Download Regression Analysis: Positive Slope Probability, Confidence Intervals, Assumption Check and more Exams Statistics in PDF only on Docsity! 5E Final practice Formulas at end are as they will be on exam. For any hypothesis test, clearly state null and alternative hypotheses. 1. Suppose two variables x and y have a negative correlation and follow the regression model y = 2 − .1x + , where is normal with a standard deviation of .7. We do a least-squares regression using 30 values for x, where the values have a mean of 4 and a sample standard deviation of 1.2. What is the probability that we get a positive slope in our regression analysis? What if we use 100 values for x, with the same mean and standard deviation them? What if the standard deviation of is only .4? Hint: Use 12.17 from p. 513. To do that use the sample st dev to find ∑ (xi − x̄)2. 2. Below is a scatter plot of data with regression output: b0 = 5.1, b1 = 4.7 and SSE = 345. The sample standard deviation of the x values are 12.3 and the mean is 5.5. The sample standard deviation of the y values are 3.4 and the mean is 10.1. There are 30 data points plotted. Find a 95% confindence interval for the the slope b1. At a 5% significance level, does the data provide evidence that there is a correlation between the variables? Find a 95% confidence interval for the mean of y given that x is equal to 7. Find a 95% confidence interval for an individual value y given that x is equal to 7. The residuals are plotted below against the independent variable x. Are the assumptions of our regres- sion model satisfied? ( Explain). 3. Which are possible regression outputs for the data depicted below. (i) b0 = .9, b1 = 3.0, SST = 820, SSE = 235, SSR = 652 (ii) b0 = 1.0, b1 = 3.1, SST = 921, SSE = 262, SSR = 659 (iii) b0 = .9, b1 = 3.1, SST = 828, SSE = 235, SSR = 593 (iv) b0 = 1.0, b1 = 3.0, SST = 828, SSE = 335, SSR = 593 1 4. Below is regression output and a scatter plot for monthly returns on Microsoft and Apple with data from September 2000 to December 2007. (Source: Yahoo) One outlying value has been removed, so there are a total of 86 data points. What are the sample correlation coeficient and coefficient of determination? Do these indicate a strong correlation for the data? Describe how those statistics match the scatter plot. At a 5% significance level, do the data provide evidence that there is a positive correlation between the returns on Microsoft and Apple? Given that the return in a given month for Microsoft is 2%, what range do you expect the return for Apple was in that month? (Use a 95% confidence level.) What if the return on Microsoft is 10%? Negative 3%? MSFT AAPL mean .215 3.416 st dev 8.183 12.388 b1 .473 SSE 11770 b0 3.314 SSR 1273 s 11.837 ∑ (xi − x̄)2 5691 sb1 .157 ∑ (xi − x̄)(yi − ȳ) 2692 2
