Determine the minimum sample size required when you want to be 99% confident that the sample mean is within one unit of the population mean and o = 19.8. Assume the population is normally distributed. ..... A 99% confidence level requires a sample size of| | (Round up to the nearest whole number as needed.)
Determine the minimum sample size required when you want to be 99% confident that the sample mean is within one unit of the population mean and o = 19.8. Assume the population is normally distributed. ..... A 99% confidence level requires a sample size of| | (Round up to the nearest whole number as needed.)
A First Course in Probability (10th Edition)
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ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
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![**Determining the Minimum Sample Size for a 99% Confidence Level**
To ensure a high level of accuracy when estimating the mean of a population, it is crucial to determine the appropriate sample size. When you want to be 99% confident that the sample mean is within one unit of the population mean, use the following information and steps:
**Given:**
- Population standard deviation (σ) = 19.8
- Confidence level = 99%
- Maximum allowable error (E) = 1 unit
**Objective:**
To find the minimum sample size (n) required for the specified confidence level.
**Instructions:**
1. Use the formula for sample size determination:
\[ n = \left( \frac{Z \cdot \sigma}{E} \right)^2 \]
Where:
- \( Z \) is the Z-score corresponding to the desired confidence level (for 99%, \( Z \approx 2.576 \)).
- \( \sigma \) is the standard deviation of the population.
- \( E \) is the maximum allowable error.
2. Substitute the given values into the formula:
\[ n = \left( \frac{2.576 \cdot 19.8}{1} \right)^2 \]
3. Calculate the value and round up to the nearest whole number to find the minimum required sample size.
It's important to note that rounding up ensures the sample size is sufficiently large to maintain the desired confidence level and accuracy.
**Problem:**
A 99% confidence level requires a sample size of ___.
(Round up to the nearest whole number as needed.)
**Action:**
Follow the outlined steps to determine the appropriate sample size for your data needs.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F02fc7836-9552-4a6b-92e4-0fb932880b81%2F5f72e682-d5bf-48cd-a720-39ad4141974e%2Falpbn5o_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Determining the Minimum Sample Size for a 99% Confidence Level**
To ensure a high level of accuracy when estimating the mean of a population, it is crucial to determine the appropriate sample size. When you want to be 99% confident that the sample mean is within one unit of the population mean, use the following information and steps:
**Given:**
- Population standard deviation (σ) = 19.8
- Confidence level = 99%
- Maximum allowable error (E) = 1 unit
**Objective:**
To find the minimum sample size (n) required for the specified confidence level.
**Instructions:**
1. Use the formula for sample size determination:
\[ n = \left( \frac{Z \cdot \sigma}{E} \right)^2 \]
Where:
- \( Z \) is the Z-score corresponding to the desired confidence level (for 99%, \( Z \approx 2.576 \)).
- \( \sigma \) is the standard deviation of the population.
- \( E \) is the maximum allowable error.
2. Substitute the given values into the formula:
\[ n = \left( \frac{2.576 \cdot 19.8}{1} \right)^2 \]
3. Calculate the value and round up to the nearest whole number to find the minimum required sample size.
It's important to note that rounding up ensures the sample size is sufficiently large to maintain the desired confidence level and accuracy.
**Problem:**
A 99% confidence level requires a sample size of ___.
(Round up to the nearest whole number as needed.)
**Action:**
Follow the outlined steps to determine the appropriate sample size for your data needs.
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