3. An average glass of milk contains 115 grams of calcium with a standard deviation of 8 grams. Using Chebyshev’s theorem, construct an interval that contains the calcium content of at least 50% of the glasses of milk.

Retain 2 decimal places of accuracy in all calculations.

To apply **Chebyshev’s theorem**, we need to recall that the theorem states that, for any dataset (regardless of its distribution), at least \(\frac{1}{k^2}\) of the data will lie within \(k\) standard deviations of the mean, where \(k > 1\).

We are asked to find an interval that contains **at least 50%** of the calcium content in the glasses of milk. According to Chebyshev's theorem, for at least 50% of the data to lie within \(k\) standard deviations of the mean, we need to solve for \(k\) in the following inequality:

\[
\frac{1}{k^2} \geq 0.50
\]

1. First, solve for \(k\):

\[
\frac{1}{k^2} \geq 0.50
\]

\[
k^2 \leq 2
\]

\[
k \geq \sqrt{2} \approx 1.41
\]

Thus, at least 50% of the data will lie within **1.41 standard deviations** of the mean.

2. Now, construct the interval around the mean. The average calcium content is \( \mu = 115 \) grams, and the standard deviation is \( \sigma = 8 \) grams. The interval will be:

\[
\text{Interval} = [ \mu - k\sigma, \mu + k\sigma ]
\]

Substituting the values:

\[
\text{Interval} = [ 115 - 1.41(8), 115 + 1.41(8) ]
\]

\[
\text{Interval} = [ 115 - 11.28, 115 + 11.28 ]
\]

\[
\text{Interval} = [ 103.72, 126.28 ]
\]

Thus, the interval that contains at least 50% of the calcium content in the glasses of milk is:

\[
\boxed{[103.72, 126.28]}
\]

### Would you like more details or clarification on any part of the solution?

### Related questions:
1. How would the interval change if we wanted to contain 75% of the data?
2. Can Chebyshev’s theorem be applied to normally distributed data, and how does it compare to the empirical rule?
3. What would the interval be if the standard deviation was larger, say 10 grams?
4. How does Chebyshev’s theorem ensure a minimum percentage, and why does it work for any distribution?
5. What is the relationship between \(k\) and the proportion of data covered by the interval?

#### Tip: Chebyshev's theorem applies to any distribution, but it tends to be less precise for normal distributions, where the empirical rule gives tighter intervals.

An average glass of milk contains 115 grams of calcium with a standard deviation of 8 grams. Using Chebyshev’s theorem, construct an interval that contains the calcium content of at least 50% of the glasses of milk.

Learn how to apply Chebyshev's theorem to find the interval containing at least 50% of the calcium content in a glass of milk, with an average of 115 grams and a standard deviation of 8 grams. This step-by-step solution uses Chebyshev's theorem to calculate the interval [103.72, 126.28].

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