CMSC 320
2026
Fardina F Alam
Hypothesis Testing: Critical Region and P-Value
(PART02)
Topics we will cover
Making the Decision
After choosing the significance level (α) and calculating the test statistic:
Use one of two methods:
1. Critical Value Method
Compare test statistic to critical value (cutoff)
2. p-value Method
Compare p-value to α
Final Decision Reject H₀ or Fail to reject H₀
Key Idea Is the result extreme enough to doubt H₀?
Critical Region & Critical Value
Critical Region (Rejection Region)
The range of test statistic values where we reject the null hypothesis because the data is too extreme.
Size depends on significance level (α)
Critical Value the boundary/cutoff point separating the critical region from other values—crossing it means rejecting the null hypothesis.
Critical Region & Critical Value
Example (Two-tailed test, α = 0.05)
Critical values: −1.96 and +1.96 Test statistic > 1.96 or < −1.96 → Reject H₀
The figure shows how the critical values mark the boundaries of two rejection regions (shaded in pink).
Note: we considered critical value 1.96 here. The critical value can be found using the appropriate table or statistical software for the distribution and degrees of freedom. THIS COURSE FOCUSES ON UNDERSTANDING THE CONCEPT RATHER THAN THE CALCULATION OF THE CRITICAL VALUE.
Critical Region Boundaries
Compare test statistic vs. critical value(s)
The critical region depends on:
The choice is determined by the research hypothesis (direction of effect).
Example: Human Heights
Ques: If an alien abduct someone, what is our most likely height?
Suppose we're investigating whether aliens are abducting humans based on their height.
If heights are normally distributed, the most likely human we got is the mode.
Set up: Null Hypothesis (H0): The avg height of abducted humans is just like the most common height in the human population.
Question: Based on the setup of the null hypothesis, where would we expect to observe rejection of the null hypothesis in our analysis?
Consider the extreme regions representing test statistic values that are unlikely to occur (far from the mean) if the null hypothesis is true.
Is it a one-tailed test or a two-tailed test?
Hypothesis tests can be one-tailed or two-tailed, depending on the alternative hypothesis Ha.
Does the alternative hypothesis (Ha) propose a deviation in either direction (not equal)?
Is Ha concerned with positive effects or right tail of the distribution(Greater Values )?
Use a two-tailed test
Use a one-tailed upper-tail test (Greater or Positive Effect)
Use a one-tailed lower-tail test (lesser or negative Effect)
Yes
No
Yes
No
One-Tailed Test
Two-Tailed Test
Example:� If α=0.05 → each tail = 0.025 → two critical values
Purpose: Test whether the parameter differs significantly in either direction.
One-tailed vs Two-tailed test
One-tailed vs Two-tailed test
Left-tailed/lower-tailed test: critical value and rejection region will be in the left tail of the distribution
Right-tailed/upper-tailed test: critical value and rejection region will be in the right tail of the distribution
For one tailed: If the sample being tested falls into the one-sided critical area, the alternative hypothesis will be accepted instead of the null hypothesis.
** μ is the true (unknown) population mean, while μ0 is the specific mean value assumed in the null hypothesis for comparison.
Two-tailed test: critical values and rejection regions are split between both the left and right tails of the distribution.
Example
One Tailed: Testing if a new drug increases blood pressure (right-tailed) or decreases blood pressure (left-tailed) compared to a baseline value
H0: μ = μ0
Two Tailed: Testing if a new drug has an effect on blood pressure, without specifying whether the effect is an increase or a decrease.
The mean blood pressure with the new drug is equal to the baseline mean.
The mean blood pressure with the new drug is different from the baseline mean (either higher or lower).
Ha: μ ≠ μ0
Example: Critical Value
You are testing whether the average weight of a batch of apples is different from 150 grams (μ). You will use a 1% significance level (α = 0.01), and you assume the population standard deviation (σ) is 10 grams. You collect a sample of 30 apples and find that the sample mean (x’) is 152 grams.
Ho: μ = 150
H1: μ ≠ 150
α = 0.01
σ=10
Choose the Appropriate Test and Test Statistics?
Determine the Critical Value: Z₁ = -2.58 and Z₂ = +2.58
(check previous slide- decided using Z table )
Since 1.095 is within the range of -2.576 to +2.576, we fail to reject the null hypothesis.
How to Find a Critical Value
Unfortunately, the formulas for finding critical values are very complex. Typically, you don’t calculate them by hand. You can use some statistical software or statistical tables (Z-table, T distribution table, Chi-square table, F-table) to find them.
ALTERNATIVE: P-Value (Another approach to evaluating the significance of test results.)
(2) P-Value
Recap
Recall: The Critical Value Method:
Now Come Alternative Approach to help us assessing the evidence against the null hypothesis (H₀): P-Value Method
The Use of P –Values in Decision Definition Making
The p-value approach: A method in hypothesis testing where we calculate how likely the observed result (or something more extreme) is if the null hypothesis is true.
Smaller p-values indicate stronger evidence against the null hypothesis.
Compare p-value with significance level α
Key Idea The p-value tells how strong the evidence is against the null hypothesis.
P-Value Procedure
Make a Decision:
P-Value - Example
Suppose you are testing the following hypothesis at a significance level (α) of 5% and you got the p-value as 3%, and your sample statistic (sample mean) is x̄ = 25
H₀: μ = 20 (The population mean is 20)
H₁: μ > 20 (The population mean greater than 20)
Given, α=5%, we are fine to reject our null hypothesis 5 out of 100 times even though it is true.
P-value is 3% which is less than α
0.03 < 0.05 → We reject the null Hypothesis (extremely strong evidence against the null hypothesis.)
What about P-Value is 6%?
Example: Now use P Value Approach
You are testing whether the average weight of a batch of apples is different from 150 grams (μ). You will use a 1% significance level (α = 0.01), and you assume the population standard deviation (σ) is 10 grams. You collect a sample of 30 apples and find that the sample mean (x’) is 152 grams.
Ho: μ = 150
H1: μ ≠ 150
α = 0.01
σ=10
Find P-Value: What is the p-value that corresponds to this z-score?
From the Z-table or calculator, the area to the right of Z = 1.095 is about 0.1368 (NO NEED TO FIND THIS DURING EXAM, THIS WILL BE GIVEN).
Since it's a two-tailed test, P = 2 * 0.1368 =0.274
Since P-Value (0.2736) > α (0.01), we fail to reject Ho
WRITING NULL AND ALTERNATIVE HYPOTHESIS
The school board claims that atleast 60% of students bring a phone at school. A teacher believe that this number is too high and randomly samples 25 students to test at at level of significance of .02
Write down H0, H, C, alpha.
PRACTICE
NEXT PART:
Different Types of Statistical Tests