Critical values define specific regions within the sampling distribution for test statistics. It plays a key role in hypothesis testing and confidence intervals. In hypothesis testing, the critical value is used to decide if a result is statistically significant or not. In confidence intervals, critical values help to calculate the upper and lower limits.
In this article, we will explore critical values, delving into their definition, various types, and the techniques used to compute them. We will solve some examples of critical value to make this concept easier for our readers.
In statistics, the critical value is the value that is compared to test statistics in hypothesis testing to identify whether the null hypothesis should be rejected or not. The null hypothesis will not be rejected if the test statistic value is less than the critical value. Reject the null hypothesis when the value of the test statistic is greater than the critical value.
Graphically, the critical value represents the boundary between the rejection region and the non-rejection region on a probability distribution. The rejection region is the area in the tails of the distribution where the test statistic is unlikely to occur if the null hypothesis is true. Non-rejection region refers to the zone in the middle of the distribution where the test statistic is likely to occur under the null hypothesis.
The formula for calculating critical values depends on the nature of the statistical test and the distribution involved. We can use either the confidence interval or the significance level to calculate the critical value. In the next selection, we will discuss different formulas for finding critical values.
Here is a general procedure for calculating critical values:
There are four main types of critical values:
iii. F - Critical value
Let’s learn how to evaluate these types with easy steps.
o Degree for numerator = n1 – 1
o Degree for denominator = n2 – 1
Here are some solved examples of finding critical values for different statistical tests:
Evaluate the t-critical value for a one-tailed test when the level of confidence is 99% and the sample size is 9.
Step 1: Find the value of alpha.
Alpha level = 1 – (99 /100)
= 1 – 0.99
Step 2: Find the degree of freedom by subtracting 1 from the sample size.
Degree of freedom (df) = 9 – 1 = 8
Step 3: Observe the t-distribution table for the one-tailed test.
Step 4: Find the degree of freedom at the leftmost column and alpha at the top row of the table. Take the intersection of this row and column, this value will be our t- t-critical value.
The value of alpha and degree of freedom intersect at 2.8965. Therefore, the - Critical value for a one-tailed test is 2.8965
Evaluate the z-critical value for a left-tailed test when the alpha level is 0.452.
Step 1: As the test is left-tailed, subtract alpha from 1.
1 – 0.452 = 0.548
Step 2: Find in 0.548 in z-distribution table (left-tailed test)
Step 3: Sum the values of intersecting row (top) and column (most left).
0.1+0.02 = 0.12
It is a left-tailed test, thus multiply it by -1.
Thus, -0.12 is the z-critical value for the left-tailed test.
In this article, we have explained the importance of critical values in hypothesis testing and confidence intervals. We discussed how to calculate them, involving factors like significance levels, test distributions, and test tails. We covered four main types (T, Z, F, and Chi-square) with practical calculation steps and provided examples for clarity.