In a regression analysis involving 27 observations, the following estimated regression equation was developed. ŷ = 25.2 + 5.5x, For this estimated regression equation SST = 1,600 and SSE = 510. (a) At a = 0.05, test whether x, is significant. State the null and alternative hypotheses. O Ho: Bo = 0 H: Bo # 0 O Ho: B1 # 0 H: B1 = 0 O Ho: Bo#0 H,: Bo = 0 O Hoi B1 = 0 H: B # 0 %3D %3D Find the value of the test statistic. (Round your answer to two decimal places.) F = Find the p-value. (Round your answer to three decimal places.) p-value = Is x, significant? O Reject Ho. We conclude that x, is significant. Reject Ho. We conclude that x, is not significant. Do not reject Ho. We conclude that x, is significant. Do not reject Ho. We conclude that x, is not significant.

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Chapter1: Starting With Matlab
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In a regression analysis involving 27 observations, the following estimated regression equation was developed:

\[
\hat{y} = 25.2 + 5.5x_1
\]

For this estimated regression equation, SST \( = 1,600 \) and SSE \( = 510 \).

### (a) At \( \alpha = 0.05 \), test whether \( x_1 \) is significant.

**State the null and alternative hypotheses.**

- \( H_0: \beta_1 = 0 \)  
  \( H_a: \beta_1 \neq 0 \)  

**Find the value of the test statistic.** (Round your answer to two decimal places.)

\[ F = \underline{\hspace{3cm}} \]

**Find the \( p \)-value.** (Round your answer to three decimal places.)

\[ p\text{-value} = \underline{\hspace{3cm}} \]

**Is \( x_1 \) significant?**

- ○ Reject \( H_0 \). We conclude that \( x_1 \) is significant.
- ○ Reject \( H_0 \). We conclude that \( x_1 \) is not significant.
- ○ Do not reject \( H_0 \). We conclude that \( x_1 \) is significant.
- ○ Do not reject \( H_0 \). We conclude that \( x_1 \) is not significant.

Suppose that variables \( x_2 \) and \( x_3 \) are added to the model, and the following regression equation is obtained:

\[
\hat{y} = 16.3 + 2.3x_1 + 12.1x_2 - 5.8x_3
\]

For this estimated regression equation SST \( = 1,600 \) and SSE \( = 100 \).
Transcribed Image Text:In a regression analysis involving 27 observations, the following estimated regression equation was developed: \[ \hat{y} = 25.2 + 5.5x_1 \] For this estimated regression equation, SST \( = 1,600 \) and SSE \( = 510 \). ### (a) At \( \alpha = 0.05 \), test whether \( x_1 \) is significant. **State the null and alternative hypotheses.** - \( H_0: \beta_1 = 0 \) \( H_a: \beta_1 \neq 0 \) **Find the value of the test statistic.** (Round your answer to two decimal places.) \[ F = \underline{\hspace{3cm}} \] **Find the \( p \)-value.** (Round your answer to three decimal places.) \[ p\text{-value} = \underline{\hspace{3cm}} \] **Is \( x_1 \) significant?** - ○ Reject \( H_0 \). We conclude that \( x_1 \) is significant. - ○ Reject \( H_0 \). We conclude that \( x_1 \) is not significant. - ○ Do not reject \( H_0 \). We conclude that \( x_1 \) is significant. - ○ Do not reject \( H_0 \). We conclude that \( x_1 \) is not significant. Suppose that variables \( x_2 \) and \( x_3 \) are added to the model, and the following regression equation is obtained: \[ \hat{y} = 16.3 + 2.3x_1 + 12.1x_2 - 5.8x_3 \] For this estimated regression equation SST \( = 1,600 \) and SSE \( = 100 \).
Use an F test and a 0.05 level of significance to determine whether \( x_2 \) and \( x_3 \) contribute significantly to the model.

State the null and alternative hypotheses.

- \( H_0: \beta_1 = 0 \)  
  \( H_a: \beta_1 \neq 0 \)

- \( H_0: \beta_1 \neq 0 \)  
  \( H_a: \beta_1 = 0 \)

- \( H_0: \) One or more of the parameters is not equal to zero.  
  \( H_a: \beta_2 = \beta_3 = 0 \)

- \( H_0: \beta_2 = \beta_3 = 0 \)  
  \( H_a: \) One or more of the parameters is not equal to zero.

Find the value of the test statistic.  
\[ \text{Value} = \_\_\_\_\_\_\_\_ \]

Find the \( p \)-value. (Round your answer to three decimal places.)  
\[ p\text{-value} = \_\_\_\_\_ \]

Is the addition of \( x_2 \) and \( x_3 \) significant?

- Do not reject \( H_0 \). We conclude that the addition of variables \( x_2 \) and \( x_3 \) is significant.

- Reject \( H_0 \). We conclude that the addition of variables \( x_2 \) and \( x_3 \) is not significant.

- Reject \( H_0 \). We conclude that the addition of variables \( x_2 \) and \( x_3 \) is significant.

- Do not reject \( H_0 \). We conclude that the addition of variables \( x_2 \) and \( x_3 \) is not significant.
Transcribed Image Text:Use an F test and a 0.05 level of significance to determine whether \( x_2 \) and \( x_3 \) contribute significantly to the model. State the null and alternative hypotheses. - \( H_0: \beta_1 = 0 \) \( H_a: \beta_1 \neq 0 \) - \( H_0: \beta_1 \neq 0 \) \( H_a: \beta_1 = 0 \) - \( H_0: \) One or more of the parameters is not equal to zero. \( H_a: \beta_2 = \beta_3 = 0 \) - \( H_0: \beta_2 = \beta_3 = 0 \) \( H_a: \) One or more of the parameters is not equal to zero. Find the value of the test statistic. \[ \text{Value} = \_\_\_\_\_\_\_\_ \] Find the \( p \)-value. (Round your answer to three decimal places.) \[ p\text{-value} = \_\_\_\_\_ \] Is the addition of \( x_2 \) and \( x_3 \) significant? - Do not reject \( H_0 \). We conclude that the addition of variables \( x_2 \) and \( x_3 \) is significant. - Reject \( H_0 \). We conclude that the addition of variables \( x_2 \) and \( x_3 \) is not significant. - Reject \( H_0 \). We conclude that the addition of variables \( x_2 \) and \( x_3 \) is significant. - Do not reject \( H_0 \). We conclude that the addition of variables \( x_2 \) and \( x_3 \) is not significant.
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