1. Home
2. MLC Tools
3. InfStat Calculators
4. CI for µ (σ is Known)

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 Calculator

Sample Mean (x̄):

Sample Size (n):

Population Standard Deviation (σ):

Confidence Level (1 - α):

Does the population follow a normal distribution? Yes No

Tips about Confidence Interval for Mean when σ is Unknown:

---

Definations

• A confidence interval for the mean when the population standard deviation (σ) is known is an estimate of the range within which the true population mean (μ) lies, with a certain level of confidence. This interval is constructed using the sample mean, the known population standard deviation, and the sample size (𝑛).

General Steps

1. Calculate the Standard Error (\(δ_\bar {x}\)):
   • The standard error for mean when population standard deviation is known is calculated as follows:
   \[δ_{\bar{x}} = \frac{σ}{\sqrt{n}}\] Give me the following values and enjoy with the MLC result.

   Standard Error Calculator

   Sample

   Standard Error (δ̅x)

   --




2. 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 degree of freedom, then I’ll calculate the corresponding \(z_{\alpha/2}\) for you!



3. Calculate the Margin of Error (\(M_{ε}\)):
   • The value margin error determines the widith of the confidence interval. \[M_{ε} = (z_{\alpha/2}) * (δ_\bar{x})\]
4. Construct the Confidence Interval:
   • The upper and lower limit of the confidence intervals are given as: \[ CI = \bar{x} \pm M_{ε} \] \[ LCI = \bar{x} - M_{ε} \] \[ UCI = \bar{x} + M_{ε} \]
5. Interpretation:
   • State the confidence interval and interpret what it means in the context of the problem. Here, I provide you a deafult style for interpreting the result of the confidence interval.
     We are + (1-α)% confident that the + true value of population mean lies between + [LCL and UCL].
     
     For example: "We are 95% confident that the true value of population mean lies between 50 and 70."

Common Errors:

1. Using the Wrong Z-Score:
   • Ensure the correct Z-score is used for the chosen confidence level.
2. Incorrect Standard Error Calculation:
   • Double-check the formula and ensure the correct values are used.
3. Assuming Population Standard Deviation is Known:
   • Only use this method if the population standard deviation (𝜎) is truly known. If it's unknown, use the t-distribution instead
4. Misinterpretation of the Confidence Interval:
   • The confidence interval does not predict where future sample means will lie but estimates where the true population mean is likely to be.

Additional Tips

1. Large Sample Sizes:
   • Larger sample sizes result in narrower confidence intervals, providing more precise estimates of the population mean.
2. Report the Confidence Level:
   • Always specify the confidence level when reporting the interval.
3. Check Assumptions:
   • Ensure the sample is randomly selected and the population distribution is approximately normal, especially for small sample sizes.
4. Contextual Understanding:
   • Interpret the confidence interval in the context of your specific problem or research question.

Your sample size is below the requirement. The population must follow a normal distribution or the sample size must be at least 30.