MATH120: Quantitative Methods - Introduction to Hypothesis Testing

Business analysts provide insights and information to help decision-makers answer these questions.
They develop hypotheses, which are tentative explanations of principles operating in nature, that can be studied and explored.

Three types of hypotheses are explored:
- Research hypotheses
- Statistical hypotheses
- Substantive hypotheses
The primary focus will be on testing statistical hypotheses, but understanding research and substantive hypotheses is also important for business decision-makers.

A research hypothesis is a statement of what the analyst believes will be the outcome of an experiment or study.
It's similar to the general definition of a hypothesis.
Business analysts often have an idea or theory based on prior experience or work.
Examples of research hypotheses:
- Older workers are more loyal to a company.
- Bigger companies spend a higher percentage of their annual budget on advertising than smaller ones.
- Using a Six Sigma quality approach to manufacturing results in greater productivity.
- Scrap metal price is a good leading indicator of the Industrial Production Index.

To formally test research hypotheses, a more formal structure using statistical hypotheses is needed.
A statistical hypothesis is a formal hypothesis structure set up with a null and an alternative hypothesis to scientifically test research hypotheses.
Null Hypothesis ( $H_0$ ): States that the “null” condition exists. There is nothing new happening, the old theory is still true, the old standard is correct, and the system is in control.
Alternative Hypothesis ( $H<em>1$ or $H</em>a$ ): States that the new theory is true, there are new standards, the system is out of control, and/or something is happening.
Example: Flour packaging with a target average of 80 oz.
- Null hypothesis ( $H_0$ ): The average weight of the flour packages is 80 oz ( $\mu = 80$ ).
- Alternative hypothesis ( $H_a$ ): The average weight is not 80 oz ( $\mu \neq 80$ ).
The alternative hypothesis is the complement of the null hypothesis, covering all other possibilities.
Statistical hypotheses can lead to one-tailed or two-tailed tests.
The hypotheses for the flour packaging example are two-tailed because there is no presumption that the packages are overfilled or underfilled.
Analysts allow for the possibility that either $\mu > 80$ oz. or $\mu < 80$ oz. could occur.

Two-Tailed Test:
- Tests both sides of the distribution.
- Uses = and ≠ in the statistical hypotheses.
- Directionless; the alternative hypothesis allows for either > or <.
One-Tailed Test:
- Tests only one side of the distribution.
- Always directional; the alternative hypothesis uses either > or <.
- Example: A company with an 18% market share believes it has increased due to marketing efforts.
  - Null hypothesis: The market share is still 18% ( $p = 0.18$ ).
  - Alternative hypothesis: The market share is now greater than 18% ( $p > 0.18$ ).

One-Tailed Tests
- Null hypothesis has an = sign
- Directional
- Cues in the problem: higher, lower, older, younger, more, less, longer, shorter
- The alternative hypothesis has either a > or < sign
- Hypothesis possibilities:
  - $H_0: \mu = k$
  - $H_1: \mu > k$
- $H_0: \mu = k$
- $H_1: \mu < k$
Two-Tailed Tests
- Null hypothesis has an = sign
- Nondirectional
- Cues in the problem: same, different, equal, not equal, in control, out-of-control
- The alternative hypothesis can have only ≠ sign
- Hypothesis possibilities:
  - $H_0: \mu = k$
  - $H_1: \mu \neq k$

When testing a statistical hypothesis, if the null hypothesis is rejected and the alternative hypothesis is accepted, a statistically significant result has been obtained.

A substantive result occurs when the outcome of a statistical study produces statistically significant results that are also important to the decision-maker.

State Hypothesis: Clearly identify what is being tested and whether the hypotheses are one-tailed or two-tailed. Assume the null hypothesis is true at the beginning of the study.
Select Test Statistic: Choose the most appropriate statistical test based on the type, number, and level of data, the statistic used, and the assumptions underlying the test.
Select Value of Alpha ( $\alpha$ ): Set the probability of committing a Type I error. Common values include 0.05, 0.01, 0.10, and 0.001.
Establish Decision Rule: Determine critical values using alpha and the test statistic (or use alpha as a critical probability value for the p-value method) before the study begins.
Gather Sample Data: Collect data through surveys, focus groups, random sampling, or secondary data sources, ensuring proper techniques of random sampling are followed.
Calculate Test Statistic: Compute the test statistic after the data is sampled.
Reach a Statistical Conclusion: Conclude whether to reject or not reject the null hypothesis based on the decision rule and the value of the test statistic.
Determine Business Implications: Decide what business implications the study results contain and whether the results are substantive. If the null hypothesis is rejected, determine if the results are substantive and what business decisions need to be made.

Based on the critical values, statistical outcomes are divided into two groups:
1. Those that cause the rejection of the null hypothesis (rejection region).
2. Those that do not cause the rejection of the null hypothesis (nonrejection region).

Type I Error: Rejecting a true null hypothesis.
*Analogy: Firing an employee for stealing when they are not actually stealing.
Type II Error: Failing to reject a false null hypothesis.
*Analogy: Not firing an employee who is stealing due to lack of sufficient evidence.

	Null True	Null False
Reject Null	Type I Error ( $\alpha$ )	Correct Decision (Power)
Fail to Reject Null	Correct Decision	Type II Error ( $\beta$ )