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QM Course Guide

Hypothesis Testing

When we take a sample from a population, we use descriptive statistics to summarize that sample data.

We can also use sample data to draw conclusions about the population.

Hypothesis testing is one way we make inferences about the population.

What is a hypothesis?

What is a hypothesis? a claim about a characteristic (p, or µ) of a population.


Population proportion

95% of students pass stats class (QM) the first time they take it. 


Population average

The average grade in your QM class is 74%.

What is a hypothesis test?

What is a hypothesis test? A standard procedure for testing a claim about a population characteristic.

People, companies and organizations make claims about all sorts of things. We perform hypothesis tests to evaluate if the claims are likely to be true.

1. Teacher

“My students score higher on standardized tests.” 

Are test scores significantly higher?

2. Mechanic

“I’ll charge you less than the other guys.”

Does this mechanic charge less?

3. University

“Our graduating students’ starting salaries are different from the average starting salary.”

Are salaries significantly different?

How to we test hypotheses?

Step 1.  Formulate the null and alternative hypotheses

Null hypothesis (H0): The starting assumption for a hypothesis test, that there is no difference between the sample and the population mean (µ) or parameter (p).

Examples of Null Hypotheses (Ho):

1. Teacher

Test scores are not significantly different from the average.

2. Mechanic

This mechanic charges about the same amount as everyone else.

3. University

Starting salaries for graduating students from this institution are the same as students from elsewhere.

Alternative hypothesis (Ha): The claim that the population mean (µ) or parameter (p) has a value that is different from that claimed in the null hypothesis.

Deciding if your test is left, right or two-tailed

1. Teacher

Test scores in this class are higher than the average for all classes.

Ha: µ > average test scores

Right-tailed test

2. Mechanic

This mechanic charges less than everyone else.

Ha: µ < average cost

Left-tailed test

3. University

Students’ starting salaries are significantly different from the average.

Ha: µ average cost

Two-tailed test

Step 2. Collect Data

Draw a sample (link word sample to section 1.a.ii) and measure:

  1. Sample size (n)
  2. Relevant sample statistic: sample mean (x̄), or sample proportion (p̂)

Step 3. Perform a statistical test

Common hypothesis tests (t-test, chi-square)

Common Hypothesis Tests include T-tests and Chi-Squared tests


The t-test compares sample means / proportions to population means / proportions, and asks how different they are. We use t-tests with quantitative data.

  • If the sample mean = the null hypothesis (no difference) the t-value is zero (0).
  • As the sample mean diverges from the population mean the t-value increases.

The larger the sample size the fewer values are in the tails of the distribution.

For example

SPSS T-test “How-To” (Embed)

Chi-squared (χ2)

Chi squared tests ask:  Are there differences between the observed frequencies of an event or characteristic, and the frequencies we expect to see?

We use chi-squared tests to look at differences between groups, usually for categorical / qualitative data.

SPSS Chi-Squared “How-To”

See the example (link example to “working through a chi-square test”) at the end of this section to learn how to work through a chi-square hypothesis test by hand.

Step 4. Possible outcomes and drawing conclusions

Reject Ho

In which case we have evidence to support Ha

Not Reject Ho

In which case we do not have evidence to support Ha

Interpreting Hypothesis Tests

  1. If the pre-chosen level of significance is higher than the p-value reject Ho.

  1. If the pre-chosen level of significance is lower than the p-value do not reject Ho. 

  1. Not rejecting Ho doesn’t make Ho true – it means your sample data doesn’t provide enough evidence to disprove it. 



The p-value is the probability that you would obtain the answer you have, assuming the null hypothesis is true.

It is often used in conjunction with a pre-determined level of statistical significance.

Critical values

Critical Values

Critical values are cut off points beyond which a test value is unlikely to lie.                                                  

Common critical values in the social sciences include 0.05 and 0.01.


But be careful! Errors can happen                                      

Errors are divided into two categories, type I and type II.

Tests rejects null

Test fails to reject null

Null(Ho) is true

Type I error, false positive

Correct decision, no effect

Null (Ho) is false

Correct decision, effect exists

Type II error, false negative

Working through a chi-square test

Chi-square (χ2)

Chi square tests ask:  Are there differences between the observed frequencies of an event or characteristic, and the frequencies we expect to see?

                Variable 1                            

   &                                 Variable 2

Soccer Team (Team A, Team B)

Height (Short, Tall)

Regular junk food consumption (yes / no)

Weight (underweight, average, overweight)


Calculating the Chi Square value

Using the soccer team example from above, we input the observed and expected values for hair colour on each team.

Observed values: Categorize and count the number of short and tall players. 

Expected values: Calculate the number of short and tall players you would expect to see if the two variables (Team and Height) are independent (don’t affect each other).

How to calculate expected values: Σ column * Σ row / n

Σ = sum

n = total number





Team A

7 (6)

4 (5)


Team B

5 (6)

6 (5)






*Note: expected values must be at least 5 for this test to work.

Use a table like the one below to help you calculate the Chi-Square (χ2) value:

 Chi-square (χ2) formula: χ2 = (observed – expected)2


Fo = Frequency observed

Fe = Frequency expected

Understanding the results

Using your pre-determined level of significance, compare your chi-square value to this example’s critical value of 3.84.

If the calculated chi sq. value is…

< critical value


> critical value


In this case?

Calculated chi sq. value

Greater or less than

Critical value



Conclusion: The height difference is independent / random.