C is for cabbage ~ A researcher wants to know if the head weight of the cabbage is a good predictor for ascorbic acid (vitamin C) content. From a random sample of 60 cabbages, the researcher recorded the ascorbic acid content in milligrams and the weight of the cabbage head in pounds. Head weight is the X variable and Ascorbic acid content is the Y variable in this scenario. R output Coefficients: Std. Error t value 25.052 -6.689 Estimate | (Intercept) 77.574 HeadWeight -7.567 Pr(>Itl) < 2e-16 *** 9.75e-09 *** 3.096 1.131 --- Residual standard error: 7.668 on 58 degrees of freedom Multiple R-squared: 0.4355, Adjusted R-squared: 0.4257 F-statistic: 44.74 on 1 and 58 DF, p-value: 9.753e-09 What is the upper bound for a 90% confidence interval for the true slope B1 of the regression line? Give your answer to 4 decimal places.

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### C is for Cabbage

A researcher aims to determine whether the head weight of cabbage is a good predictor of its ascorbic acid (vitamin C) content. Using a random sample of 60 cabbages, the researcher recorded the ascorbic acid content in milligrams and the head weight in pounds. 

In this scenario:
- **Head weight** is the **X variable** 
- **Ascorbic acid content** is the **Y variable**

#### R Output

| Coefficients | Estimate | Std. Error | t value | Pr(>|t|) |
|--------------|----------|------------|---------|---------|
| (Intercept)  | 77.574   | 3.096      | 25.052  | <2e-16 *** |
| HeadWeight   | -7.567   | 1.131      | -6.689  | 9.75e-09 *** |

Additional Information:
- **Residual standard error**: 7.668 on 58 degrees of freedom
- **Multiple R-squared**: 0.4355
- **Adjusted R-squared**: 0.4257
- **F-statistic**: 44.74 on 1 and 58 DF
- **p-value**: 9.753e-09

### Question

What is the upper bound for a 90% confidence interval for the true slope \(\beta_1\) of the regression line? Give your answer to 4 decimal places.

**Your Answer:**

*Answer box for users to input their response*

#### Explanation of the R Output:
- **Estimate**: This is the estimated value of the coefficients in the regression equation.
  - The intercept is 77.574, meaning this is the estimated ascorbic acid content when the head weight is zero.
  - The coefficient for head weight is -7.567, indicating that for every one-pound increase in cabbage head weight, the ascorbic acid content decreases by 7.567 milligrams.
  
- **Std. Error**: The standard error of the coefficient estimates.
  
- **t value**: The t-statistic for the hypothesis test that the coefficient is different from zero.
  
- **Pr(>|t|)**: The p-value for the hypothesis test. A value less than 0.05 typically indicates that the coefficient is significantly different from zero
Transcribed Image Text:### C is for Cabbage A researcher aims to determine whether the head weight of cabbage is a good predictor of its ascorbic acid (vitamin C) content. Using a random sample of 60 cabbages, the researcher recorded the ascorbic acid content in milligrams and the head weight in pounds. In this scenario: - **Head weight** is the **X variable** - **Ascorbic acid content** is the **Y variable** #### R Output | Coefficients | Estimate | Std. Error | t value | Pr(>|t|) | |--------------|----------|------------|---------|---------| | (Intercept) | 77.574 | 3.096 | 25.052 | <2e-16 *** | | HeadWeight | -7.567 | 1.131 | -6.689 | 9.75e-09 *** | Additional Information: - **Residual standard error**: 7.668 on 58 degrees of freedom - **Multiple R-squared**: 0.4355 - **Adjusted R-squared**: 0.4257 - **F-statistic**: 44.74 on 1 and 58 DF - **p-value**: 9.753e-09 ### Question What is the upper bound for a 90% confidence interval for the true slope \(\beta_1\) of the regression line? Give your answer to 4 decimal places. **Your Answer:** *Answer box for users to input their response* #### Explanation of the R Output: - **Estimate**: This is the estimated value of the coefficients in the regression equation. - The intercept is 77.574, meaning this is the estimated ascorbic acid content when the head weight is zero. - The coefficient for head weight is -7.567, indicating that for every one-pound increase in cabbage head weight, the ascorbic acid content decreases by 7.567 milligrams. - **Std. Error**: The standard error of the coefficient estimates. - **t value**: The t-statistic for the hypothesis test that the coefficient is different from zero. - **Pr(>|t|)**: The p-value for the hypothesis test. A value less than 0.05 typically indicates that the coefficient is significantly different from zero
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