QM Course Guide

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

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.

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):

Alternative hypothesis (H_a): 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

https://dataz4s.com/statistics/one-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

https://www.scribbr.com/statistics/statistical-tests/

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

T-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.   

https://statisticsbyjim.com/hypothesis-testing/t-tests-t-values-t-distributions-probabilities/

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

Interpreting Hypothesis Tests

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

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

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

P-Values

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. 

    https://www.statisticshowto.com/probability-and-statistics/find-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.

https://pregnancymb.blogspot.com/2019/10/type-1-error-pregnancy-test.html

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?

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

*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

             expected

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…

In this case?

Conclusion: The height difference is independent / random.