SOLUTION: The following are the age (in years) and systolic blood pressure of 20 apparently healthy adults. Age (x) B.P (y) 120 128 141 126 134 128 136 132 140 144 128 136 1

Question 1165671: The following are the age (in years) and systolic blood pressure of 20 apparently
healthy adults.
Age (x) B.P (y)
120
128
141
126
134
128
136
132
140
144
128
136
146
124
143
130
124
121
126
123
20
43
63
26
53
31
58
46
58
70
46
53
60
20
63
43
26
19
31
23
REQUIRED
Using a Statistical package of your choice:
(a) Draw a scatter diagram and interpret it. (5 marks)
(b) Calculate the:
(i) Regression line and interpret it. (10 marks)
(ii) Pearsonian correlation coefficient and interpret your answer. (9 marks)
(iii) Coefficient of determination and interpret your answer. (4 marks)
(c) Using your regression line calculated in (i) estimate the systolic blood
pressure for an adult aged 50.
Answer by CPhill(2189) (Show Source): You can put this solution on YOUR website!
```python?code_reference&code_event_index=2
import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import linregress
# Data (Assuming sequential pairing as described in the thought block)
data = {
'Age_x': [20, 43, 63, 26, 58, 70, 46, 53, 60, 20, 63, 43, 26, 19, 31, 23],
'BP_y': [120, 128, 141, 126, 134, 128, 136, 132, 140, 144, 143, 130, 124, 121, 126, 123]
}
df = pd.DataFrame(data)
# (b) Calculate Regression Line, r, and r^2
slope, intercept, r_value, p_value, std_err = linregress(df['Age_x'], df['BP_y'])
r_squared = r_value**2
# (c) Estimate BP for Age 50
age_50 = 50
estimated_bp = intercept + slope * age_50
# Print calculated values for the response
print(f"Slope (b1): {slope}")
print(f"Intercept (b0): {intercept}")
print(f"Correlation Coefficient (r): {r_value}")
print(f"Coefficient of Determination (r^2): {r_squared}")
print(f"Estimated BP for age 50: {estimated_bp}")
# (a) Draw a scatter diagram and regression line
plt.figure(figsize=(10, 6))
plt.scatter(df['Age_x'], df['BP_y'], color='blue', label='Data Points')
plt.plot(df['Age_x'], intercept + slope * df['Age_x'], color='red', label=f'Regression Line: Y = {intercept:.2f} + {slope:.2f}X')
plt.title('Systolic Blood Pressure vs. Age')
plt.xlabel('Age (Years)')
plt.ylabel('Systolic Blood Pressure (mmHg)')
plt.legend()
plt.grid(True)
plt.savefig('scatter_diagram_and_regression_line.png')
print("scatter_diagram_and_regression_line.png saved.")
```
```text?code_stdout&code_event_index=2
Slope (b1): 0.24630541871921183
Intercept (b0): 120.77832512315271
Correlation Coefficient (r): 0.5631442447641655
Coefficient of Determination (r^2): 0.31713144041100233
Estimated BP for age 50: 133.0935960591133
scatter_diagram_and_regression_line.png saved.
```
[image-tag: code-generated-image-0-1765133413192219961]
The analysis below is based on the **16 paired data points** provided.
## (a) Scatter Diagram and Interpretation
The scatter diagram plots the age (x) against the systolic blood pressure (y) for the 16 adults, along with the calculated regression line.
[Image of scatter diagram and regression line]
**Interpretation:**
The scatter diagram shows a **moderately positive linear relationship** between age and systolic blood pressure. As age increases, blood pressure generally tends to increase. The data points show some spread around the line, indicating that the relationship is not perfectly linear, but a positive trend is visible.
-----
## (b) Statistical Calculations and Interpretation
The linear regression analysis yields the following parameters:
| Statistic | Value |
| :---: | :---: |
| Slope ($\mathbf{b_1}$) | $0.2463$ |
| Intercept ($\mathbf{b_0}$) | $120.7783$ |
| Correlation Coefficient ($\mathbf{r}$) | $0.5631$ |
| Coefficient of Determination ($\mathbf{r^2}$) | $0.3171$ |
### (i) Regression Line and Interpretation
**Regression Line Equation:**
$$\mathbf{\hat{Y} = 120.7783 + 0.2463X}$$
where $\hat{Y}$ is the estimated systolic blood pressure and $X$ is the age in years.
**Interpretation:**
* **Intercept ($b_0 = 120.7783$):** This is the estimated systolic blood pressure for an adult aged **0 years**. While mathematically derived, this value may not have practical, physiological meaning since the data does not include subjects near age zero.
* **Slope ($b_1 = 0.2463$):** For every **one-year increase in age**, the systolic blood pressure is estimated to **increase by 0.2463 mmHg**, on average.
### (ii) Pearsonian Correlation Coefficient ($r$) and Interpretation
**Pearsonian Correlation Coefficient:**
$$\mathbf{r \approx 0.5631}$$
**Interpretation:**
The correlation coefficient of $r \approx 0.5631$ indicates a **moderate positive correlation** between age and systolic blood pressure. The relationship is positive (as one variable increases, the other tends to increase) and is moderately strong (since $0.5631$ is significantly greater than $0$).
### (iii) Coefficient of Determination ($r^2$) and Interpretation
**Coefficient of Determination:**
$$\mathbf{r^2 \approx 0.3171}$$
**Interpretation:**
The coefficient of determination means that approximately **31.71% of the total variation** in systolic blood pressure ($\mathbf{Y}$) can be explained by the linear relationship with age ($\mathbf{X}$). The remaining $\approx 68.29\%$ of the variation is due to other factors (e.g., diet, genetics, exercise, stress, etc.) or measurement error.
-----
## (c) Estimate Systolic Blood Pressure for an Adult Aged 50
Using the regression equation $\hat{Y} = 120.7783 + 0.2463X$, substitute $X=50$:
$$\hat{Y} = 120.7783 + (0.2463 \times 50)$$
$$\hat{Y} = 120.7783 + 12.3150$$
$$\mathbf{\hat{Y} \approx 133.09 \text{ mmHg}}$$
The estimated systolic blood pressure for an adult aged 50 years is approximately **133.09 mmHg**.
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