Suppose that prices of a gallon of milk at various stores in one town have a mean of $3.53
 with a standard deviation of $0.14
.  Using Chebyshev's Theorem, state the range in which at least 75%
 of the data will reside.  Please do not round your answers.

Chebyshev's Theorem states that for any dataset, at least $ \frac{1}{k^2} $ of the data values must lie within $ k $ standard deviations of the mean, where $ k > 1 $. To find the range where at least 75% of the data resides, we follow these steps:

### Step 1: Use the formula from Chebyshev's Theorem.
We are given that at least 75% of the data must fall within a certain range. So, we solve for $ k $ such that:

\[
1 - \frac{1}{k^2} = 0.75
\]

This implies:

\[
\frac{1}{k^2} = 0.25 \quad \text{or} \quad k^2 = 4
\]

Thus:

\[
k = 2
\]

### Step 2: Calculate the range using $ k = 2 $.
Chebyshev's Theorem tells us that at least 75% of the data falls within 2 standard deviations of the mean. The mean price of milk is $ 3.53 $, and the standard deviation is $ 0.14 $. Therefore, the range is:

\[
\text{Lower bound} = \mu - 2\sigma = 3.53 - 2(0.14) = 3.53 - 0.28 = 3.25
\]
\[
\text{Upper bound} = \mu + 2\sigma = 3.53 + 2(0.14) = 3.53 + 0.28 = 3.81
\]

### Conclusion:
At least 75% of the data will fall within the range $[3.25, 3.81]$.

Would you like more details or have any questions?

Here are 5 related questions:
1. What is the interpretation of Chebyshev's Theorem for $k = 3$?
2. How would the range change if the standard deviation were larger?
3. Can Chebyshev’s Theorem apply to non-normally distributed data?
4. How would the range change if the mean price were $4 instead of $3.53?
5. How does Chebyshev's Theorem differ from the Empirical Rule?

**Tip:** Chebyshev's Theorem is useful because it applies to any data distribution, not just normal distributions!

Suppose that prices of a gallon of milk at various stores in one town have a mean of $3.53 with a standard deviation of $0.14. Using Chebyshev's Theorem, state the range in which at least 75% of the data will reside. Please do not round your answers.

Using Chebyshev's Theorem, learn how to find the range in which at least 75% of milk prices fall. Given a mean of $3.53 and a standard deviation of $0.14, this problem demonstrates the theorem's application with clear calculations. Suitable for students in Grades 10-12.

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