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\(\sigma_\bar{x}\) calculator \(\sigma_\bar{x}\) finite population \(\sigma_{\bar{x_1} - \bar{x_2}}\) calculator

\(\sigma_\bar{p}\) calculator \(\sigma_\bar{p}\) Finite Population \(\sigma_{\bar{p_1} - \bar{p_2}}\) calculator

Standard Error Calculator with Finite Population Correction

Sample Size (n)

Population Size (N)

Is the population standard deviation known? Yes No

Standard deviation

Result

Please enter all values.

Tips about Hypothesis testing for Mean when σ known:

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Definations

• The standard error of the mean (\(\sigma_{\bar{x}}\)) measures the precision of the sample mean as an estimate of the population mean. When dealing with finite populations, the standard error must be adjusted to account for the fact that sampling without replacement affects the variability of the sample mean.
• Finite Population Correction (FPC) is used to adjust the standard error when the sample size is a significant proportion of the population size. The formula for the standard error with the finite population correction is:
• When population standard deviation (σ) is known: \[ \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} \times \sqrt{\frac{N - n}{N - 1}} \]
• When population standard deviation is not known (using sample standard deviation s) \[ \sigma_{\bar{x}} = \frac{s}{\sqrt{n}} \times \sqrt{\frac{N - n}{N - 1}} \]
Where:
\(n\) = Sample Size
\(N\) = Population size
\(\sigma\) = Population standard deviation
\(s\) = Sample standard deviation 2

Common Errors:

1. Ignoring the Correction Factor:
   • Failing to apply the finite population correction when the sample size is a significant proportion of the population size can lead to an underestimation of the standard error.
2. Incorrect Sample Size:
   • Using an incorrect sample size or population size in the formula will result in inaccurate standard error calculations.
3. Misunderstanding Proportions:
   • Assuming that the sample size is negligible compared to the population size when it’s not, leading to incorrect standard error calculations.
4. Not Updating for Correction:
   • If the population standard deviation is unknown and the sample standard deviation is used, it’s important to update the formula and include the finite population correction.

Additional Tips

1. Check Proportions:
   • Always check the ratio of sample size to population size (\(𝑛/𝑁)\). If this ratio is 5% or more, applying the finite population correction is important.
2. Understand the Impact:
   • The finite population correction factor becomes less significant as the sample size becomes small relative to the population. In large populations or small sample sizes, the correction factor’s impact is minimal.
3. Use Correct Formulas:
   • Ensure you use the correct formula depending on whether the population standard deviation is known or not. For unknown standard deviation, use the sample standard deviation and adjust accordingly.