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Confidence Interval Calculators

CI for µ (σ Known) CI for µ (σ Unknown) CI for µ1 - µ2 (σ Known)

CI for µ1 - µ2 (σ Unknown) CI for P CI for (P1-P2)

Confidence Interval for Proportion Calculator

Number of successes (x):

Sample size (n):

Sample proportion (𝑝̂):

Confidence Level (1 - α):

Tips about Confidence Interval for the differnace between Proportions:

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Definations

• A confidence interval for a proportion is a range of values, derived from sample statistics, that is likely to contain the true population proportion. It provides an estimate of the parameter along with an associated confidence level, typically 95% or 99%, indicating the degree of certainty in the estimate.
• A wider interval indicates more uncertainty in the estimate, whereas a narrower interval indicates more precision.

General Steps

• Define the Sample Proportion::
  

  \[ \hat{p} = \frac{x}{n}, \] where: \(\hat{p}\) is the sample proportion \(x\) is the number of successes in the sample and \(n\) is the sample size

• Compute the Standard Error of the Difference (\(δ_{\hat{p}}\)):
  

  To compute the standard error for the proportions, use the following formula:

  \[ δ_{\hat{p}} = \sqrt{\frac{\hat{p} (1 - \hat{p}}{n}} \]

• Determine the \(z_{\alpha/2}\) for the Desired Confidence Level:
  

  In the table below, I’ve listed common confidence levels (90%, 95%, 99%) along with their corresponding \(z_{\alpha/2}\):

  Confidence Level (1-α)	\(z_{\alpha/2}\)
  90%	1.645
  95%	1.960
  99%	2.576
  	N/A

  Feel free to provide a confidence level in percentage, and I’ll calculate the corresponding \(z_{\alpha/2}\) for you!

• Calculate the Margin of Error (\(M_{ε}\)):
  The value margin error determines the widith of the confidence interval.
  \[M_{ε} = (z_{\alpha/2}) * (δ_{\hat{p}})\]
• Construct the Confidence Interval:
  \[ CI = \hat{p} \pm M_{ε} \] \[ LCI = \hat{p} - M_{ε} \] \[ UCI = \hat{p} + M_{ε} \]
• Interpretation:
  Here, I provide you a deafult style for interpreting the result of the confidence interval.
  
  We are + (1-α)% confident that the + true population proportions lies between + [LCL and UCL].
  
  For example: "We are 95% confident that the true population in proportions lies between -0.3042 and 0.1042."

Common Mistakes

• Small Sample Size:
  If the sample size is too small, the normal approximation to the binomial distribution may not be valid. Ensure np≥5 and 𝑛(1−𝑝)≥5
• Misinterpreting the Confidence Level:
  A 95% confidence level means that if we were to take 100 different samples and compute a confidence interval for each sample, we would expect about 95 of the intervals to contain the true population proportion.
• Ignoring the Assumptions:
  The method assumes a random sample and a binomial distribution of the data. Violating these assumptions can lead to incorrect intervals.
• Confusing Sample Proportion and Population Proportion:
  The confidence interval estimates the population proportion, not the sample proportion, which is already known.

Additional Steps for Improvement

• Use Exact Methods for Small Samples:
  For small sample sizes, use exact methods like the Clopper-Pearson interval instead of the normal approximation.
• Adjust for Finite Population:
  If the sample is a significant fraction of the population, adjust the standard error using a finite population correction..
• Graphical Representation::
  Visualizing the confidence interval along with the data can provide better insight and communicate the uncertainty effectively.