Use the following results obtained from a simple linear regression analysis with 12 observations.
  

=37.2895-(1.2024)X

R2=.6744 sb1=.2934


Test to determine if there is a significant negative relationship between the independent and dependent variable at α= .01

To test for the significance of the relationship between the independent variable (X) and the dependent variable (Y) at a significance level (α) of 0.01, you can perform a hypothesis test using the t-test statistic.

Here are the steps to conduct the hypothesis test:

Step 1: State the null hypothesis (H0) and alternative hypothesis (Ha):
H0: β1 = 0 (There is no significant relationship between X and Y)
Ha: β1 ≠ 0 (There is a significant relationship between X and Y)

Step 2: Calculate the t-statistic:
The t-statistic can be calculated using the formula:
t = (β1 - 0) / sb1
where β1 is the coefficient of X obtained from the regression analysis and sb1 is the standard error of β1.

Given that β1 = -1.2024 and sb1 = 0.2934, we can substitute these values into the formula to calculate the t-statistic:
t = (-1.2024 - 0) / 0.2934

Step 3: Find the critical value:
Since the significance level (α) is given as 0.01, we need to find the critical value for a two-tailed test with α/2 = 0.005. You can refer to the t-distribution table or use statistical software to find the critical value.

Step 4: Compare the t-statistic with the critical value:
If the absolute value of the t-statistic is greater than the critical value, we reject the null hypothesis and conclude that there is a significant relationship between the independent and dependent variables.

I will help you provide the critical value from the t-distribution table, but I need to know the degrees of freedom (df). The degrees of freedom in simple linear regression are calculated as n-2, where n is the number of observations. In this case, n=12, so df = 12-2 = 10. Please give me a moment while I find the critical value.

To determine if there is a significant negative relationship between the independent (X) and dependent (Y) variable, we can perform a hypothesis test using the given information.

H0: The slope coefficient (β1) is equal to zero (no relationship)
H1: The slope coefficient (β1) is less than zero (negative relationship)

To test the hypothesis, we can use the t-test statistic, which can be calculated using the following formula:

t = (b1 - β1) / sb1

where:
b1 = the estimated slope coefficient from the regression analysis (-1.2024 in this case)
sb1 = the standard error of the slope coefficient (0.2934 in this case)

Substituting the values into the formula:

t = (-1.2024 - 0) / 0.2934
t = -1.2024 / 0.2934
t ≈ -4.101

Using the t-distribution table or a statistical software, we can find the critical t-value at a significance level of α = 0.01 for a one-tailed test (since we are testing for a negative relationship). The degrees of freedom for this test is (n - 2), where n is the number of observations (12 in this case).

For α = 0.01 and degrees of freedom (df) = (12 - 2) = 10, the critical t-value is approximately -2.764.

Since the calculated t-value (-4.101) is more extreme (further in the rejection region) than the critical t-value (-2.764) in the left tail, we can reject the null hypothesis (H0).

Hence, there is sufficient evidence to suggest that there is a significant negative relationship between the independent and dependent variable at α = 0.01 level.