A researcher has collected two samples of data, and the sample variances are 2.16² and 4.82². It would be appropriate to use the two-sample t-test with a common variance.

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The purpose of a two-sample t-test is to determine if the difference between the two sample populations is statistically significant—essentially asking if the two samples were taken from the same large population, and therefore are both accurate predictors for the study, or if they are subdivided to a point where they are no longer interchangeable.

Let's say these statistics generated information on how many minutes it takes to receive food at two separate local restaurants. The first thing is to identify the null and alternate hypotheses—and they are standard for every T-test. The null hypothesis is that the time to receive food is the same (we'll say 25 minutes), while the alternate hypothesis is that they are not the same—the mean waiting time of restaurant A is different than 25 minutes, the mean waiting time of restaurant B, or vice versa.

The next step is to find the pooled standard deviation, which combines the standard deviation (or variances) of both of the samples. Since we don't know the sample size for each group, we'll have to make some assumptions or put it algebraically.

Pooled variance, or Vp, is defined as [(m - 1)*V1 + (n - 1)*V2]/(m + n - 2),where m is the size of sample A, n is the size of sample B, and V1 and V2 are the corresponding variances.

Once we know that number, we can begin the two-sample T-test to determine if they are the same. To calculate the t-statistic, you subtract the null hypothesis (the difference of the means is 0) from the difference in the sample means and divide that number by the adjusted standard deviation.

Adjust standard deviation is the square root of the pooled variance multiplied by the square root of the sum of the inverses of the different sample sizes.

Once this is complete, you will find the t-score which gives the correlated p-value from the chart, which you can find on the included link. Once you know that value, you can decide whether or not to accept the null hypothesis.

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