SOLUTION: Raw material used in the production of synthetic fiber is stored in a place which has no humidity control. Measurements of the relative humidity in the storage place and the moistu

Question 1162465: Raw material used in the production of synthetic fiber is stored in a place which has no humidity control. Measurements of the relative humidity in the storage place and the moisture content of a sample of the raw material (both in percentages) on 12 days yielded the following results:
Humidity x 42 C5 50 4C 48 62 C1 C6 44 C9 55 48
Moixture content y 12 8 14 9 11 16 7 9 12 10 1C 11
(i) Make a scatter plot to verify that it is reasonable to assume that the regression of y on x is linear (ii) Fit a straight line by the method of least squares, (iii) Estimate the moisture content when humidity is 45%. (iv) Determine the correlation coefficient.
Answer by CPhill(2264) (Show Source): You can put this solution on YOUR website!
```python?code_reference&code_event_index=2
import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import linregress
# Data with C interpreted as 3
x = np.array([42, 35, 50, 43, 48, 62, 31, 36, 44, 39, 55, 48])
y = np.array([12, 8, 14, 9, 11, 16, 7, 9, 12, 10, 13, 11])
# (i) Scatter plot
plt.scatter(x, y, color='blue', label='Data Points')
plt.xlabel('Relative Humidity x (%)')
plt.ylabel('Moisture Content y (%)')
plt.title('Scatter Plot of Humidity vs. Moisture Content')
# (ii) Least Squares Fit
slope, intercept, r_value, p_value, std_err = linregress(x, y)
line = slope * x + intercept
plt.plot(x, line, color='red', label=f'Fit: y={intercept:.2f} + {slope:.2f}x')
plt.legend()
plt.grid(True)
plt.savefig('scatter_plot.png')
# (iii) Estimate at x = 45%
x_est = 45
y_est = intercept + slope * x_est
# (iv) Correlation coefficient
correlation = r_value
print(f"Slope (b): {slope}")
print(f"Intercept (a): {intercept}")
print(f"Estimate for x=45: {y_est}")
print(f"Correlation coefficient (r): {correlation}")

```
```text?code_stdout&code_event_index=2
Slope (b): 0.26903206940247587
Intercept (a): -0.9495077492933035
Estimate for x=45: 11.156935373818111
Correlation coefficient (r): 0.9144291969589217
```
[image-tag: code-generated-image-0-1776522998299002780]
To analyze the relationship between relative humidity ($x$) and moisture content ($y$), we first interpreted the characters in the data (where 'C' appeared to represent '3').
**The clean dataset used for analysis:**
* **Humidity ($x$):** 42, 35, 50, 43, 48, 62, 31, 36, 44, 39, 55, 48
* **Moisture ($y$):** 12, 8, 14, 9, 11, 16, 7, 9, 12, 10, 13, 11
---
### (i) Scatter Plot
The scatter plot shows a strong, positive linear trend. As relative humidity increases, the moisture content of the raw material also tends to increase. The points cluster closely around a straight line, making the assumption of a linear regression model very reasonable.
### (ii) Least Squares Regression Line
Using the method of least squares, we calculated the regression parameters:
* **Slope ($b$):** $0.2690$
* **Intercept ($a$):** $-0.9495$
The equation of the fitted straight line is:
$$\hat{y} = -0.9495 + 0.2690x$$
### (iii) Estimate for 45% Humidity
To estimate the moisture content when the relative humidity is 45%, we substitute $x = 45$ into our regression equation:
$$\hat{y} = -0.9495 + 0.2690(45)$$
$$\hat{y} = -0.9495 + 12.105$$
$$\hat{y} \approx 11.16\%$$
**Estimated moisture content:** **11.16%**
### (iv) Correlation Coefficient ($r$)
The correlation coefficient measures the strength and direction of the linear relationship.
* **$r \approx 0.9144$**
Since $r$ is very close to $1$, it indicates a **very strong positive linear correlation** between humidity and moisture content.
RELATED QUESTIONS
Raw material used in the production of synthetic fiber is stored in a place which has no (answered by Boreal)

) Raw material used in the production of synthetic fiber is stored in a place which has... (answered by ikleyn)

Raw material used in the production of synthetic fiber is stored in a place which has no (answered by ikleyn)

) Raw material used in the production of synthetic fiber is stored in a place which has (answered by Edwin McCravy,ikleyn)

Raw material used in the production of synthetic fiber is stored in a place which has no (answered by Boreal)

The control department of a light bulb manufacturer randomly picks 4400 light bulbs from... (answered by stanbon)

The control department of a light bulb manufacturer randomly picks 4400 light bulbs from (answered by Boreal)

An aerospace parts factory has two separate production lines making fasteners; each... (answered by stanbon)