Distribution Introduction

What is a distribution curve? Generically, it shows the probability of a value falling between any two numbers. Many distributions follow a bell-shaped curve, with the peak in the center. The area under a distribution curve represents the probability of a value falling within the range. As you can see in Figure 1, even though the range of each shaded area is 15, the probability that a value falls within each range varies depending on the size of the shaded area. So, when the curve is higher, the x-values are more likely to occur. It is by these areas, not by the y-values directly, that a distribution curve represents the frequency distribution of the x-values.

t-test introduction

A two-sample t-test is an inferential test that determines if there is a significant difference between the means of two data sets. In other words, this t-test decides if the two data sets come from the same population (Figure 2A) or from different populations (Figure 2B). For example, imagine testing the blood pressure side effects of a new drug. The mean systolic blood pressure of a group of 20 people not administered the drug (given a placebo) was 120.8 with a standard deviation of 11.2, and the mean systolic blood pressure of another group of 20 people who received the actual drug therapy was 130.6 with a standard deviation of 13.4. Do these two sample means represent Case I, with the samples coming from one population, OR is the difference in the means large enough to indicate they come from two different populations (Case II)? A t-test uses probability to decide between these two cases.

Case I represents the null hypothesis (HO: µ1 = µ2) indicating that the mean of group one equals the mean of group two; both samples come from the same population. This would signify that the drug had no effect on blood pressure. The difference in the means is small, suggesting that they come from the same population. Case II represents the alternate hypothesis (HA1≠ µ2), indicating that the mean of group one does not equal the mean of group two; the two sample means are from different populations. The difference in the means is too large to come from one population in most cases. Hence the means are probably coming from two different populations. A t-test decides which of these hypotheses to accept.

In Figure 2B, the difference in the sample means is larger, therefore, it is likely that the means come from two different populations. However, look at Figure 2C. It is possible that the two means could come from the same population and have the same difference. It is not likely because the probability (area under curve) of getting a small sample mean (x1) or a large sample mean (x2) from population 1 is small. If you accept the alternate hypothesis (HA1≠ µ2), indicating the means come from two different populations (Case II); it is more likely you will be correct. But you could be wrong. There is not a high probability, but the null hypothesis (HO: µ1 = µ2) could be true (Case III). How many times out of 100 are you willing to be wrong?

Alpha Level (α)

An alpha level represents the number of times out of 100 you are willing to be incorrect if you reject the null hypothesis. If you choose an alpha level of 0.05, 5 times out of 100 you will be incorrect if you reject the null hypothesis. Those five times, both means would come from the same population (Case III). But that's about it. 95 times out of 100, you will be correct because it is more likely that the means come from two different populations (Case II). The difference in the means is large enough that it is most likely that the means come from two different populations.

t-distribution's relation to t-test

For the t-test, as in all hypothesis testing, the computations are done assuming the null hypothesis is true. The t-distribution's curve represents the distribution of the differences of means around 0. Why? Well (again, assuming the null hypothesis is true), if the difference in means is too far from 0 to be likely, the null hypothesis is then rejected and the alternative hypothesis is accepted. (The alpha level discussed above sets the level at which the results are unlikely enough to reject the null hypothesis).

To actually perform a t-test, the difference of the two sample means is used to compute the t-statistic. The method of computing this value can be found by clicking here. The t-statistic value for the drug study turns out to be 2.51. Next, the t-critical value will be determined by statistical software. To find it, only α and the degrees of freedom (df) are needed. The df is simply the number of data points in both data sets minus 2 and is needed because there is a (slightly) different t-distribution curve for each df. The t-critical value is the cutoff between retaining or rejecting the null hypothesis. Whenever the t-statistic is farther from 0 than the t-critical value, the null hypothesis is rejected; otherwise, the null hypothesis is retained. For the drug study, df is 38 and the t-critical value is 2.33 if the alpha level is 0.05. The t-critical and t-statistic values are x-values on the graph of the t-distribution, as you can see in Figure 4. If the t-statistic value is greater than the t-critical, meaning that it is beyond it on the x-axis (a blue x), then the null hypothesis is rejected and the alternate hypothesis is accepted. However, i
f the t-statistic had been less than the t-critical value (a red x), the null hypothesis would have been retained.

P-values

Instead of comparing the t-critical and t-statistical values to determine significant difference, you may also compare the alpha level and p-values. In Figure 4, the alpha level would be the area under the curve to the right of the positive t-critical and to the left of the negative t-critical (all gray and light blue). Together, these areas total the alpha-level, 0.05. The p-value is the area under the curve to the right of the purple t-statistic plus the area to the left of the negative, purple t-statistic (the light blue only). For the drug study, this area equals 0.0329. Because the p-value is then less than the alpha level, the alternate hypothesis is accepted. However, if the p-value was greater than the alpha level, p>α, (the blue covered the gray), the null hypothesis would be retained.

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