Hypothesis Testing

Question 1

There are 3 approaches to tackle inferentail hypothesis testing. What are they?

Answer 1

T-test for the slope

F-test for the slope

Confidence interval test for the slope

Question 2

Using the sample data set pictured, how would you use the confidence interval approach to determine if the relationship between x and y is significant using the following hypothesis:

H₀: β1=0 (there is no relationship between X and Y)
H₁: β1≠0 (there is relationship between X and Y)

(Hint: up to step 3 of 5)

Based on our finding of the confidence interval, would we reject or not reject the null hypothesis?

Answer 2

Step 1: State the hypothesis

H₀: β1=0 (there is no relationship between X and Y)
H₁: β1≠0 (there is relationship between X and Y)

Step 2: State the alpha. (Most commonly used is 0.05)
α=0.05

Step 3: Determine the coefficients and t critical values for the model

Step 4: Calculate the confidence interval for the relationship using the following formula:

B̂₁−t_a/2∗SB̂₁ ≤ β₁ ≤ B̂₁+t_a/2∗SB̂₁

2. 245 - 3.1824 x 0.292 ≤ 0 ≤ 2.245 + 3.1824 x 0.292
3. 3157 ≤ 0 ≤ 3.1736

Step 5: Decision

Reject H₀ because the hypothesized β₁ = 0 does not fall within the confidence interval of the slope.

Question 3

Using the sample data set pictured, how would you calculate the F critical values for the regression model with the following hypothesis:

H₀: β1=0 (there is no relationship between X and Y)
H₁: β1≠0 (there is relationship between X and Y)

(Hint: up to step 4 of 5)

Based on our finding of the critical value, would we reject or not reject the null hypothesis?

Answer 3

Step 1: State the hypothesis

H₀: β1=0 (there is no relationship between X and Y)
H₁: β1≠0 (there is relationship between X and Y)

Step 2: State the alpha. (Most commonly used is 0.05)
α=0.05

Step 3: Calculate the test statistic using the formula, reference the ANOVA summary.

MSR/MSE

863. 104/14.565
864. 257

Step 4: Calculate the critical values

Using the Casio Calculator

DIST -> F -> InvF
Inputs:
Area: 0.05
n:df: 1
d:df: 3
Output:
10.128

Using R

qf(0.05,1,3,lower.tail = F)

Step 5: Decision

Reject H₀ because test stat > critical value
Reject H₀ because critical value < test stat

Question 4

Using the sample data set pictured, how would you calculate the F test statistic for the regression model with the following hypothesis:

H₀: β1=0 (there is no relationship between X and Y)
H₁: β1≠0 (there is relationship between X and Y)

(Hint: up to step 3 of 5)

Answer 4

Step 1: State the hypothesis

H₀: β1=0 (there is no relationship between X and Y)
H₁: β1≠0 (there is relationship between X and Y)

Step 2: State the alpha. (Most commonly used is 0.05)
α=0.05

Step 3: Calculate the test statistic using the formula, reference the ANOVA summary.

MSR/MSE

863. 104/14.565
864. 257

Question 5

Using the sample data set pictured, how would you calculate the T test statistics for the regression model with the following hypothesis:

H₀: β1=0 (there is no relationship between X and Y)
H₁: β1≠0 (there is relationship between X and Y)

(Hint: up to step 3 of 5)

Answer 5

Step 1: State the hypothesis

H₀: β1=0 (there is no relationship between X and Y)
H₁: β1≠0 (there is relationship between X and Y)

Step 2: State the alpha. (Most commonly used is 0.05)
α=0.05

Step 3: Calculate the test statistic using the formula

(B̂₁−β₁)/SB̂₁
( 2.245 − 0 ) / 0.292
2.245 / 0.292
7.698

Where:

B̂<sub>1</sub>= Coefficient Estimate for X
β<sub>1</sub>= Hypothesis value being tested
SB̂<sub>1</sub>= Coefficient standard error estimate for X

Question 6

Using the sample data set pictured, how would you calculate the T critical values for the regression model with the following hypothesis:

H₀: β1=0 (there is no relationship between X and Y)
H₁: β1≠0 (there is relationship between X and Y)

(Hint: up to step 4 of 5)

Based on our findings, would we reject or not reject the null hypothesis?

Answer 6

Step 1: State the hypothesis

H₀: β1=0 (there is no relationship between X and Y)
H₁: β1≠0 (there is relationship between X and Y)

Step 2: State the alpha. (Most commonly used is 0.05)
α=0.05

Step 3: Calculate the test statistic using the formula

(B̂₁−β₁)/SB̂₁
( 2.245 − 0 ) / 0.292
2.245 / 0.292
7.698

Step 4: Calculate the critical values for T.

Using Casio calculator:
DIST -> t -> Invt
Inputs:
Area = α/2 (two-tailed test) = 0.025
df: n - 2 = 5 - 2 = 3
Outputs:
-3.1824 and 3.1824

Using R:

qt(.975,3) = 3.182446 or qt(.025,3) = -3.182446

Step 5: Decision

Reject H₀ because test stat is outside of the range of critical values

Question 7

What is the pictured formula used to derive?

Answer 7

The prediction interval estimate for an individual y, given Xp.

Where Xp is the given value of X.

Question 8

What is the picture formula used to derive?

Answer 8

The confidence interval estimate for the mean value of y, given Xp.

Where Xp is the given value of X.