Arizona State University Statistics Questions

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Mathematics

Arizona State University

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Homework 2 Question 1 Suppose we wanted to predict the selling price of a house using its size in a certain area of a city. A random sample of six houses were selected from the area. The data is presented in the following table with size given in hundreds of square feet, and sale price in thousands of dollars. Size (Xi) 10 13 16 19 22 25 Price (Yi) 70 95 85 125 130 120 𝑎) Find the least squares estimate for the regression line 𝑌𝑖 = 0 + 1 𝑋𝑖 + 𝜖𝑖 . 𝑏) What would be your estimate of the sale price for a 2,000 square foot house? 𝑐) Estimate the standard deviation of the error term 𝜖𝑖 𝑑) Test the null hypothesis that 1 = 0 (perform a two-sided test), using 𝛼 = 0.05. Is the model useful? 𝑒) Perform the regression using R, and give the 𝑝-value to the test in part 𝑑). Verify that the p-value agrees with your conclusion in part 𝑑). Question 2 With the growth of internet service providers, a researcher decides to examine whether there is a correlation between cost of internet service per month (rounded to the nearest dollar) and degree of customer satisfaction (on a scale of 1 - 10 with a 1 being not at all satisfied and a 10 being extremely satisfied). The researcher only includes programs with comparable types of services. A sample of the data is provided below. dollars satisfaction 11 6 18 8 17 10 15 4 9 9 5 6 12 3 19 5 22 2 25 10 𝑎) Find the equation of the fitted regression line. 𝑏) Estimate the standard deviation of the error term i 𝑐) Calculate a 90% confidence interval for 1 𝑑) Test the null hypothesis that 1 = 0 (perform a two-sided test), using 𝛼 = 0.1. Is the model useful? 𝑒) Perform the regression using R, and give the 𝑝-value to the test in part 𝑑). Verify that the 𝑝-value agrees with your conclusion in part 𝑑).
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Explanation & Answer

Please have a look at the answer.

1-a) The least squares estimate for the regression line
Yi = β0 + β1Xi + εi
can be given as:
β1 =

𝑁 ∑ 𝑥𝑦−∑ 𝑥 ∑ 𝑦
𝑁 ∑ 𝑥 2 −(∑ 𝑥)2

β0 =

∑ 𝑦−β1 ∑ 𝑥
𝑁

So, we get the least square regression line as:
Yi = 38333.33 + 37.619Xi + εi
1-b) For a 2,000 square feet house, the estimated sale price is:
Yi = 38333.33 + 37.619*2000 = $113571.3
1-c) The error term is given as:
εi = Yi – Ŷi
X

Y
1000
1300
1600
1900
2200
2500

70000
95000
85000
125000
130000
120000

Yi
εi
75952.33 -5952.33
87238.03 7761.97
98523.73 -13523.7
109809.4 15190.57
121095.1 8904.87
132380.8 -12380.8

The standard deviation of error terms (RMSE)...


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