A linear regression study gave the results shown below for a pair of sets of 20 observations
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A linear regression study gave the results shown below for a pair of sets of 20 observations

For the theory section, show your work and do not round off your middle work.

Keep your final answer to four (4) decimal places, where relevant.
1. A linear regression study gave the results shown below for a pair of sets of 20 observations.
           ∑x = 300;  ∑y = 2800; ∑(x-x ̅) = 600; ∑(y-y ̅ )2 = 900;
           ∑(x-x ̅) (y-y ̅ ) = 400  
a Calculate the estimates of the slope and y intercept.
b What is the regression equation to predict y from x?
c Calculate r and r2.
d At the 1% level of significance, can you conclude that there is a significant positive correlation between the two variables? 

2. A statistics student wonders whether he can predict the annual salary of a first-year employee from their grade point average in a Bachelor of Science (Statistics Major). He obtained data from 10 alumni of his department. The data is given below. 
GPA (x) Annual salary ($) (y)
2.7         40,000
3.2         43,000
3.6         49,000
3.3         45,000
3.1         48,000
2.5         36,000
2.7         39,000
3.3         42,000
2.9         30,000
2.6         22,000
a Find ∑x, ∑y, ∑xy, ∑x2, ∑y2, x ̅y ̅
b Use the summary statistics from part (a) and formulas you may have learned in your previous statistics course OR use the simple linear regression procedures demonstrated in the textbook for this course to obtain the regression equation to predict y from x. 
c Interpret the slope and the y intercept with reference to the context.
d What is the predicted annual salary of a person with a GPA of 3.6? What is the residual?
e Calculate the coefficient of determination. Use the coefficient of determination to justify whether the model is a good fit. 
f Use the test of slope to see whether there is a significant relation between the two variables x and y.
g Use the test of correlation to see whether there is a significant positive correlation coefficient.
h Explain in words why the test statistics for tests (f) and (g), above, are the same. 
i Construct a 95% two-sided confidence interval for slope β.
j Construct a 95% two-sided confidence interval for correlation coefficient ρ.

3. In a study to determine if response time, y, could be modeled as a linear function of the temperature, a process was run at each of four temperatures three times, a total of twelve pairs of observations. The two AOV tables shown below are (I) for fitting y as a simple linear function of x, and (II) for AOV using the values of x to define the “treatments.”
Analysis of Regression
                     Sum of             Mean
Source             DF          Squares            Square     
Regression    ______       290.40000        290.40000     
Error             ______        43.60000           4.36000
Total             ______      334.00000
Analysis of Variance
                                 Sum of         Mean
Source      DF              Squares         Square       
Groups   _______      318.0000000   106.0000000    
Error      _______        16.0000000       2.0000000
Total      _______      334.0000000
a Complete the tables by providing the correct degrees of freedom.
b Is there significant evidence that the slope of the line is not zero? Explain.
c Is there significant evidence that the straight line does not provide an appropriate fit? Explain.
d Write a short note explaining what the inverse regression problem is, and why it is important in statistical analysis.
Hint
Statistics"Linear regression models are used to show or predict the relationship between two variables or factors, where the factor which is being predicted is known as the dependent variable and the factors that are used to predict the value of the dependent variable are known as the independent variables. Also, in simple linear regression, each observation consists of two values, one is for the ...

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