The following table shows students’ test scores on the first two tests in an introductory chemistry class.
 
 Chemistry Test Scores
First test, x	93
45
73
90
58
89
85
45
64
63
87
79
Second test, y	76
56
64
74
56
81
77
39
65
66
72
74
 
Step 1 of 2 :  Find an equation of the least-squares regression line. Round your answer to three decimal places, if necessary.


To find the equation of the least-squares regression line, we use the formula for the line:

\[
y = mx + b
\]

Where:
- \(m\) is the slope, calculated as:
\[
m = \frac{\text{Cov}(x, y)}{\text{Var}(x)} = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}
\]
- \(b\) is the y-intercept, calculated as:
\[
b = \bar{y} - m\bar{x}
\]

Let's calculate this step-by-step using the given data:

### 1. Calculate the means \(\bar{x}\) and \(\bar{y}\)
\[
\bar{x} = \frac{\sum x_i}{n}, \quad \bar{y} = \frac{\sum y_i}{n}
\]

### 2. Calculate \((x_i - \bar{x})(y_i - \bar{y})\) and \((x_i - \bar{x})^2\) for all data points, and sum them up.

### 3. Use the formulas for \(m\) and \(b\).

I’ll calculate this now.import numpy as np
import pandas as pd

# Data from the problem
x = np.array([93, 45, 73, 90, 58, 89, 85, 45, 64, 63, 87, 79])  # First test scores
y = np.array([76, 56, 64, 74, 56, 81, 77, 39, 65, 66, 72, 74])  # Second test scores

# Step 1: Calculate means of x and y
x_mean = np.mean(x)
y_mean = np.mean(y)

# Step 2: Calculate covariance and variance
cov_xy = np.sum((x - x_mean) * (y - y_mean))
var_x = np.sum((x - x_mean)**2)

# Step 3: Calculate slope (m) and intercept (b)
m = cov_xy / var_x
b = y_mean - m * x_mean

m, b(0.619005440971783, 21.737188409464757)(0.619005440971783, 21.737188409464757)successThe equation of the least-squares regression line is:

\[
y = 0.619x + 21.737
\]

Let me know if you would like further details or have any questions! Here are some related questions to expand your understanding:

1. How do we interpret the slope and intercept of the regression line in this context?
2. What is the importance of using the least-squares method in regression analysis?
3. How would you use this regression equation to predict a student’s score on the second test based on their first test score?
4. What assumptions are made when performing linear regression analysis?
5. How can we measure the strength of the linear relationship between the two variables?

**Tip:** To check the reliability of the regression model, compute the correlation coefficient or \( R^2 \) value. It helps assess the fit of the model to the data.

The following table shows students’ test scores on the first two tests in an introductory chemistry class. Find an equation of the least-squares regression line. Round your answer to three decimal places, if necessary.

Learn how to calculate the least-squares regression line for chemistry test scores. This problem involves linear regression using the least-squares method, essential formulas, and step-by-step calculations for test data. Suitable for students in Grades 10-12.

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