Find y. y= Y₁ 19 27 15 34 29
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
Related questions
Question
![We are conducting a hypothesis test to determine if there is a significant relationship between \( x \) and \( y \). We will use the \( F \) test statistic. The hypotheses to be tested are:
\[
H_0: \beta_1 = 0
\]
\[
H_a: \beta_1 \neq 0
\]
The \( F \) test statistic is calculated using the mean square due to regression (MSR) and the mean square due to error (MSE). The MSE is calculated using the SSE (sum of squares due to error) and \( n \), the number of observations:
\[
F = \frac{MSR}{MSE} \quad \text{where} \quad MSE = \frac{SSE}{n-2}
\]
Combining these into a single formula, we have:
\[
F = \frac{MSR}{\frac{SSE}{n-2}}
\]
We previously found SSE = 177.3 and \( n = 5 \). To calculate MSR, which is the sum of squares due to regression (SSR) divided by the number of independent variables, note that there is only one independent variable \( x \), so MSR = SSR. The formula to find SSR is:
\[
SSR = \sum (\hat{y}_j - \overline{y})^2
\]
where \( \hat{y}_j \) is the predicted value of the dependent variable for the \( i \)th observation and \( \overline{y} \) is the mean of the observed dependent variables.
The given data follows:
\[
\begin{array}{|c|c|c|c|c|c|}
\hline
x_j & 4 & 5 & 12 & 17 & 22 \\
\hline
y_i & 19 & 27 & 15 & 34 & 29 \\
\hline
\end{array}
\]
Find \( \overline{y} \):
\[
\overline{y} = \text{Insert Answer Here}
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe2fdea76-24b5-4e92-ab93-0d275ab88ec3%2F030f74f0-3fe6-4893-99cd-5a9f6880d37a%2Fqa1s9r_processed.png&w=3840&q=75)
Transcribed Image Text:We are conducting a hypothesis test to determine if there is a significant relationship between \( x \) and \( y \). We will use the \( F \) test statistic. The hypotheses to be tested are:
\[
H_0: \beta_1 = 0
\]
\[
H_a: \beta_1 \neq 0
\]
The \( F \) test statistic is calculated using the mean square due to regression (MSR) and the mean square due to error (MSE). The MSE is calculated using the SSE (sum of squares due to error) and \( n \), the number of observations:
\[
F = \frac{MSR}{MSE} \quad \text{where} \quad MSE = \frac{SSE}{n-2}
\]
Combining these into a single formula, we have:
\[
F = \frac{MSR}{\frac{SSE}{n-2}}
\]
We previously found SSE = 177.3 and \( n = 5 \). To calculate MSR, which is the sum of squares due to regression (SSR) divided by the number of independent variables, note that there is only one independent variable \( x \), so MSR = SSR. The formula to find SSR is:
\[
SSR = \sum (\hat{y}_j - \overline{y})^2
\]
where \( \hat{y}_j \) is the predicted value of the dependent variable for the \( i \)th observation and \( \overline{y} \) is the mean of the observed dependent variables.
The given data follows:
\[
\begin{array}{|c|c|c|c|c|c|}
\hline
x_j & 4 & 5 & 12 & 17 & 22 \\
\hline
y_i & 19 & 27 & 15 & 34 & 29 \\
\hline
\end{array}
\]
Find \( \overline{y} \):
\[
\overline{y} = \text{Insert Answer Here}
\]
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