Independent Samples -The Difference of the Means

Independent Means:

Sums and Differences of Independent Variables:

Independent variables can be joined to form new variables. The variance and mean of the combination can be found from the variances and means of the original variables.

Combination of Variables      In English (Melodic Mathematics)
μx+y = μx + μy                     The mean of sum is the sum of means.
μx-y = μx - μ                      The mean of a difference is the difference of means.
σ2x+y = σ2x + σ2y               The variance of sum is the sum of variances.
σ2x-y = σ2x - σ2y                 The variance of difference is the sum of variances.

The Difference of the Means: μ1 – μ2

As we are joining two variables by subtraction, the most important rules from the table above is that the mean of difference is the difference of the means and variance of difference is the sum of variances.

It is significant to note that the variance of difference is the sum of variances, not the standard deviation of difference is the sum of standard deviations. Whenever we go to determine the standard error, we must combine variances to do so. Also, you are probably wondering why the variance of difference is the sum of variances rather than the difference of variances. As the values are squared, the negative associated with second variable becomes positive, and it becomes the sum of variances. As well, variances cannot be negative, and when you took the difference of variances, it could be negative.

Population Variances Known:

Whenever the population variances are known, the difference of means consists of a normal distribution. The variance of difference is the sum of variances divided by sample sizes. This makes sense, optimistically, as according to the central limit theorem, the variance of sampling distribution of the sample means is the variance divided by the sample size, so what we are doing is add up the variance of each mean altogether. The test statistic is as shown below.

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Population Variances Unknown, however both sample sizes large:

Whenever the population variances are not known, the difference of means consists of a Student's t distribution. Though, when both sample sizes are big enough, then you will be utilizing the normal row from t-table, thus your book lumps this below the normal distribution, instead of t-distribution. This provides us the chance to work with the problem devoid of knowing when the population variances are equal or not. The test statistic is as shown, and is similar to above, apart from the sample variances are used rather than the population variances.

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Population Variances Unknown, unequal with small sample sizes:

Now the question that arises in your mind is that how do you know if the variances are equal or not if you do not know what they are. Some of the books teach F-test to test the equality of two variances, and when your book does that, and then you must use the F-test to see. The other books (that is, statisticians) argue that if you do the F-test first to see if the variances are equivalent and then employ the similar level of significance to perform the t-test to test the difference of the means, that the overall level of significance is not same. Therefore the Bluman text states the student whether or not the variances are equivalent and the Triola text.

As you do not know the population variances, you are going to be employing a Student's t-distribution. As the variances are unequal, there is no attempt made to average them altogether as we will in the next condition. The degree of freedom is the smaller of two degrees of freedom (n-1 for each). The ‘min’ function means take the minimum of the two values. Or else, the formula is similar as we employed with big sample sizes.

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Population Variances Unknown however equal with small sample sizes:

When the variances are equivalent, then an effort is made to average them altogether. Now, equal doesn’t mean similar. This is possible for two variances to be statistically equivalent however be numerically distinct. We will determine a pooled estimate of variance that is simply the weighted mean of variance. Weighting factors are the degrees of freedom.

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Once the pooled estimation of the variance is computed, this mean (or average) variance is utilized in the place of individual sample variances. Or else, the formula is similar as before. The degree of freedom is the sum of individual degrees of freedom.

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