Math Problem Statement

Find the range, variance, and standard deviation for the time (in minutes) spent in rush-hour traffic by a sample of 14 drivers: 52, 70, 60, 49, 46, 48, 53, 51, 70, 71, 59.

Solution

From the image, it looks like you're working on a problem involving statistics. Specifically, you are asked to calculate the range, variance, and standard deviation of the times (in minutes) spent in rush-hour traffic by a sample of 14 drivers. Here's a breakdown of the tasks and solutions based on the dataFrom the image, it looks like you're working on a problem involving statistics. Specifically, you are asked to calculate the range, variance, and standard deviation of the times (in minutes) spent in rush-hour traffic by a sample of 14 drivers. Here's a breakdown of the tasks and solutions based on the

Part 1: Find the Range

You’ve correctly identified the range as 25. The range is calculated by subtracting the smallest value from the largest value: Range=7146=25\text{Range} = 71 - 46 = 25

Part 2: Find the Variance

It appears that the value you entered, 539.4, was incorrect, and the correct variance is 82.7. Variance (σ2\sigma^2) measures how spread out the numbers are.

To find the variance, you would use the formula: Variance=1n1(xixˉ)2\text{Variance} = \frac{1}{n-1} \sum (x_i - \bar{x})^2 Where:

  • nn is the number of data points (14 in this case).
  • xix_i represents each individual data point.
  • xˉ\bar{x} is the mean of the data points.

The correct variance given in your problem is 82.7.

Part 3: Find the Standard Deviation

The standard deviation is the square root of the variance: Standard Deviation=Variance=82.7\text{Standard Deviation} = \sqrt{\text{Variance}} = \sqrt{82.7} Let me calculate the correct standard deviation for you.

The standard deviation is approximately 9.1.

Would you like more details on how these values were calculated or have any questions?

Here are five additional related questions you might find helpful:

  1. How is the variance formula derived?
  2. What is the difference between population variance and sample variance?
  3. Why is the range a less reliable measure of spread than variance or standard deviation?
  4. How would the results change if there were an outlier in the data?
  5. What are some common applications of standard deviation in real-world data analysis?

Tip: Standard deviation is a useful measure to understand how much variation exists in a dataset. The larger the standard deviation, the more spread out the data.

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Math Problem Analysis

Mathematical Concepts

Descriptive Statistics
Range
Variance
Standard Deviation

Formulas

Range = max(data) - min(data)
Variance (σ^2) = Σ(xi - μ)^2 / (n - 1)
Standard Deviation (σ) = sqrt(Variance)

Theorems

None explicitly, but variance and standard deviation are foundational in statistics.

Suitable Grade Level

High School (Grades 9-12)