, R, L, 2002, Statistical Inference, Duxbury Press. Cox, D, R, and Hinkley, D, V, 2000, Theoretical Statistics, Chapman and Hall Ltd. Liese, F, and Miescke, K, 2008, Statistical Decision Theory: Estimation, Testing, and Selection, Springer.