2. The raw material used in the production of a certain synthetic fiber is stored in a location without a humidity control. Measurements of relative humidity in the storage location and the moisture content of a sample of the raw material were taken over 15 days with the following data (in percentages) resulting. relative humidity 46 53 29 61 36 39 47 49 52 38 55 32 57 54 44 Moisture content 12 15 7 17 10 11 11 12 14 9 16 8 18 14 12 Estimate regression line for above given data. ANS: regression line is Y = −2.51 + 0.32x

To estimate the regression line for the given data, we need to compute the line of best fit for the relationship between relative humidity \( x \) and moisture content \( y \).

Let's walk through the process of finding the regression line \( Y = a + bx \), where:
- \( a \) is the y-intercept,
- \( b \) is the slope of the line.

### Steps to Calculate the Regression Line:
1. **Calculate the Mean of \( x \) and \( y \):**
   - Let \( \bar{x} \) be the mean of relative humidity values.
   - Let \( \bar{y} \) be the mean of moisture content values.

2. **Calculate the Slope \( b \):**
   The formula for the slope \( b \) is:
   \[
   b = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}
   \]

3. **Calculate the Y-Intercept \( a \):**
   Using the formula:
   \[
   a = \bar{y} - b\bar{x}
   \]

After going through these calculations with the provided data, the regression line comes out to be:
\[
Y = -2.51 + 0.32x
\]
This means that for each 1% increase in relative humidity, the moisture content in the raw material increases by approximately 0.32%.

Would you like to see the detailed calculations for each step?

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### Related Questions

1. What is the significance of the slope in this regression line?
2. How does humidity control affect the moisture content in synthetic fiber production?
3. What other environmental factors might impact moisture content besides humidity?
4. How can we predict moisture content if the humidity level is outside the range of our data?
5. Could a non-linear regression model be more appropriate for this data?

### Tip
Always check if your data points meet the assumptions of linear regression, especially linearity and independence, before relying heavily on a linear model.

The raw material used in the production of a certain synthetic fiber is stored in a location without a humidity control. Measurements of relative humidity in the storage location and the moisture content of a sample of the raw material were taken over 15 days with the following data (in percentages) resulting. relative humidity 46 53 29 61 36 39 47 49 52 38 55 32 57 54 44 Moisture content 12 15 7 17 10 11 11 12 14 9 16 8 18 14 12 Estimate regression line for above given data. ANS: regression line is Y = −2.51 + 0.32x

Learn how to estimate the regression line for the relationship between relative humidity and moisture content in raw materials used in synthetic fiber production. The regression line for the given data is Y = −2.51 + 0.32x. This problem involves key concepts in linear regression and statistics. Suitable for students in Grades 9-12.

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