The critical value is the cut-off point to determine whether to accept or reject the null hypothesis for your sample distribution.
The critical value approach provides a standardized method for hypothesis testing, enabling you to make informed decisions based on the evidence obtained from sample data.
After calculating the test statistic using the sample data, you compare it to the critical value(s) corresponding to the chosen significance level (α).
The critical value(s) represent the boundary beyond which you reject the null hypothesis. You will have rejection regions and non-rejection region as follows:
Two-sided test
A two-sided hypothesis test has 2 rejection regions, so you need 2 critical values on each side. Because there are 2 rejection regions, you must split the significance level in half.
Each rejection region has a probability of α / 2, making the total likelihood for both areas equal the significance level.
In this test, the null hypothesis H0 gets rejected when the test statistic is too small or too large.
Left-tailed test
The left-tailed test has 1 rejection region, and the null hypothesis only gets rejected when the test statistic is too small.
Right-tailed test
The right-tailed test is similar to the left-tailed test, only the null hypothesis gets rejected when the test statistic is too large.
Now that you understand the definition of critical values, let’s look at how to use critical values to construct a confidence interval.
Using Critical Values to Construct Confidence Intervals
Confidence Intervals use the same Critical values as the test you’re running.
If you’re running a z-test with a 95% confidence interval, then:
- For a two-sided test, The CVs are -1.96 and 1.96
- For a one-tailed test, the critical value is -1.65 (left) or 1.65 (right)
To calculate the upper and lower bounds of the confidence interval, you need to calculate the sample mean and then add or subtract the margin of error from it.
Lower Bound = Sample Mean - Margin of Error
Upper Bound = Sample Mean + Margin of Error
To get the Margin of Error, multiply the critical value by the standard error:
Standard Error = Standard Deviation / √(Sample Size)
Margin of Error = Critical Value * Standard Error
Let’s see an example. Suppose you are estimating the population mean with a 95% confidence level.
You have a sample mean of 50, a sample size of 100, and a standard deviation of 10. Using a z-table, the critical value for a 95% confidence level is approximately 1.96.
Calculate the standard error:
Standard Error = 10 / √(100) = 1
Determine the margin of error:
Margin of Error = 1.96 * 1 = 1.96
Compute the lower bound and upper bound:
Lower Bound = 50 - 1.96 = 48.04
Upper Bound = 50 + 1.96 = 51.96
The 95% confidence interval is (48.04, 51.96). This means that we are 95% confident that the true population mean falls within this interval.
Finding the Critical Value
The formula to find critical values depends on the specific distribution associated with the hypothesis test or confidence interval you’re using.
Here are the formulas for some commonly used distributions.
Standard Normal Distribution (Z-distribution):
The critical value for a given significance level (α) in the standard normal distribution is found using the cumulative distribution function (CDF) or a standard normal table.
Critical Value = z(α)
z(α) represents the z-score corresponding to the desired significance level α.
Student’s t-Distribution (t-distribution):
The critical value for a given significance level (α) and degrees of freedom (df) in the t-distribution is found using the inverse cumulative distribution function (CDF) or a t-distribution table.
Critical Value = t(α, df)
t(α, df) represents the t-score corresponding to the desired significance level α and degrees of freedom df.
Chi-Square Distribution (χ²-distribution):
The critical value for a given significance level (α) and degrees of freedom (df) in the chi-square distribution is found using the inverse cumulative distribution function (CDF) or a chi-square distribution table.
Critical Value = χ²(α, df)
where χ²(α, df) represents the chi-square value corresponding to the desired significance level α and degrees of freedom df.
F-Distribution:
The critical value for a given significance level (α), degrees of freedom for the numerator (df₁), and degrees of freedom for the denominator (df₂) in the F-distribution is found using the inverse cumulative distribution function (CDF) or an F-distribution table.
Critical Value = F(α, df₁, df₂)
F(α, df₁, df₂) represents the F-value corresponding to the desired significance level α, df₁, and df₂.
As you can see, the specific formula to find critical values depends on the distribution and the parameters associated with the problem at hand.
Usually, you don’t calculate the critical values manually as you can use statistical tables or statistical software to determine the critical values.
I will update this tutorial with statistical tables that you can use later.
Conclusion
The critical value is as a threshold where you make a decision based on the observed test statistic and its relation to the significance level.
It provides a predetermined point of reference to objectively evaluate the strength of the evidence against the null hypothesis and guide the acceptance or rejection of the hypothesis.
If the test statistic falls in the critical region (beyond the critical value), it means the observed data provide strong evidence against the null hypothesis.
In this case, you reject the null hypothesis in favor of the alternative hypothesis, indicating that there is sufficient evidence to support the claim or relationship stated in the alternative hypothesis.
On the other hand, if the test statistic falls in the non-critical region (within the critical value), it means the observed data do not provide enough evidence to reject the null hypothesis.
In this case, you fail to reject the null hypothesis, indicating that there is insufficient evidence to support the alternative hypothesis.