MATLAB: An Introduction with Applications
6th Edition
ISBN: 9781119256830
Author: Amos Gilat
Publisher: John Wiley & Sons Inc
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![## Statistical Calculations and Hypothesis Testing
### Consider the following data:
\[
\begin{array}{|c|c|c|c|c|c|}
\hline
x_i & 3 & 12 & 6 & 20 & 14 \\
\hline
y_i & 55 & 40 & 55 & 5 & 15 \\
\hline
\end{array}
\]
### (a) Mean Square Error (MSE)
Compute the mean square error using the equation:
\[ s^2 = \text{MSE} = \frac{\text{SSE}}{n - 2} \]
- Result: **72.92**
### (b) Standard Error of the Estimate
Compute the standard error of the estimate using the equation:
\[ s = \sqrt{\text{MSE}} = \sqrt{\frac{\text{SSE}}{n - 2}} \]
- Result: **8.539**
### (c) Estimated Standard Deviation of \( b_1 \)
Compute the estimated standard deviation of \( b_1 \) using the equation:
\[ s_{b_1} = \frac{s}{\sqrt{\sum(x_i - \bar{x})^2}} \]
- Result: **0.636**
### (d) Hypothesis Testing
Use the t-test to evaluate the following hypotheses (\(\alpha = 0.05\)):
- \( H_0: \beta_1 = 0 \)
- \( H_a: \beta_1 \neq 0 \)
#### Test Statistic
- Value: **-5.106**
#### p-value
- Result: **0.0145**
#### Conclusion
- **Reject \( H_0 \). We conclude that the relationship between x and y is significant.**](https://content.bartleby.com/qna-images/question/f7b76bbb-1122-4c75-9c9e-2038b7c47d4e/a3298a90-94ab-4279-a1fe-47a52036f8df/hcz3gfj_processed.png)
Transcribed Image Text:## Statistical Calculations and Hypothesis Testing
### Consider the following data:
\[
\begin{array}{|c|c|c|c|c|c|}
\hline
x_i & 3 & 12 & 6 & 20 & 14 \\
\hline
y_i & 55 & 40 & 55 & 5 & 15 \\
\hline
\end{array}
\]
### (a) Mean Square Error (MSE)
Compute the mean square error using the equation:
\[ s^2 = \text{MSE} = \frac{\text{SSE}}{n - 2} \]
- Result: **72.92**
### (b) Standard Error of the Estimate
Compute the standard error of the estimate using the equation:
\[ s = \sqrt{\text{MSE}} = \sqrt{\frac{\text{SSE}}{n - 2}} \]
- Result: **8.539**
### (c) Estimated Standard Deviation of \( b_1 \)
Compute the estimated standard deviation of \( b_1 \) using the equation:
\[ s_{b_1} = \frac{s}{\sqrt{\sum(x_i - \bar{x})^2}} \]
- Result: **0.636**
### (d) Hypothesis Testing
Use the t-test to evaluate the following hypotheses (\(\alpha = 0.05\)):
- \( H_0: \beta_1 = 0 \)
- \( H_a: \beta_1 \neq 0 \)
#### Test Statistic
- Value: **-5.106**
#### p-value
- Result: **0.0145**
#### Conclusion
- **Reject \( H_0 \). We conclude that the relationship between x and y is significant.**
Transcribed Image Text:**ANOVA Table for Hypothesis Testing**
**Objective**: Use the F test to assess the hypotheses at a 0.05 level of significance. This involves an analysis of variance (ANOVA) table.
### ANOVA Table
| Source of Variation | Sum of Squares | Degrees of Freedom | Mean Square | F | p-value |
|---------------------|----------------|--------------------|-------------|------|---------|
| Regression | 1901.25 | 1 | 1901.25 | 26.07| 0.015 |
| Error | 218.75 | 3 | 72.92 | | |
| Total | 2120 | 4 | | | |
### Calculations
- **Test Statistic**: \( F = 26.07 \)
- **p-value**: \( p = 0.015 \)
### Conclusion Options
1. Do not reject \( H_0 \). We cannot conclude that the relationship between \( x \) and \( y \) is significant.
2. Reject \( H_0 \). We cannot conclude that the relationship between \( x \) and \( y \) is significant.
3. Do not reject \( H_0 \). We conclude that the relationship between \( x \) and \( y \) is significant.
4. **Reject \( H_0 \). We conclude that the relationship between \( x \) and \( y \) is significant.**
**Selected Conclusion**: Option 4 is chosen, indicating that there is a significant relationship between \( x \) and \( y \) since the p-value (0.015) is less than the significance level (0.05).
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