
A bakery sells cake donuts, marketed as “Monster Donuts”. One donut is advertised as weighing 25g, and company policy requires that 95% of Monster Donuts are within 4% of this weight.
A hungry statistics professor, realizing they can justify buying snacks if it’s for ‘educational purposes’, purchases
30 Monster Donuts to investigate
the variation in their weight.
Bonus problem: show that, if 𝜇𝜇 = 25, the company requires 𝜎𝜎2 ≈ 0.26 to satisfy their policy.

To solve the bonus problem and show that if \(\mu = 25\), the company requires \(\sigma^2 \approx 0.26\) to satisfy their policy, we need to analyze the given conditions using statistical principles.

1. **Understanding the Policy**:
   - The weight of each Monster Donut should be within 4% of 25 grams 95% of the time.
   - 4% of 25 grams is \(0.04 \times 25 = 1\) gram.
   - Therefore, 95% of the donuts should weigh between \(25 - 1 = 24\) grams and \(25 + 1 = 26\) grams.

2. **Interpreting the Requirement**:
   - The weights of the donuts are normally distributed with mean \(\mu = 25\) grams.
   - We need to find the standard deviation \(\sigma\) such that 95% of the weights lie within the interval [24, 26].

3. **Using the Properties of the Normal Distribution**:
   - For a normal distribution, approximately 95% of the data lies within 1.96 standard deviations of the mean. This is known as the empirical rule or the 68-95-99.7 rule.
   - Therefore, we can set up the equation for the interval:
     \[
     \mu - 1.96\sigma \leq 24 \quad \text{and} \quad \mu + 1.96\sigma \geq 26
     \]

4. **Solving for \(\sigma\)**:
   - Since \(\mu = 25\), we substitute and solve for \(\sigma\):
     \[
     25 - 1.96\sigma = 24 \implies 1.96\sigma = 1 \implies \sigma = \frac{1}{1.96} \approx 0.51
     \]

5. **Calculating \(\sigma^2\)**:
   - The variance \(\sigma^2\) is the square of the standard deviation:
     \[
     \sigma^2 = (0.51)^2 \approx 0.26
     \]

Thus, to satisfy the company policy, the variance \(\sigma^2\) must be approximately 0.26.

Would you like more details on any specific part of the explanation? Here are some additional questions you might want to explore:

1. How is the standard deviation derived from the z-score?
2. What are the implications of the normal distribution properties in quality control?
3. How can the professor test the company's compliance with the policy using the sample of 30 donuts?
4. What statistical tests can be applied to verify the variance?
5. How would a change in the policy affect the required standard deviation?

**Tip:** In any quality control scenario, understanding the relationship between the z-score and standard deviation is crucial for setting and maintaining product standards.

Explore how statistical principles determine the variance required for Monster Donuts to meet weight policy. Given that the mean weight is 25g, discover the necessary standard deviation through detailed calculations and implications of normal distribution properties. Suitable for advanced undergraduate students and professionals interested in quality control and statistical analysis.

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