Calculation of population Means of two independent sample

Confidence Interval to Estimate μ₁ − μ₂

In a hypothesis test, when the sample evidence leads us to reject the null hypothesis, we conclude that the population means differ or that one is larger than the other. An obvious next question is how much larger? In practice, when the sample mean difference is statistically significant, our next step is often to calculate a confidence interval to estimate the size of the population mean difference.

The confidence interval gives us a range of reasonable values for the difference in population means μ₁ − μ₂. We call this the two-sample T-interval or the confidence interval to estimate a difference in two population means. The form of the confidence interval is similar to others we have seen.

                (samplestatistic)±(marginoferror)

               (samplestatistic)±(criticalT−value)(standarderror)

Sample Statistic

Since we’re estimating the difference between two population means, the sample statistic is the difference between the means of the two independent samples: \[x_1-x_2\]

Critical T-Value

The critical T-value comes from the T-model, just as it did in “Estimating a Population Mean.” Again, this value depends on the degrees of freedom (df). For two-sample T-test or two-sample T-intervals, the df value is based on a complicated formula that we do not cover in this course. We either give the df or use technology to find the df.

Standard Error

The estimated standard error for the two-sample T-interval is the same formula we used for the two-sample T-test. (As usual, s₁ and s₂denote the sample standard deviations, and n₁ and n₂ denote the sample sizes.)

           \[ \sqrt{{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}}\] 

  Putting all this together gives us the following formula for the two-sample T-interval.

      \[(x_1-x_2)± T\ \sqrt{{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}}\]

For Example:

 \[x_1\]=850

     \[x_2\]    =719

        T=1.6790

     SE(standard error)=72.47

\[X_1+X_2\]±T* standard error=(850−719)±(1.6790)(72.47)≈131±122

expressing this an interval gives us(9,253)

Assumptions 

In order to test whether there is a difference between population means, we are going to make three assumptions:

1. The two populations have the same variance. This assumption is called the assumption of homogeneity of variance.
2. The populations are normaly distributed
3. Each value is sampled independentely from each other value. This assumption requires that each subject provide only one value. If a subject provides two scores, then the scores are not independent.