You are a researcher studying the lifespan of a certain species of bacteria. A preliminary sample of 40 bacteria reveals a sample mean of I = 64 hours with a standard deviation of s=4.8 hours. You would like to estimate the mean lifespan for this species of bacteria to within a margin of error of 0.65 hours at a 99% level of confidence. What sample size should you gather to achieve a 0.65 hour margin of error? Round your answer up to the nearest whole number. n = bacteria
You are a researcher studying the lifespan of a certain species of bacteria. A preliminary sample of 40 bacteria reveals a sample mean of I = 64 hours with a standard deviation of s=4.8 hours. You would like to estimate the mean lifespan for this species of bacteria to within a margin of error of 0.65 hours at a 99% level of confidence. What sample size should you gather to achieve a 0.65 hour margin of error? Round your answer up to the nearest whole number. n = bacteria
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
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![### Estimating the Lifespan of a Bacterial Species
As a researcher studying the lifespan of a certain species of bacteria, you have collected some preliminary data. Specifically, you have a sample of 40 bacteria, which revealed an average lifespan (\(\bar{x}\)) of 64 hours with a standard deviation (\(s\)) of 4.8 hours.
To enhance your research accuracy, you aim to estimate the mean lifespan of this bacterial species within a margin of error of 0.65 hours at a 99% confidence level.
#### Required Sample Size Calculation
To determine the required sample size (\(n\)) to achieve a 0.65-hour margin of error, use the following steps:
1. **Identifying Parameters:**
- Sample mean (\(\bar{x}\)) = 64 hours
- Standard deviation (\(s\)) = 4.8 hours
2. **Desired Estimation:**
- Margin of Error (E) = 0.65 hours
- Confidence Level = 99%
Given these parameters, you need to calculate the sample size \(n\). The formula to calculate the sample size for estimating a population mean with a specified margin of error is:
\[ n = \left( \frac{Z \cdot s}{E} \right)^2 \]
Where:
- \(Z\) is the z-score corresponding to the desired confidence level.
- \(s\) is the sample standard deviation.
- \(E\) is the margin of error.
For a 99% confidence level, the Z-value is approximately 2.576.
3. **Calculating Sample Size:**
Substitute the values into the sample size formula:
\[ n = \left( \frac{2.576 \cdot 4.8}{0.65} \right)^2 \]
4. **Rounding Up:**
Round your answer up to the nearest whole number to ensure the margin of error criterion is met.
Given these instructions, you can now calculate and gather the appropriate sample size to ensure precise and reliable estimates for the lifespan of this bacterial species.
**Sample Size Needed:** \(n = \) [Calculated Value] bacteria
Note: This is a fundamental part of sample size determination in statistical analysis, crucial for ensuring the accuracy and reliability of research findings.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faa28b18f-3e2d-4592-9d09-7b963ee49630%2F0d8ce669-3a56-4547-b6ed-cb9ef6c84fdf%2F13jmd8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Estimating the Lifespan of a Bacterial Species
As a researcher studying the lifespan of a certain species of bacteria, you have collected some preliminary data. Specifically, you have a sample of 40 bacteria, which revealed an average lifespan (\(\bar{x}\)) of 64 hours with a standard deviation (\(s\)) of 4.8 hours.
To enhance your research accuracy, you aim to estimate the mean lifespan of this bacterial species within a margin of error of 0.65 hours at a 99% confidence level.
#### Required Sample Size Calculation
To determine the required sample size (\(n\)) to achieve a 0.65-hour margin of error, use the following steps:
1. **Identifying Parameters:**
- Sample mean (\(\bar{x}\)) = 64 hours
- Standard deviation (\(s\)) = 4.8 hours
2. **Desired Estimation:**
- Margin of Error (E) = 0.65 hours
- Confidence Level = 99%
Given these parameters, you need to calculate the sample size \(n\). The formula to calculate the sample size for estimating a population mean with a specified margin of error is:
\[ n = \left( \frac{Z \cdot s}{E} \right)^2 \]
Where:
- \(Z\) is the z-score corresponding to the desired confidence level.
- \(s\) is the sample standard deviation.
- \(E\) is the margin of error.
For a 99% confidence level, the Z-value is approximately 2.576.
3. **Calculating Sample Size:**
Substitute the values into the sample size formula:
\[ n = \left( \frac{2.576 \cdot 4.8}{0.65} \right)^2 \]
4. **Rounding Up:**
Round your answer up to the nearest whole number to ensure the margin of error criterion is met.
Given these instructions, you can now calculate and gather the appropriate sample size to ensure precise and reliable estimates for the lifespan of this bacterial species.
**Sample Size Needed:** \(n = \) [Calculated Value] bacteria
Note: This is a fundamental part of sample size determination in statistical analysis, crucial for ensuring the accuracy and reliability of research findings.
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