Critical Values, Test Statistic, & p-Values

~By far, these are the most troublesome & confusing concepts for beginning students in elementary statistics.

~They are found in different ways depending on whether we are using the standard normal curve or a t-curve.  However, their meaning basically remains the same in both cases.

~Let's look at the basic meaning for a critical value(s).

~When a hypothesis test is performed, a significance level is stated. This is alpha (a).  Based on the significance level & the nature of the test (one or two tailed test), rejection regions are established.

~At the boundary of each rejection region, on the horizontal axis (z or t value), is the critical value(s). So, the significance level determines your critical values.  They are fixed, if using the standard normal curve, but vary, if using t-curves (depends on your t-curve).

~For example, for a significance level of .05, a hypothesis test (using the standard normal curve) will give z=1.645 (for a right-tailed test), z= -1.645 (for a left-tailed test), or z=1.960 & z= -1.960 (for a two-tailed test). These can be found easily using a TI-83 using 2nd Vars, menu 3, by inserting (1-.05), (.05), & (1-.05/2) respectively.

~If we are using t curves instead of the standard normal curve, the critical value(s) will depend on the significance level and on the specific
t-curve we are using (selected by the degrees of freedom, which is n-1).

~Unfortunately, the TI-83 does not give these directly, however, there is an indirect way of getting them for a given confidence level and sample size.
Here's how: For example, say you would like the positive t critical value where the area in the right tail is .03. That would be consistent with a 94% confidence level in a T interval. There will .03 in the left tail, as well. Now, go to Stat, tests, menu 8 (TInterval) and enter 0 for the mean, and √n for the sample standard deviation, where n is the sample size. Enter .94 for the confidence level. Note that the df=n-1. Then find this T interval. It will show the two critical t-values at each end of the interval. This method is based on the nature of a confidence interval written as (mean-E, mean+E) where E is the margin of E (for t-values). (see formula 3 below)

They can also be located by table (t-distribution). To find them, simply select the proper curve (degrees of freedom) in the left column, then read over under the proper heading (determined by the significance level). The significance level will dictate the amount of area in one or both tails.

~For example, if we are using a t-curve with n=20, the degrees of freedom would be n-1=19.  For the significance level of .05 with a right-tailed test (all the area in one tail), we would locate df=19, then read over (under Area in one tail of .05), to get the critical value of t=1.729.

~You will notice that the critical value in the very last row (large sample size) agrees with the value calculated using the standard normal curve, since, as the sample size increases, t-curves approach the standard normal curve.  Interesting observation.

~However, you really don't need to find critical values when performing a hypothesis test (using the p-value method). All you need to do is to compare the p-value to the significance level to make your decision.

~The Test Statistic is a value coming from our sample. If we are using the standard normal curve, it is a z-value.  For a t-curve, it is a t-value.

~Formulas for the Test Statistic:  All based on Z = (x-m)/s

1) For Proportions: z=(x-np)/√(npq)=[(x/n)-p]/[√(npq)]/n

                                  = (p cap- p_o)/√(p_oq_o/n), where p_o is your null hypothesis

2) For Means (s known):  z=(x bar-m_o)/[s/√(n)]

3) For Means (s unknown):  t = (x bar - m_o)/[S/√(n)], where m_o comes from the null hypothesis in both cases

~The location of this test statistic is the key. If it falls inside of a rejection region (determined by the significance level), then we will reject the null hypothesis. If not, we do not reject the null hypothesis.

~The most common mistake beginning students make is that confuse the Critical Value with the Test Statistic. The Critical Value(s) come from the signifance level, but the Test Statistic comes from our sample.

~The area to the right of the test statistic (right-tailed test) is the p-value. That's why we reject the null hypothesis if the p-value is less than the significance level, since, in this case, the test statistic will lie to the right of the critical value.  (area to the right of the critical value is the significance level, for this case). (area is in one tail)

~For a left-tailed test, the area to the left of the test statistic would be the p-value. (area is in one tail)

~For a two-tailed test, the p-value would be TWICE the area to the right or left of the test statistic (depending whether the test statistic is positive or negative, respectively). (area is in two tails)

~The TI-83 gives us an easy way to find the Test Statistic & p-value.
Simply choose the proper test (either a Z-test or T-test) (from Stat, tests, menu 1 or 2) & fill in the necessary information, then calculate. The display will show the test statistic & p-value underneath it.

~For t-curves, if all you know is the sample size and the test statistic and would like to know the p-value, the TI-83/84 Plus calculator will find it, go to 2nd Vars, menu 5, tcfd and find the area to the right or left of the test statistic for one-tailed tests or twice the area to the right or left of the test statistic for a two-tailed test. The techniques used here are similar to those used for finding areas under the standard normal curve , however, there will be 3 inputs. For example, for a right tailed test, insert the test statistic, 10000, degrees of freedom. 

~Example: for a right-tail test for a sample size of n=21 & a test statistic of t=3.500, enter tcdf(3.500, 10000, 20).

~If you were to use a t-distribution table to find the p-value, it could give you some problems.  Namely, the test statistic may not be listed in the heart of the table. So, you'll have to estimate it by looking at values on either side of it.  Then, read the heading for a range of areas for these estimated values.  That will give you a range for the p-value. In most cases, that should be enough to decide whether or not the p-value is less  than the significance level (that way you can still make your decision).

~There are cases where the test statistic is too small to be listed. In that case, the p-value would be greater than the largest area heading on the right. Similarly, if the test statistic is too large to be in the table, the p-value would be smaller than the smallest area heading on the left.