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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 Mean when σ is Unknown

Sample Mean (x̄):

Sample Size (n):

Sample Standard Deviation (s):

Confidence Level (1 - α):

Does the population follow a normal distribution? Yes No

Tips about Confidence Interval for Mean when σ is Unknown:

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Definations

• A confidence interval for the mean when the population standard deviation is unknown is an estimated range of values which is likely to include the population mean. Since the population standard deviation (σ) is unknown, the sample standard deviation (s) is used, and the t-distribution is applied instead of the standard normal distribution.

General Steps

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

   Sample

   Standard Error (δ̅x)	--
   Degrees of Freedom	--



2. Determine the \(t_{\alpha/2}\) for the Desired Confidence Level:

   • In the table below, I’ve listed common confidence levels (90%, 95%, 99%) along with their corresponding degrees of freedom (df) and \(t_{\alpha/2}\) values:

     Confidence Level (1-α)	Degrees of Freedom (df)	\(t_{\alpha/2}\)
     90%	30	1.697
     95%	30	2.042
     99%	45	2.750
     		N/A

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



3. Calculate the Margin of Error (\(M_{ε}\)):
   • The value margin error determines the widith of the confidence interval. \[M_{ε} = (t_{\alpha/2}) * (δ_\bar{x})\]
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:
   • 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. Incorrect t-Value:
   • Not using the correct t-value for the chosen confidence level and degrees of freedom.
   • Unlike to z-distribution, t-distribution determine the critical values based on the degree of freedom.
2. Misunderstanding Standard Error:
   • Confusing the formula for the standard error of the mean.
3. Assumption Violations:
   • Assuming the data follows a normal distribution without checking, especially for small sample sizes.
4. Rounding Errors:
   • Over-rounding intermediate calculations, leading to less accurate final intervals.
5. Small Sample Size:
   • Using very small samples can lead to inaccurate confidence intervals due to high variability in the t-values.

Additional Tips

1. Check Normality:
   • For small sample sizes, ensure that the sample data is approximately normally distributed.
2. Interpret Correctly:
   • Clearly interpret and report the confidence interval in the context of your data. For example, “We are 95% confident that the true population mean lies between X and Y.”
3. Consider Sample Size:
   • Larger sample sizes yield more precise confidence intervals. Aim for a sufficiently large sample to minimize uncertainty.
4. Practice:
   • ORegularly practice different scenarios to build confidence in calculating and interpreting confidence intervals.

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