Let's analyze and solve the given problem. Here’s the breakdown:

Given Data:

• Mean mass of the discs ($\mu$) = 175.0 g
• Standard deviation ($\sigma$) = 0.9 g
• Acceptable range: $174.8 \, \text{g} \leq \bar{x} \leq 175.2 \, \text{g}$
• Confidence level = 99% (assumed from part b).

Part (a): Confidence interval and margin of error

The margin of error (ME) for a confidence interval is determined using the formula: $\text{ME} = z \cdot \frac{\sigma}{\sqrt{n}}$ Where:

• $z$: critical value for the desired confidence level (e.g., $z = 2.576$ for 99% confidence).
• $\sigma$: population standard deviation.
• $n$: sample size.

We would solve this if the sample size is provided or inferred.

Part (b): Sample size for a specific margin of error

The formula for determining sample size is: $n = \left( \frac{z \cdot \sigma}{\text{ME}} \right)^2$ Here:

• Desired margin of error ($\text{ME}$) = 0.2 g
• $z = 2.576$ (critical value for 99% confidence)
• $\sigma = 0.9 \, \text{g}$

Substitute values: $n = \left( \frac{2.576 \cdot 0.9}{0.2} \right)^2$

First, calculate the numerator: $2.576 \cdot 0.9 = 2.3184$

Now divide by the margin of error: $\frac{2.3184}{0.2} = 11.592$

Square this value: $n = (11.592)^2 = 134.38$

Since sample size must be a whole number, round up: $n = 135$

Answer to part (b): Approximately 135 discs must be measured.

Conclusion:

(a) Requires sample size data to calculate the confidence interval.

(b) Approximately 135 discs should be measured to ensure the mean mass is within ±0.2 g with 99% confidence.

Would you like me to calculate part (a) further (if sample size is known), or explain anything else? Let me know!

Related Questions:

1. What is the formula to calculate $z$-scores for a confidence interval?
2. How is the standard error of the mean calculated in quality control?
3. How does increasing the sample size affect the margin of error?
4. What assumptions are made when calculating confidence intervals for normal distributions?
5. Can we use these formulas if the data is not normally distributed?

Tip: Always round up sample size calculations to ensure the desired confidence level is maintained.