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use the picture above of the critical values for the pearson correlation coefficient to answer the question. find the regression equation, letting the systolic reading be the independent (x) variable. if one of these people has a systolic blood pressure of 137 mm Hg. What is the best predicted diastolic blood pressure? Round regression equation to two decimal places as needed. Round best predicted diastolic blood pressure to one decimal place as needed. The systolic blood pressure data in respective order is as follows : 122, 134, 143, 140, 135, 113, 110, 145. the diastolic blood pressure data in respective order is as follows : 62, 71, 80, 80, 100, 59, 82, 105.

To find the regression equation and the best predicted diastolic blood pressure given the systolic data, we will go through these steps:

### Step 1: Compute the correlation coefficient \( r \)
We will calculate the correlation coefficient \( r \) between systolic and diastolic blood pressure. Using Pearson's correlation formula:

\[
r = \frac{n\sum xy - (\sum x)(\sum y)}{\sqrt{[n\sum x^2 - (\sum x)^2][n\sum y^2 - (\sum y)^2]}}
\]

Where:
- \( n \) is the number of data points,
- \( x \) is the systolic blood pressure values, and
- \( y \) is the diastolic blood pressure values.

The systolic and diastolic values are:
- \( x \): 122, 134, 143, 140, 135, 113, 110, 145
- \( y \): 62, 71, 80, 80, 100, 59, 82, 105

### Step 2: Calculate the slope and intercept of the regression line
The regression line is represented as:
\[
y = a + bx
\]
Where:
- \( b = \frac{n\sum xy - (\sum x)(\sum y)}{n\sum x^2 - (\sum x)^2} \) is the slope, and
- \( a = \bar{y} - b\bar{x} \) is the intercept.

### Step 3: Predict the diastolic pressure for a systolic pressure of 137 mm Hg
Once the regression equation is determined, substitute \( x = 137 \) into the equation to predict the diastolic pressure.

Let's proceed by calculating each of these values.### Step 1: Pearson's Correlation Coefficient
The calculated correlation coefficient \( r = 0.579 \), indicating a moderate positive correlation between systolic and diastolic blood pressure.

### Step 2: Regression Equation
The regression equation, rounded to two decimal places, is:
\[
y = -11.20 + 0.70x
\]
Where \( x \) is the systolic blood pressure, and \( y \) is the predicted diastolic blood pressure.

### Step 3: Predicted Diastolic Pressure for a Systolic Pressure of 137 mm Hg
Using the equation \( y = -11.20 + 0.70(137) \), the best predicted diastolic blood pressure is approximately:
\[
y = 84.6 \, \text{mm Hg}
\]

Would you like further details on any of these calculations, or do you have additional questions?

Here are some related questions to consider:
1. How does changing the systolic value affect the diastolic prediction?
2. What are the implications of a correlation coefficient close to 1 or -1?
3. What other methods can be used to predict diastolic pressure based on systolic pressure?
4. How can the accuracy of this regression model be evaluated?
5. What are the limitations of using linear regression for this dataset?

**Tip:** In regression analysis, checking the residuals (differences between actual and predicted values) helps to evaluate the model's accuracy.

Use the picture above of the critical values for the Pearson correlation coefficient to answer the question. Find the regression equation, letting the systolic reading be the independent (x) variable. If one of these people has a systolic blood pressure of 137 mm Hg, what is the best predicted diastolic blood pressure? Round the regression equation to two decimal places as needed. Round best predicted diastolic blood pressure to one decimal place as needed. The systolic blood pressure data in respective order is as follows: 122, 134, 143, 140, 135, 113, 110, 145. The diastolic blood pressure data in respective order is as follows: 62, 71, 80, 80, 100, 59, 82, 105.

Learn how to find the regression equation using systolic and diastolic blood pressure data and predict a diastolic pressure for a systolic value of 137 mm Hg. This detailed solution covers key statistical concepts such as Pearson's correlation and linear regression. Suitable for college-level statistics students.

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