Hypothesis Testing

Dr. Thiyanga S. Talagala
Department of Statistics, Faculty of Applied Sciences
University of Sri Jayewardenepura, Sri Lanka

What is a hypothesis?

In statistics, a hypothesis is a formal statement or claim about a population parameter (like a mean, proportion, or variance) or the relationship between variables that can be tested using statistical methods.

Question 1

A researcher claims that the average monthly expenditure of female students is higher than that of male students at the University of Sri Jayewardenepura during the month of April. To evaluate this claim, he gathers expenditure data from all the students at the university for the month of April. How should he proceed to test the claim?

Why Hypothesis Testing is Usually Not Needed with Population Data?

  • Population Parameters are Known: With complete population data, you can calculate the exact values (e.g., population mean, standard deviation) instead of estimating them from a sample.

  • No Uncertainty: Hypothesis testing is used to manage uncertainty and variability introduced by sampling. With population data, there is no sampling uncertainty to account for.

Purpose of Hypothesis Testing

  • Hypothesis testing is specifically designed for making inferences about a population based on sample data.

  • The goal of hypothesis testing is to use sample data to determine whether there is enough evidence to reject the null hypothesis in favor of the alternative hypothesis, while controlling for error.

Tests of Hypotheses

Question 2

A wildlife researcher claims that the average height of one-week-old male elephants is greater than 2.0 meters. The researcher collects data from a sample of 30 male elephants. The sample has a mean height of 2.1 meters and a standard deviation of 0.5 meters. Test this claim at the 0.05 significance level.

  1. Write the null hypothesis and alternative hypothesis for this situation.

Null Hypothesis (H0): This is the statement you’re initially assuming to be true. It often represents no effect, no difference, or no relationship.

Alternative Hypothesis (H1 or Ha): This is the statement you’re trying to find evidence for. It contradicts the null hypothesis.

How did authors of statistical hypothesis tests come up with their test statistics?

Design a test statistics from scratch

Common methods

  1. Likelihood ratio method

  2. Method of moments

  3. Score (Lagrange multiplier)

  4. Ward test

  5. Bayesian methods

  6. Permutation and resampling methods

  7. Maximum likelihood estimation

  8. Method of least squares

Key Requirements

  • Test statistics are calculated from the sample data.

  • We should be able to derive its probability distribution under the null hypothesis (exactly or approximately).

In this example out test statisic is

\[T = \frac{\bar{X}-\mu}{s/\sqrt{n}}\]

Understanding the decision making process in hypothesis testing

We look at the distribution of the test statistic under the null hypothesis to understand how the test statistic behaves when the null hypothesis is true.

\[T = \frac{\bar{X}-1}{s/\sqrt{n}} \sim t_{n-1}\]

Errors in Hypothesis Testing

  • The Type I error is the probability of rejecting the null hypothesis when it is actually true.

\[\alpha = 𝑃(\text{Reject } H_0|H_0 \text{ is true})\]

  • Type II error is the probability of failing to reject the null hypothesis when the alternative hypothesis is actually true.

\[\beta = 𝑃(\text{Falling to Reject } H_0|H_0 \text{ is false})\]

This is very similar to

• False negative test = type I error

• False positive test = type II error

Significance level

\[\alpha = 𝑃(\text{Reject } H_0|H_0 \text{ is true})\]

The more serious the type I error, the smaller the significance level should be.

Power

\[\text{Power} = 1-\beta\]

The power of a test is the probability of rejecting the null hypothesis when it is false; in other words, it is the probability of avoiding a type II error.

No rejection region that will simultaneously make both \(\alpha\) and \(\beta\) small.

Factors that impact power

  1. Sample size

  2. Effect size: Effect size quantifies the magnitude of the difference between the null hypothesis and the true value.

\(\frac{\mu - \mu_0}{\sigma}\)

  1. Significance level

  2. Variance

  3. One-tailed vs Two-tailed

TRUE or FASLE

Power increases when:

\(n\) increases.

\(d\) (effect size) increases.

Variance decreases.

\(\alpha\) increases.

A one-tailed test is appropriate.

Question 2: Solution - inclass discussion

  • If the null hypothesis is not rejected, we do not accept the alternative: we do not have enough evidence to conclude that the alternative hypothesis is true. This does not mean that the null hypothesis is true; it simply indicates that the sample data does not provide strong enough evidence against it at the chosen significance level.

  • In other words, the null hypothesis represents what we assume to be true unless there is strong evidence to suggest otherwise.

Question 3

A wildlife researcher claims that the average height of one-week-old male elephants is higher than that of one-week-old female elephants at an orphanage. The researcher collected data from thirty one-week-old male elephants and thirty one-week-old female elephants in order to test this claim.

Data set
male 1.70 1.9 2.8 2.00 2.10 2.90 2.20 1.40 1.70 1.80 2.60 2.20 2.20 2.10 1.70 2.90 2.2 1.00 2.40 1.80 1.50 1.90 1.50 1.60 1.70 1.20 2.40 2.10 1.40 2.60
female 1.93 1.2 2.4 2.38 2.32 2.19 2.05 1.44 1.19 1.12 0.81 1.29 0.23 3.67 2.71 0.38 1.1 1.03 2.28 1.42 1.75 1.47 1.46 2.87 1.27 3.02 0.05 2.08 1.62 1.72

Suggest alternative chart types for visualizing data that can be easily sketched by hand using a pencil or pen.

Solution - In class discussion

Question 4

As reported by a NEWS channel, the mean serum high density (HDL) cholesterol of female 20 - 29 years old is 53. A doctor claims that the HDL Cholesterol level of female 20 - 29 years old is greater than 53. He uses the following data, randomly gathered from 22 individuals.

65, 47, 51, 54, 70, 55, 44, 48, 36, 53, 45, 34, 59, 45, 54, 50, 40, 60, 53, 53, 54, 55

It is known from past research that the distribution of the HDL cholesterol is normally distributed and the corresponding population variance is 81. Test the claim that the HDL level is greater than 53 at \(\alpha\) = 0.01 level of significance.

Question 5

A chemist wants to measure the bias in a pH meter. She uses the meter to measure the pH in 14 neutral substances (pH=7) and obtains the data below.

7.01, 7.04, 6.97, 7.00, 6.99, 6.97, 7.04, 7.04, 7.01, 7.00, 6.99, 7.04, 7.07, 6.97

Is there sufficient evidence to support the claim that the pH meter is not correctly calibrated at the α = 0.05 level of significance?

Question 6

A dietician hopes to reduce a person’s cholesterol level by using a special diet supplemented with a combination of vitamin pills. Twenty (20) subjects were pre-tested and then placed on diet for two weeks. Their cholesterol levels were checked after the two week period. The results are shown below. Cholesterol levels are measured in milligrams per decilitre.

before: 210, 235, 208, 190, 172, 244, 211, 235, 210, 190, 175, 250, 200, 270, 222, 203, 209, 220, 250, 280)

after: 190, 170, 210, 188, 173, 195, 228, 200, 210, 184, 196, 208, 211, 212, 205, 221, 240, 250, 230, 220

Test the claim that the Cholesterol level before the special diet is greater than the Cholesterol level after the special diet at α = 0.01 level of significance.