1. Home
2. MLC Tools
3. InfStat Calculators
4. CI for µ1 - µ2 (σ's are Unknown)

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 Unknown

Sample Mean 1 (x̄₁): Sample Mean 2 (x̄₂): Sample Size 1 (n₁): Sample Size 2 (n₂): Confidence Level (1 - α): Sample Standard Deviation 1 (s₁): Sample Standard Deviation 2 (s₂): Variance Assumption: Equal Variances Unequal Variances Do the population distributions follow a normal distribution? Yes No

Results

Tips about Confidence Interval for the difference between Means:

---

Definations

• Sample mean (x̄)​ The average value of a sample.
• Confidence Interval (CI): A range of values, derived from a data sample, that is likely to contain the value of an unknown population parameter.
• Sample Standard devaitions (s): A measure of the amount of variation or dispersion in a set of sample values.
• Difference Between Means: The difference between the means of two independent samples, often represented as (\( \bar{x}_1 - \bar{x}_2\))

General Steps

• Calculate the Standard Error (\(δ_{\bar{x}_1 - \bar{x}_2}\))::
  The standard error (SE) for the difference between two means when population standard deviations are unknown is calculated based on the following two conditions:
  
  1. When we assume the two popoulation variances are equal: we first required to calculate the pooled standard deviation
  \[ p_o = \sqrt{\frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2}} \] Then the standard error will be \[ δ_{\bar{x}_1 - \bar{x}_2} = p_o \sqrt{\frac{1}{n_1} + \frac{1}{n_2}} \] and the degree of fredom is as follows: \[ df = n_1 + n_2 - 2 \] 2. When the population variance are not assumed to be equal: The standard error is calculated as \[ δ_{\bar{x}_1 - \bar{x}_2} = \sqrt{\left( \frac{s_1^2}{n_1} \right) + \left( \frac{s_2^2}{n_2} \right)} \] and the degree of freedom will be: \[ df \approx \frac{\left( \frac{s_1^2}{n_1} + \frac{s_2^2}{n_2} \right)^2}{\frac{\left( \frac{s_1^2}{n_1} \right)^2}{n_1 - 1} + \frac{\left( \frac{s_2^2}{n_2} \right)^2}{n_2 - 1}} \] Give me the following values and enjoy with the MLC result.

  Sample 1	Sample 2
  Equal Variance Not Equal Variance

  Standard Error (δ̅x1 - ̅x2)	--
  Degrees of Freedom	--



• 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!

• Calculate the Margin of Error (\(M_{ε}\)):
  The value margin error determines the widith of the confidence interval.
  \[M_{ε} = (t_{\alpha/2}) * (δ_{\bar{x}_1 - \bar{x}_2})\]
• 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:

• Assuming Equal Variances Incorrectly: One common mistake is assuming that the variances of the two populations are equal when they are not. This can lead to incorrect confidence intervals.
• Using the Wrong t-Distribution: When population variances are unknown, you should use the t-distribution rather than the standaed normal distribution, especially for small sample sizes.
  
  Using the z-distribution instead of the t-distribution can lead to incorrect intervals, especially when sample sizes are small.
• Incorrect Degrees of Freedom: When variances are assumed to be equal, the degrees of freedom is typically \(n_1 + n_2 - 2\)
  
  When variances are not assumed to be equal, you need to use the Welch-Satterthwaite equation to estimate the degrees of freedom, which is more complex and often a source of error.
• Not Checking Assumptions: Both methods assume that the samples are independent and normally distributed. Not checking these assumptions can lead to incorrect conclusions.