A Scientist Is Interested In The Average Height Of All Plant
A Scientist Is Interested In The Average Height Of All Plants Of A Par
A scientist is interested in determining the appropriate sample size needed to estimate the average height of all plants of a particular type treated with an experimental chemical. The scientist aims for a 90% confidence level that the true mean height falls within a specified margin of error. Based on a prior pilot study, the estimated standard deviation of plant heights is 5.35 cm. The maximum acceptable error for the confidence interval is 1 cm. The goal is to calculate the minimum number of plants that must be sampled to meet these criteria, assuming the Central Limit Theorem applies to the sample mean.
Paper For Above instruction
To determine the required sample size for estimating the mean plant height with a specified confidence level and margin of error, we utilize principles of inferential statistics. The problem specifies that the width of the confidence interval should not exceed 1 cm with a confidence level of 90%, and that the estimated standard deviation from a pilot study is 5.35 cm.
Understanding the Components
The key parameters include:
- Confidence level: 90%
- Margin of error (E): 1 cm
- Estimated population standard deviation (σ): 5.35 cm
In sampling distributions, when estimating the mean from a normally distributed population or a large enough sample (by the Central Limit Theorem), the confidence interval is given by:
\[ \bar{x} \pm z_{\alpha/2} \times \frac{\sigma}{\sqrt{n}} \]
where:
- \( \bar{x} \) is the sample mean
- \( z_{\alpha/2} \) is the z-value associated with the confidence level
- \( \sigma \) is the population standard deviation
- \( n \) is the sample size
Determining the Z-value
For a 90% confidence interval, the significance level is:
\[ \alpha = 1 - 0.90 = 0.10 \]
Thus:
\[ \alpha/2 = 0.05 \]
From standard normal distribution tables, the z-value corresponding to a 95th percentile (since 5% in each tail) is:
\[ z_{0.05} \approx 1.645 \]
Calculating the Sample Size
The margin of error for the confidence interval (E) should be:
\[ E = z_{\alpha/2} \times \frac{\sigma}{\sqrt{n}} \]
Rearranging:
\[ n = \left( \frac{z_{\alpha/2} \times \sigma}{E} \right)^2 \]
Plugging in the known values:
\[ n = \left( \frac{1.645 \times 5.35}{1} \right)^2 = \left( 8.80575 \right)^2 \approx 77.54 \]
Since sample size must be a whole number and cannot be fractional, the scientist should round up:
\[ n = 78 \]
Conclusion
The scientist needs to sample at least 78 plants to ensure a 90% confidence that the confidence interval for the mean height is within 1 cm, based on the estimated standard deviation from the pilot study. Proper sample size determination is crucial in experimental design, ensuring sufficient statistical power and reliable inferences about the population parameter.
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