What is Rejection Region?

S3-SA3-0344

Grade Level:

Class 9

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

The Rejection Region, also known as the critical region, is a set of values for the test statistic in hypothesis testing that leads to rejecting the null hypothesis. If our calculated test statistic falls into this region, it means the observed data is very unlikely if the null hypothesis were true, so we decide to reject it.

Simple Example

Quick Example

Imagine you're checking if a new cricket bat helps batsmen score more runs. You set a 'rejection region' for average runs per match. If the new bat helps batsmen score, say, 20% or more runs on average (which is very high), then this 20% or higher range becomes your rejection region. If your experiment shows batsmen with the new bat score 25% more runs, you reject the idea that the new bat makes no difference.

Worked Example

Step-by-Step

PROBLEM: A mobile company claims its new phone battery lasts 10 hours on average. You suspect it's less. You test 30 phones and find the average battery life is 9.5 hours with a standard deviation of 1.5 hours. You decide to reject the company's claim if the average battery life is significantly less than 10 hours at a 5% significance level (one-tailed test).

Step 1: State the Null (H0) and Alternative (H1) Hypotheses.
H0: Average battery life (μ) = 10 hours
H1: Average battery life (μ) < 10 hours

Step 2: Choose the significance level (α).
α = 0.05 (5%)

Step 3: Determine the critical value for a one-tailed Z-test at α = 0.05.
For a left-tailed test at α = 0.05, the critical Z-value is approximately -1.645.

Step 4: Define the Rejection Region.
The rejection region is Z < -1.645.

Step 5: Calculate the test statistic (Z-score).
Z = (sample mean - population mean) / (standard deviation / sqrt(sample size))
Z = (9.5 - 10) / (1.5 / sqrt(30))
Z = -0.5 / (1.5 / 5.477)
Z = -0.5 / 0.274
Z = -1.825

Step 6: Compare the test statistic with the critical value.
Our calculated Z = -1.825.
The critical Z = -1.645.
Since -1.825 < -1.645, our calculated Z-score falls into the rejection region.

Step 7: Make a decision.
We reject the null hypothesis.

ANSWER: The rejection region for this problem is Z < -1.645. Our calculated Z-score of -1.825 falls within this region, leading us to reject the company's claim.

Why It Matters

Understanding the rejection region is crucial for making informed decisions based on data, whether it's in AI/ML for training models or in Data Science for analyzing user behavior. Engineers use it to test new product designs, and economists use it to validate theories. It helps professionals like statisticians and data scientists decide if observed results are truly significant or just due to chance.

Common Mistakes

MISTAKE: Confusing the rejection region with the acceptance region. | CORRECTION: The rejection region is where you reject the null hypothesis; the acceptance region is where you fail to reject it. They are complementary.

MISTAKE: Not considering if the test is one-tailed or two-tailed when finding the critical value. | CORRECTION: For a one-tailed test, the entire significance level (alpha) is in one tail. For a two-tailed test, alpha is split equally between both tails.

MISTAKE: Incorrectly using the significance level (alpha) to find the critical value. | CORRECTION: Always use the correct alpha value (e.g., 0.05 for 5%) to look up the critical Z or T value from tables or calculate it using software.

Practice Questions

Try It Yourself

QUESTION: If the critical Z-value for a right-tailed test is 1.96, what is the rejection region? | ANSWER: Z > 1.96

QUESTION: For a two-tailed test with a significance level of 0.01 (1%), what are the critical Z-values? | ANSWER: Z < -2.576 and Z > 2.576 (approximately)

QUESTION: A teacher claims that her students score an average of 75 marks in a test. You sample 40 students and find their average score is 72 with a standard deviation of 8. If you want to test if the average score is significantly less than 75 at a 10% significance level (one-tailed), what is the critical Z-value and what is the rejection region? | ANSWER: Critical Z-value is approximately -1.28. The rejection region is Z < -1.28.

MCQ

Quick Quiz

Which of the following statements about the rejection region is TRUE?

It is the area where we accept the null hypothesis.

It contains values of the test statistic that are very likely if the null hypothesis is true.

If the test statistic falls into this region, we reject the null hypothesis.

Its size is always fixed at 5% of the total area.

The Correct Answer Is:

The rejection region is specifically defined as the range of values for the test statistic that leads us to reject the null hypothesis. Options A and B describe the acceptance region, and Option D is incorrect as the size (significance level) can vary.

Real World Connection

In the Real World

In quality control for a smartphone factory in India, engineers might use the rejection region concept. For example, if a batch of phone screens is supposed to have a defect rate of less than 1%, they might set up a hypothesis test. If their sample shows a defect rate that falls into the 'rejection region' (meaning it's significantly higher than 1%), they will reject the entire batch, preventing faulty products from reaching customers.

Key Vocabulary

Key Terms

NULL HYPOTHESIS: The statement being tested, usually that there is no effect or no difference. | ALTERNATIVE HYPOTHESIS: The statement we accept if we reject the null hypothesis, suggesting there is an effect or difference. | TEST STATISTIC: A value calculated from sample data during a hypothesis test. | SIGNIFICANCE LEVEL (ALPHA): The probability of rejecting the null hypothesis when it is actually true (Type I error). | CRITICAL VALUE: The boundary value(s) that define the rejection region.

What's Next

What to Learn Next

Now that you understand the rejection region, you should learn about 'p-value'. The p-value is another way to make decisions in hypothesis testing and is closely related to whether your test statistic falls into the rejection region. It will help you understand how strong the evidence against the null hypothesis is.