Hypothesis Testing and Estimation: Null and Alternative Hypothesis, Type I and II Errors, Z Test, T Test and Chi-Square for SSC

Statistical inference lets us draw conclusions about a population from a sample, using estimation and testing of hypotheses. These are advanced topics in the SSC CGL Statistics syllabus, but the questions are usually formula-based and the key terms are easy to learn. This guide explains the hypotheses, the errors, the common tests, the critical values and the confidence interval, with solved examples and practice questions.

Estimation

• Point estimate: a single value used to estimate a population parameter (for example the sample mean for the population mean).
• Interval estimate (confidence interval): a range of values likely to contain the parameter.
• Confidence interval for the mean: x̄ ± z × (σ/√n).
• For a 95% confidence level z = 1.96, and for 99% z = 2.58.
• A good estimator is unbiased, consistent, efficient and sufficient.

Terms in Hypothesis Testing

Type I and Type II Errors

Steps in Testing a Hypothesis

1. State the null and alternative hypotheses.
2. Choose the level of significance (α).
3. Select the test statistic (z, t, chi-square or F).
4. Find the critical value and the critical region.
5. Calculate the value of the test statistic from the sample.
6. Compare and decide: reject H₀ if the calculated value falls in the critical region.

Common Tests

Critical Values of z

Solved Examples

Example 1. A sample of 100 has a mean of 52. The population standard deviation is 10. Test H₀: μ = 50 against H₁: μ ≠ 50 at the 5% level.
z = (52 − 50)/(10/√100) = 2/1 = 2. Since 2 > 1.96, reject H₀.

Example 2. Find the 95% confidence interval for the mean in Example 1.
52 ± 1.96 × 1 = (50.04, 53.96).

Example 3. Two observed frequencies are 18 and 22, and the expected frequencies are 20 each. Find χ².
(18 − 20)²/20 + (22 − 20)²/20 = 4/20 + 4/20 = 0.4.

Example 4. A sample of 16 items is tested with the t test. How many degrees of freedom are there?
n − 1 = 15.

Example 5. If the level of significance is 5%, what is the probability of a Type I error?
0.05.

Tricks

• Type I error = reject a true H₀ (false alarm); Type II = accept a false H₀ (missed detection).
• If |z| calculated > z critical, reject H₀.
• Use the t test for small samples and the z test for large samples.
• The degrees of freedom of the t test for one sample = n − 1.
• A smaller level of significance makes it harder to reject H₀.

Quick Tips

• Learn the definitions of Type I and Type II errors.
• Learn the critical values 1.96, 2.58 and 1.645.
• Use the t test when n < 30 and σ is unknown.
• Remember that the chi-square statistic is Σ(O − E)²/E.

Quick Revision Points

• Type I error probability = α.
• Power of a test = 1 − β.
• 95% CI uses z = 1.96; 99% CI uses z = 2.58.
• t test: degrees of freedom = n − 1.
• Chi-square = Σ(O − E)²/E.

Practice Questions With Answers

1. What is the error of rejecting a true null hypothesis called?
   Answer: Type I error
2. What is the critical value of z for a 5% two-tailed test?
   Answer: 1.96
3. What is the power of a test?
   Answer: 1 − β
4. Which test is used for a small sample with unknown σ?
   Answer: t test
5. A sample of 25 has x̄ = 40, μ = 38 and σ = 5. Find z.
   Answer: 2
6. What is the degrees of freedom for a t test with n = 20?
   Answer: 19
7. Which test is used for goodness of fit?
   Answer: Chi-square test
8. If α = 0.01, what is the Type I error probability?
   Answer: 0.01
9. What is the 99% confidence level z value?
   Answer: 2.58
10. What is the statement of no difference called?
    Answer: Null hypothesis

How to Practise This Chapter

Practise 10 questions every day and write the steps on paper. Then attempt timed sets from our practice papers and the previous year papers page. See all topics on the Statistics notes page, read the previous topic Sampling Theory and Methods.

Frequently Asked Questions

What is a null hypothesis?

A statement of no difference or no effect, which is tested.

What is an alternative hypothesis?

The statement accepted when the null hypothesis is rejected.

What is a Type I error?

Rejecting a true null hypothesis.

What is a Type II error?

Accepting a false null hypothesis.

What is the level of significance?

The probability of a Type I error, usually 5% or 1%.

What is the z test?

A test for the mean when the sample is large or the population standard deviation is known.

What is the t test?

A test for the mean when the sample is small and the population standard deviation is unknown.

What is the chi-square test?

A test for goodness of fit and independence of attributes.

What is a confidence interval?

A range of values within which the population parameter is likely to lie.

What is the critical value for a 5% two-tailed z test?

±1.96.

What is the p-value?

The probability of obtaining the observed result or more extreme results if the null hypothesis is true.

Is this topic asked in SSC?

Yes, in the Statistics paper of SSC CGL.