Question

Simple Linear Regression: Suppose a simple linear regression analysis provides the following results:

b0 = 6.000,    b1 = 3.000,    sb0 = 0.750,

sb1 = 0.500,  se = 1.364

and n = 24. Use this information to answer the following questions.

(a) State the model equation.

ŷ = β₀ + β₁x

ŷ = β₀ + β₁x + β₂s_b1   

ŷ = β₀ + β₁x₁ + β₂x₂

ŷ = β₀ + β₁s_b1

ŷ = β₀ + β₁s_b1

x̂ = β₀ + β₁s_b1

x̂ = β₀ + β₁y

(b) Test for a linear relationship between x and y. Use a 5% level of significance.

State the hypotheses to be tested.

H₀: β₀ = 0
H_a: β₀ ≠ 0

H₀: β₃ = 0
H_a: β₃ ≠ 0    

H₀: β₂ = 0
H_a: β₂ ≠ 0

H₀: β₁ = 0
H_a: β₁ ≠ 0

H₀: β₄ = 0
H_a: β₄ ≠ 0

Interpret the hypotheses you specified above.

H₀: None of the explanatory variables are important in explaining/predicting x.
H_a: At least one explanatory variable is important in explaining/predicting x.

H₀: There is a linear relationship between x and y.
H_a: There is no linear relationship between x and y  

H₀: There is no linear relationship between x and y.
H_a: There is a linear relationship between x and y

H₀: All of the explanatory variables are important in explaining/predicting x.
H_a: None of the explanatory variables are important in explaining/predicting x.

State the decision rule.

Reject H₀ if p > 0.025.
Do not reject H₀ if p ≤ 0.025.

Reject H₀ if p < 0.05.
Do not reject H₀ if p ≥ 0.05.     

Reject H₀ if p > 0.05.
Do not reject H₀ if p ≤ 0.05.

Reject H₀ if p < 0.025.
Do not reject H₀ if p ≥ 0.025.

State the appropriate test statistic name, degrees of freedom, test statistic value, and the associated p-value (Enter your degrees of freedom as a whole number, the test statistic value to three decimal places, and the p-value to four decimal places).

t ( ?   ) = , p = ( ? )  

State your decision.

?Reject the null hypothesis: There is a linear relationship between y and x.

?Reject the null hypothesis: There is not a linear relationship between y and x.     

?Do not reject the null hypothesis: There is a linear relationship between y and x.

?Do not reject the null hypothesis: There is not a linear relationship between y and x.

(c) What would be a typical size error of prediction when you use this regression model? (Round your answer to three decimal places.)

???

Choose the correct interpretation of the typical size error of prediction you identified above by mentally inserting the value into the blanks below.

When using this model to estimate parameters, we expect to be _______ units closer to the true value, on average.

When using this model to estimate parameters, we expect to be off by _______ units, on average.     

When using this model to make predictions, we expect to be _______ units closer to the true value, on average.

When using this model to make predictions, we expect to be off by _______ units, on average.

Regardless of your conclusions above concerning the quality of the model, use the model to answer the following questions.

(d) Use the model to make a prediction when x = 5.5. (Round your answer to three decimal places.)

???

Imagine that the actual value is 19.720 when x = 5.5. Calculate the residual. (Round your answer to three decimal places.)

???

Interpret the residual you calculated immediately above by mentally inserting the ABSOLUTE VALUE of the residual into the blanks below.

?Our prediction was _______ units lower than the actual target value when x = 5.5. Our prediction was an overestimate.

?Our prediction was _______ units higher than the actual target value when x = 5.5. Our prediction was an overestimate.     

?When using this model to make predictions, we expect to be off by _______ units, on average.

?When using this model to make predictions, we expect to be _______ units closer to the true value, on average.

?Our prediction was _______ units lower than the actual target value when x = 5.5. Our prediction was an underestimate.

?Our prediction was _______ units higher than the actual target value when x = 5.5. Our prediction was an underestimate.

(e) Use the model to make a prediction when x = 6.0. (Round your answer to three decimal places.)
???

Imagine that the actual value is 26.125 when x = 6.0. Calculate the residual. (Round your answer to three decimal places.)
???

Interpret the residual you calculated immediately above by mentally inserting the ABSOLUTE VALUE of the residual into the blanks below.

?When using this model to make predictions, we expect to be off by _______ units, on average.

?Our prediction was _______ units lower than the actual target value when x = 5.5. Our prediction was an underestimate.    

? When using this model to make predictions, we expect to be _______ units closer to the true value, on average.

?Our prediction was _______ units higher than the actual target value when x = 5.5. Our prediction was an underestimate.

?Our prediction was _______ units higher than the actual target value when x = 5.5. Our prediction was an overestimate.

?Our prediction was _______ units lower than the actual target value when x = 5.5. Our prediction was an overestimate.

Answer 1

(a) State the model equation.​​​​​

ŷ = β₀ + β₁x

(b) Test for a linear relationship between x and y. Use a 5% level of significance.

H₀: β₀ = 0
H_a: β₀ ≠ 0

H₀: β₁ = 0
H_a: β₁ ≠ 0

Interpret the hypotheses you specified above.

H₀: None of the explanatory variables are important in explaining/predicting x.
H_a: At least one explanatory variable is important in explaining/predicting x.

State the decision rule.

Reject H₀ if p < 0.025.
Do not reject H₀ if p ≥ 0.025.

State the appropriate test statistic name, degrees of freedom, test statistic value, and the associated p-value

t = b₁ / s_b1

_{t =}  3.000/ 0.500

t= 6.00

degrees of freedom= n-1 = 24-1 = 23

The two-tailed P value is less than 0.0001 at t= 6

State your decision.

Reject the null hypothesis: There is a linear relationship between y and x.

c) What would be a typical size error of prediction when you use this regression model? (Round your answer to three decimal places.)

Standard Error (s_e) = 1.364

Interpret the residual you calculated

when using this model to make predictions, we expect to be off by __1.364_____ units, on average.

Use the model to make a prediction when x = 5.5

y= 6 + 3* x

y= 6 + 3* 5.5

y= 6+ 16.5 = 22.5

Imagine that the actual value is 19.720 when x = 5.5. Calculate the residual?

residual = y(estimate) - y(actual)

residual =  22.5 - 19.720 = 2.780

Our prediction was __2.780_____ units higher than the actual target value when x = 5.5. Our prediction was an overestimate.     

(e) Use the model to make a prediction when x = 6.0. (Round your answer to three decimal places.)
???

y= 6 + 3* x

y= 6 + 3* 6

y= 6+ 18 = 24

Imagine that the actual value is 26.125 when x = 6.0. Calculate the residual.

residual = y(estimate) - y(actual)

residual =  24 - 26.125 = -2.125

Our prediction was __-2.125_____ units lower than the actual target value when x = 6. Our prediction was an underestimate.