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Assignment: STA510 Business StatisticsPART A:Question 1 – Case Studya)The equation of the least squares regression line is given as:ŷ= b0+ b1xwhere b0= the y-interceptb1= the slope of the regression lineAccording to the excel sheet, the label A19 gives the intercept of the regression line and label A20 givesthe street score on the dexterity test (i.e. slope)Hence, the equation of the least squares regression line isŷ = 183.2621 + 122.3153xwhere ŷ = estimated selling price of the housex = rating of the street from 0 (lowest appeal) to 10 (highest appeal)The slope of the regression equation is positive, suggesting a direct relationship between rating of thestreet and the selling price of the house. This indicates due to the high level of the slope (b1= 122.3153)the price of the house increases significantly with the street rating.b)The standard error of estimate can be directly obtained from the given excel sheet label A7Hence,the standard error of estimate (Se) = 225.5847The large value of the standard error of estimate shows a greater amount of scatter in the price data.This means that with the increase in the rating of the house from 0 (lowest appeal) to 10 (highestappeal), the price of the house increases significantly.
c)t-test for the slope (b1) of the regression line:Null hypothesis H0: b1= b10(the slope is b10)Alternative hypothesis H1: b1≠ b10(the slope is not b10)Test statisticst = (b1- b10)/sb1where b1= slope of the regression lineb10= hypothesized value for slope of regression linesb1= the estimated standard deviation of the slopeTesting H0: b1= 0 versus H1: b1≠ 0For the n = 120 houses in the test sample, we have already calculated the standard error ofestimate (Se) = 225.5847 and the standard deviation of the slope (sb1) can be obtained from theexcel sheet label C20 as 10.77271.Let us assume b10= 0The observed value of the test statistic isObserved t = (b1- 0)/sb1= (122.3153 – 0)/10.77271= 11.3542Now, we see that the calculated test statistic (t = 11.3542) is also equal to the t statisticscalculated during the test.Referring to the t-distribution table, for the 0.05 level of significance and number of degrees offreedom (df) = n – 2 = 120 – 2 = 118, the critical values of t are – 1.960 to + 1.960.Thus, at the 0.05 level, the test statistic is outside the critical values and so we are able to rejectthe null hypothesis that the slope of the regression line could be zero.95% confidence interval for the slope of the regression line:For n = 120 data points, the number of degrees of freedom for the t distribution will bedf = n – 2 =118.Referring to the t distribution table, t = 1.960 will correspond to 95% confidence interval:From b1- tsb1to b1+ tsb1or 122.3153 – 1.960*10.77271 to 122.3153 + 1.960*10.77271or from101.2 to 143.43We have 95% confidence that the slope (b1) of the regression line is in the interval bounded by101.2 and 143.43. Since b1= 0 is not within the 95% interval, we can conclude, at the 0.05 level,that a linear relationship exist between house price and the street rating score.
d)The coefficient of determination, r2can be obtained from the excel sheet label B5 asr2= 0.522107 or 52.2107%The coefficient of determination, r2gives an idea of how well the regression line fits the data inthe sample test. But here the r2is only 52.2107% which means that many of the data are aboveor below the regression line by significant amount (Nagelkerke, 1991). This is obvious from thefact that the selling price of the house is dependent on various factors viz. bay views, GrandStreet, size of the house, size of the block, design of and fitting in the house, etc. These factorsplay a significant role in increasing the price of the house.So, yes the street rating appears to be a good predictor of the house prices keeping in mind thegiven factors.e)The information that has been gathered from the above calculations give a clearunderstanding that the sample that was obtained from the 120 houses gives a clear picture ofthe large variation in the selling price of the house which depends on various factors viz. bayviews, Grand Street, size of the house, size of the block, design of and fitting in the house, etc.The large value of standard error of the estimate (Se) shows a greater amount of variation in theselling price of the house data (Irwin, 1949). This shows that all the data points does not fit onthe line and are above and below the line by huge margin.The slope of the regression line, b1= 122.3153 lies between the critical values 101.2 and 143.43which shows that the equation of regression line is a good estimation of the data and the sellingprice of the house.Moreover, the coefficient of determination is around 52.2107% which gives us an idea of thevariation of the prices of the house.Keeping these factors in mind, one can say that the statistics obtained from the sample data area fairly good estimation of calculating the price of a given house with its street rating.