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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 Two-Sample Mean σ's are Known

Sample Mean 1 (x̄₁):

Sample Size 1 (n₁):

Population Standard Deviation 1 (σ₁):

Sample Mean 2 (x̄₂):

Sample Size 2 (n₂):

Population Standard Deviation 2 (σ₂):

Confidence Level (1 - α):

Do the populations follow a normal distribution? Yes No

Tips about Confidence Interval for the difference between Proportions:

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Definations

• Confidence Interval (CI): A range of values, derived from a data sample, that is likely to contain the value of an unknown population parameter.
• Difference Between Means: (s) The difference between the average values (means) of two different populations, often represented as (\( \bar{x}_1 - \bar{x}_2\)).
• Population Standard Deviation (σ): A measure of the amount of variation or dispersion in a population. When it is known, the standard error is computed directly using σ.

General Steps

• Determine the Standard Error of the Difference (\(δ_{\bar{x}_1 - \bar{x}_2}\))::
  The standard error for the difference between two means when population standard deviations are uknown is calculated as follows
  \[ δ_{\bar{x}_1 - \bar{x}_2} = \sqrt{\left( \frac{σ_1^2}{n_1} \right) + \left( \frac{σ_2^2}{n_2} \right)} \] Where
  \(σ_1\) and \(σ_2\) are the population standard deviations.
  \(n_1\) and \(n_2\) are sample sizes.
  
  
Give me the following values and enjoy with the MLC result.

Sample 1	Sample 2

Standard Error (σΔx̄)

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• 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.
  \[M_{ε} = (z_{\alpha/2}) * (δ_{\bar{x}_1 - \bar{x}_2})\]
• Construct the Confidence Interval:
  \[ CI = (\bar{x}_1 - \bar{x}_2) \pm M_{ε} \] \[ LCI = (\bar{x}_1 - \bar{x}_2) - M_{ε} \] \[ UCI = (\bar{x}_1 - \bar{x}_2) + M_{ε} \]
• Interpretation:
  Here, I provide you a deafult style for interpreting the result of the confidence interval.
  
  We are + (1-α)% confident that the + true difference in population mean lies between + [LCL and UCL].
  
  For example: "We are 95% confident that the true difference in proportions lies between 50 and 70."

Common Errors

Based on my experience with previous students, I’ve noticed some common mistakes related to this confidence intervals. It’s essential to exercise caution while performing these calculations. Here are some key points to keep in mind:

• Incorrect z-Value:
  Using the wrong z-value for the desired confidence level. For example, using 1.645 instead of 1.96 for a 95% confidence interval.
• Misusing Population Standard Deviations:
  Confusing population standard deviations (σ) with sample standard deviations (s). When population standard deviations are known, use σ.
• Ignoring Sample Sizes:
  Failing to properly account for the sample sizes in the standard error calculation. Larger samples provide more precise estimates and narrower confidence intervals.
• Misinterpreting the Interval:
  The confidence interval provides a range of values that, with a certain level of confidence, contains the true difference between means. It does not mean that the true difference is equally likely to be anywhere within the range.
• Incorrect Assumptions:
  Assuming normality of the sampling distribution of the difference between means when the sample sizes are small and the underlying population distribution is not normal.

Additional Tips:

1. Check Assumptions:
   • Ensure that the samples are independent and that the population distributions are normal, especially for small sample sizes.
2. Use Correct Z-Value:
   • For common confidence levels, use:
   • 90%: Z = 1.645
   • 95%: Z = 1.96
   • 99%: Z = 2.576
3. Sensitivity Analysis:
   • Consider how changes in sample size, standard deviations, or confidence level might affect the confidence interval. This can provide insights into the robustness of your estimates.
4. Report Findings Clearly:
   • Clearly state the confidence level and the calculated interval. For example, “We are 95% confident that the true difference between the population means lies between X and Y.”

Your sample sizes are below the requirement. The populations must follow a normal distribution or the sample sizes must be at least 30.