A distribution of measurements has a mean of 65 and a standard deviation of 7. You know nothing else about the size or shape of the data. Use this information to find the proportion of measurements in the given interval. (Round your answer to two decimal places.)

between 44 and 86

Since the problem doesn't provide information about the distribution shape, we can use **Chebyshev's Theorem** to solve this problem. Chebyshev’s Theorem applies to any distribution, regardless of its shape, and provides a way to calculate the minimum proportion of data that falls within a specified number of standard deviations from the mean.

### Step-by-step solution:

1. **Given Information**:
   - Mean, \( \mu = 65 \)
   - Standard deviation, \( \sigma = 7 \)
   - We are asked to find the proportion of measurements between 44 and 86.

2. **Determine the number of standard deviations from the mean**:
   We calculate how far the values 44 and 86 are from the mean in terms of standard deviations.
   
   - For the lower bound (44): 
     \[
     \frac{65 - 44}{7} = \frac{21}{7} = 3 \text{ standard deviations below the mean}
     \]
   - For the upper bound (86):
     \[
     \frac{86 - 65}{7} = \frac{21}{7} = 3 \text{ standard deviations above the mean}
     \]

   The interval [44, 86] represents measurements that are within 3 standard deviations of the mean.

3. **Apply Chebyshev's Theorem**:
   Chebyshev’s Theorem states that at least \( 1 - \frac{1}{k^2} \) of the data falls within \( k \) standard deviations of the mean, where \( k \) is the number of standard deviations.

   Here, \( k = 3 \), so the proportion is:
   \[
   1 - \frac{1}{3^2} = 1 - \frac{1}{9} = 1 - 0.1111 = 0.8889
   \]
   
   Therefore, at least **88.89%** of the data falls within 3 standard deviations of the mean, which means at least **0.89** of the data is between 44 and 86.

### Final Answer:
The proportion of measurements between 44 and 86 is approximately **0.89** (rounded to two decimal places).

Would you like further clarification or more details?

### 5 Related Questions:
1. How would the result change if the standard deviation were smaller, say 5?
2. What is the significance of Chebyshev’s Theorem in statistics?
3. How does this result compare with the Empirical Rule (68-95-99.7 Rule) for a normal distribution?
4. What would the proportion be if we were considering 2 standard deviations from the mean instead of 3?
5. How would we calculate the proportion if the interval was asymmetric around the mean?

### Tip:
Chebyshev’s Theorem is helpful when you don't know the distribution shape, but if you know the distribution is normal, use the Empirical Rule for more precise proportions.

A distribution of measurements has a mean of 65 and a standard deviation of 7. You know nothing else about the size or shape of the data. Use this information to find the proportion of measurements in the given interval: between 44 and 86. (Round your answer to two decimal places.)

Find the proportion of measurements within the interval [44, 86] using Chebyshev's Theorem, given a mean of 65 and a standard deviation of 7. Learn how to apply the theorem to calculate the minimum proportion of data within a specified range. Suitable for college-level statistics students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation