d. Now test whether square footage has a significant positive linear relationship with price. Use a = 0.05 level of significance. Write the null and alternative hypothesis, state the test statistic and p-value, and make a conclusion. e. Find and interpret a 95% confidence interval for the mean price when sqft=2000. f. Find and interpret a 95% prediction interval for the price when sqft=2000. g. What would happen to the interval from part f. if the confidence level is decreased to 90%? Explain in a sentence or two. h. Run a summary of your dataset (in R, this is: summary(dataset) ). Would it make sense to predict the sale price of a house that is 8500 square feet?Round to the nearest 2nd decimal place (nearest hundredth) when possible. Use a : 0.05 signicance level where needed. Each part is worth 2 points. Total of 98 points. 1. Say a regression equation if t to data, and the correlation coefcient estimate ,3 ,between X and Y is 0.5. State if true or false. a. The slope of the regression line is 0.5. b. The regression model with X explains 50% of the variation in Y, c. 25% of the variation in Y is explained by X. d. 50% of the variation in Y is explained by X. e. X is positively associated with Y. f. Even if X and Y have a non linear relationship, the value of 0.5 can be used to measure the association between X and Y. 2. Suppose a simple linear model is t to predict Y: weight in kilograms using X 2 height in centimeters of an adult. But say a new simple linear model is t using Y: height in centimeters and X 2 weight in kilograms (that is to say Y and X have reversed). State whether each of the following would be the same for this new model as it was for the original model, or it would be different and explain in a sentence or two. a. The value of R. b. The value of Hz. i. What would happen to the interval from part f. if the condence level is increased to 100%? Explain in a sentence or two. j. What is the estimate of 05? Interpret this quantity in context of the problem. k. Say we have two houses. House one is 1500 sqft (call this X1) and house two is 3500 sqft (X2). Does it make sense for both these houses to have the same estimate of 05? Explain in context of the problem. 4. This question will use the skin cancer data set that is on the class website. Say it is of interest if latitude is predictive of mortality rate due to skin cancer. Fit a simple linear model where X =Lat and Y=Mort. a. Using a a = 0.05 signicance level, conduct a formal statistical test of whether latitude has a linear association with the mortality. b. Find and interpret a 99% condence interval for the mean mortality rate when Lat=40. c. Find and interpret a 99% prediction interval for the mortality rate when Lat=40. d. What can you say about the center of the condence interval and prediction interval. Is it the same? Explain in a sentence or two why or why not it would be the same. e. How does the width of the condence interval compare to the prediction interval. Explain in a sentence or two. c. The estimate of 1. d. The estimate of g. e. The test statistic to test if the correlation between explanatory and response is equal to U. f. The residuals for each observation. 3. This question will use the MidwestSales data set that is on the class website. This dataset involves houses being sold in the midwest. The response variable is the price the house was sold (in dollars) and explanatory variables such as the square footage of the house, the number of bedroom, the year the house was built, etc. Run the following linm to import the dataset and then to give names to the variables. Remember to replace the ".fMidwestSalestxt", in the read.table call with your own path to where you saved the dataset. Say you named the dataset "MidwestSales": # Import MidHestSales dataset MidHestSales - read.tab1e[".Midwest3ales.txt", fill-TRUE, header-FALSE) it This dataset does not have names, so we will add names to the variables names {MidHestSales)-c{ Ilidll ' "price", Ilsqftll ' lubed" , "bath" ' "BL" , "garage" , "pool" ' "yearn ' "quality" , "style" , "lot" ' Ilhuyll) a. Fit a linear model where the square footage of the house is used to predict the sale price. (X =sqft and Y=price). Write the estimated regression equation. b. Interpret the estimate of the slope. c. Test whether square footage has a signicant linear relationship with price. Use a = 0.05 level of signicance. Write the null and alternative hypothesis, state the test statistic and pvalue, and make a conclusion