Math Problem Statement

Cholesterol levels were collected from patients two days after they had a heart attack (Ryan, Joiner & Ryan, Jr, 1985). Retrieved from http://www.statsci.org/data/general/cholest.html.

Cholesterol Levels 270 236 210 142 280 272 160 220 226 242 186 266 206 318 294 282 234 224 276 282 360 310 280 278 288 288 244 236 Compute the variance. Round the answer to two decimal places.

Solution

To calculate the variance of the cholesterol levels, we follow these steps:

Step 1: Calculate the mean (average)

The mean is calculated by summing all the cholesterol levels and then dividing by the number of data points.

Data points: 270, 236, 210, 142, 280, 272, 160, 220, 226, 242, 186, 266, 206, 318, 294, 282, 234, 224, 276, 282, 360, 310, 280, 278, 288, 288, 244, 236

Step 2: Calculate each squared deviation from the mean

After finding the mean, for each data point, subtract the mean and square the result.

Step 3: Calculate the variance

The variance is the average of these squared deviations. Since we're working with a sample, we divide by n1n-1 (where nn is the number of data points) to get the sample variance.

Let's calculate that now.

numpy as np

Cholesterol levels data

cholesterol_levels = [270, 236, 210, 142, 280, 272, 160, 220, 226, 242, 186, 266, 206, 318, 294, 282, 234, 224, 276, 282, 360, 310, 280, 278, 288, 288, 244, 236]

Calculate the mean

mean_cholesterol = np.mean(cholesterol_levels)

Calculate the variance (sample variance, divide by n-1)

variance_cholesterol = np.var(cholesterol_levels, ddof=1)

Round to 2 decimal places

variance_cholesterol_rounded = round(variance_cholesterol, 2)

variance_cholesterol_roundedThe variance of the cholesterol levels is 2276.29 (rounded to two decimal places).

Would you like more details or have any questions?

Here are some related questions to consider:

  1. How does variance relate to standard deviation, and how can you calculate it from variance?
  2. What does the high variance in cholesterol levels suggest about the data?
  3. How can you interpret the results in a medical context, given that this is data from heart attack patients?
  4. How might sample variance differ from population variance, and why do we use n1n-1 in the formula for sample variance?
  5. How can you calculate the mean and variance for data with missing values?

Tip: Variance gives a sense of the spread of data points from the mean. A higher variance means the data points are more spread out, while a lower variance indicates that the data points are closer to the mean.

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Math Problem Analysis

Mathematical Concepts

Statistics
Variance
Descriptive Statistics

Formulas

Variance formula: σ² = Σ(xi - μ)² / (n-1)

Theorems

Sample Variance Theorem

Suitable Grade Level

Grades 10-12