In the manufacturing process of a pharmaceutical product, repeated measurements of the concentration of the active ingredient follow a Normal distribution with mean μ equal to the true product concentration, and with standard deviation = 0.008 g/l. How many measurements would be needed to estimate the true concentration within ±0.001 g/l with 99% confidence? Give your answer rounded up to the nearest whole number. n=
In the manufacturing process of a pharmaceutical product, repeated measurements of the concentration of the active ingredient follow a Normal distribution with mean μ equal to the true product concentration, and with standard deviation = 0.008 g/l. How many measurements would be needed to estimate the true concentration within ±0.001 g/l with 99% confidence? Give your answer rounded up to the nearest whole number. n=
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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![**Text Transcription for Educational Website:**
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In the manufacturing process of a pharmaceutical product, repeated measurements of the concentration of the active ingredient follow a Normal distribution with mean \( \mu \) equal to the true product concentration, and with standard deviation \( \sigma = 0.008 \, \text{g/l} \).
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**Question:**
How many measurements would be needed to estimate the true concentration within \( \pm 0.001 \, \text{g/l} \) with 99% confidence? Give your answer rounded up to the nearest whole number.
\[ n = \] _______________
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**Explanation:**
This question involves the calculation of sample size for estimating the mean concentration of a pharmaceutical product with a specified precision and confidence level. The provided parameters include:
- Standard deviation (\( \sigma \)) of the concentration measurements: 0.008 g/l
- Desired precision (\( E \)): \( \pm 0.001 \, \text{g/l} \)
- Confidence level: 99%
To determine the required sample size, the formula for sample size calculation in the context of a normal distribution is typically used:
\[ n = \left( \frac{Z_{\text{confidence}} \cdot \sigma}{E} \right)^2 \]
Where:
- \( n \) is the sample size.
- \( Z_{\text{confidence}} \) is the Z-value for the desired confidence level (e.g., for 99% confidence, \( Z \approx 2.576 \)).
- \( \sigma \) is the standard deviation.
- \( E \) is the margin of error or precision.
The solution involves solving for \( n \) and rounding up to the nearest whole number.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe24fc82a-b7ed-4d5f-9001-130a2252ab39%2Fc64f2c3d-a85b-471c-8369-a5281a3d8625%2F9z09r0n_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Text Transcription for Educational Website:**
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In the manufacturing process of a pharmaceutical product, repeated measurements of the concentration of the active ingredient follow a Normal distribution with mean \( \mu \) equal to the true product concentration, and with standard deviation \( \sigma = 0.008 \, \text{g/l} \).
---
**Question:**
How many measurements would be needed to estimate the true concentration within \( \pm 0.001 \, \text{g/l} \) with 99% confidence? Give your answer rounded up to the nearest whole number.
\[ n = \] _______________
---
**Explanation:**
This question involves the calculation of sample size for estimating the mean concentration of a pharmaceutical product with a specified precision and confidence level. The provided parameters include:
- Standard deviation (\( \sigma \)) of the concentration measurements: 0.008 g/l
- Desired precision (\( E \)): \( \pm 0.001 \, \text{g/l} \)
- Confidence level: 99%
To determine the required sample size, the formula for sample size calculation in the context of a normal distribution is typically used:
\[ n = \left( \frac{Z_{\text{confidence}} \cdot \sigma}{E} \right)^2 \]
Where:
- \( n \) is the sample size.
- \( Z_{\text{confidence}} \) is the Z-value for the desired confidence level (e.g., for 99% confidence, \( Z \approx 2.576 \)).
- \( \sigma \) is the standard deviation.
- \( E \) is the margin of error or precision.
The solution involves solving for \( n \) and rounding up to the nearest whole number.
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