We are to conduct a hypothesis test to determine if there is a significant relationship between x and y, but now we will use the F test statistic. Again, we will test the following hypotheses. Ho: P₁ = 0 Ha: B₁ * 0 The F test statistic is calculated as follows using the mean square due to regression, MSR, and the mean square due to error, MSE. Recall the MSE is using SSE, the sum of squares due to error, and n, the number of observations. F = MSR MSE where MSE = SSE n-2 Combining these into a single formula, we have the following. F = MSR SSE n-2 We previously found SSE = 141.3 and n = 5, so we just need to calculate MSR. This will be the sum of squares due to regression, SSR, divided by the number of independent variables. For this test, there is only one independent variable, x, so MSR = SSR. Recall the formula to find SSR where ŷ, is the predicted value of the dependent variable for the ith observation and y is the mean of the observed dependent variables. SSR = - Σ(9₁-7) ² The given data follow. X; 4 5 12 17 22 Y₁ 19 27 17 34 29 Find y. y =
We are to conduct a hypothesis test to determine if there is a significant relationship between x and y, but now we will use the F test statistic. Again, we will test the following hypotheses. Ho: P₁ = 0 Ha: B₁ * 0 The F test statistic is calculated as follows using the mean square due to regression, MSR, and the mean square due to error, MSE. Recall the MSE is using SSE, the sum of squares due to error, and n, the number of observations. F = MSR MSE where MSE = SSE n-2 Combining these into a single formula, we have the following. F = MSR SSE n-2 We previously found SSE = 141.3 and n = 5, so we just need to calculate MSR. This will be the sum of squares due to regression, SSR, divided by the number of independent variables. For this test, there is only one independent variable, x, so MSR = SSR. Recall the formula to find SSR where ŷ, is the predicted value of the dependent variable for the ith observation and y is the mean of the observed dependent variables. SSR = - Σ(9₁-7) ² The given data follow. X; 4 5 12 17 22 Y₁ 19 27 17 34 29 Find y. y =
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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![### Hypothesis Testing Using the F Test
We are to conduct a hypothesis test to determine if there is a significant relationship between \( x \) and \( y \). We will use the F test statistic to perform this test. The hypotheses are stated as follows:
\[
H_0 : \beta_1 = 0
\]
\[
H_a : \beta_1 \neq 0
\]
The F test statistic is computed using the mean square due to regression (MSR) and the mean square due to error (MSE). Recall that MSE is derived from SSE, the sum of squares due to error, and \( n \), the number of observations:
\[
F = \frac{\text{MSR}}{\text{MSE}} \quad \text{where} \quad \text{MSE} = \frac{\text{SSE}}{n-2}
\]
Combining these into a single formula, we get:
\[
F = \frac{\text{MSR}}{\frac{\text{SSE}}{n-2}}
\]
We previously found SSE = 141.3 and \( n = 5 \). Therefore, we need to calculate MSR. This will be the sum of squares due to regression (SSR) divided by the number of independent variables. For this test, there is only one independent variable, \( x \), so:
\[
\text{MSR} = \text{SSR}
\]
To calculate SSR, we use the formula:
\[
\text{SSR} = \sum (\hat{y_i} - \bar{y})^2
\]
where \( \hat{y_i} \) is the predicted value of the dependent variable for the \( i \)th observation, and \( \bar{y} \) is the mean of the observed dependent variables.
#### Given Data:
\[
\begin{array}{c|ccccc}
x_i & 4 & 5 & 12 & 17 & 22 \\ \hline
y_i & 19 & 27 & 17 & 34 & 29
\end{array}
\]
**Find \( \bar{y} \):**
\[ \bar{y} = \boxed{} \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F98fd5880-bd7f-4b90-9f43-35abbd9a1544%2F130a4021-9b47-46f3-92a0-a9060ed03798%2F1mi4gwf_processed.png&w=3840&q=75)
Transcribed Image Text:### Hypothesis Testing Using the F Test
We are to conduct a hypothesis test to determine if there is a significant relationship between \( x \) and \( y \). We will use the F test statistic to perform this test. The hypotheses are stated as follows:
\[
H_0 : \beta_1 = 0
\]
\[
H_a : \beta_1 \neq 0
\]
The F test statistic is computed using the mean square due to regression (MSR) and the mean square due to error (MSE). Recall that MSE is derived from SSE, the sum of squares due to error, and \( n \), the number of observations:
\[
F = \frac{\text{MSR}}{\text{MSE}} \quad \text{where} \quad \text{MSE} = \frac{\text{SSE}}{n-2}
\]
Combining these into a single formula, we get:
\[
F = \frac{\text{MSR}}{\frac{\text{SSE}}{n-2}}
\]
We previously found SSE = 141.3 and \( n = 5 \). Therefore, we need to calculate MSR. This will be the sum of squares due to regression (SSR) divided by the number of independent variables. For this test, there is only one independent variable, \( x \), so:
\[
\text{MSR} = \text{SSR}
\]
To calculate SSR, we use the formula:
\[
\text{SSR} = \sum (\hat{y_i} - \bar{y})^2
\]
where \( \hat{y_i} \) is the predicted value of the dependent variable for the \( i \)th observation, and \( \bar{y} \) is the mean of the observed dependent variables.
#### Given Data:
\[
\begin{array}{c|ccccc}
x_i & 4 & 5 & 12 & 17 & 22 \\ \hline
y_i & 19 & 27 & 17 & 34 & 29
\end{array}
\]
**Find \( \bar{y} \):**
\[ \bar{y} = \boxed{} \]
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