Is there sufficient evidence using the observed data to conclude that a positive linear relationship exists between X and Y? Use a 5% significance level

1) Five observations collected in a regression study are as follows:

a) Is there sufficient evidence using the observed data to conclude that a positive linear relationship exists between X and Y? Use a 5% significance level.

b) Compute the 95% confidence interval for the slope of the population regression line.

2) The following data were collected on two variables X and Y.

a) Specify the null and alternate hypotheses for testing that the population correlation is negative.

b) Calculate the estimate of the population correlation coefficient between X and Y.

c) Calculate the coefficient of determination.

d) Using a 5% significance level, test the hypothesis in part a).

3) Data for X and Y are given below.

a) Calculate the least squares line.

b) Determine a 90% confidence interval for the mean value of Y using X = 10.

c) Determine a 90% prediction interval for Y using X = 10.

4) A real estate agent wishes to estimate the monthly rental for apartments based on the size (square feet) and the location of the apartments. She chose the following model: y = β₀ + β₁x₁ + β₂x₂ + Ɛ where, x₁ = square footage of the apartment; and x₂ = 1 if located in the town center, and 0 if not located in the town center. This linear regression model was fitted to a sample of size 50 to produce the following regression equation: yhat = 145 + 1.2x1 + 300x2.

a) Predict the average monthly rent for an apartment located in the town center that has 1,500 square feet.

b) Predict the average monthly rent for an apartment located in the suburbs that has 1,500 square feet.

c) Interpret b₂.