Hypothesis Testing Notes

Overview

• Hypothesis testing is used to determine whether a statement about a population parameter should be rejected.
• Null Hypothesis (H0): A tentative assumption about a population parameter.
  • Example Question: A company claims that its batteries last for 500 hours. What would be the null hypothesis to test this claim?
• Alternative Hypothesis (Ha): The opposite of the null hypothesis.
  • Example Question: If the null hypothesis is that the average height of women is 5'4", what could be an alternative hypothesis?
• The hypothesis testing procedure uses sample data to test the two competing statements indicated by H0 and Ha.

Developing Null and Alternative Hypotheses

• Formulating hypotheses requires careful structuring to ensure the test conclusion provides the desired information.
• The context of the situation is very important in determining how the hypotheses should be stated.
• Sometimes it's easier to identify the alternative hypothesis first, while other times the null hypothesis is easier to define initially.
• Correct hypothesis formulation requires practice.

Alternative Hypothesis as a Research Hypothesis

• Many hypothesis testing applications involve gathering evidence to support a research hypothesis.
• It is often best to begin with the alternative hypothesis, making it the conclusion the researcher hopes to support.
• The research hypothesis is considered true if the sample data provides sufficient evidence to reject the null hypothesis.
• Example 1: A new teaching method is believed to be better than the current one.
  • Alternative Hypothesis: The new teaching method is better.
  • Null Hypothesis: The new method is no better than the old method.
  • Example Question: If a new teaching method is implemented, what hypothesis would a researcher aim to support?
• Example 2: A new sales force bonus plan is developed to increase sales.
  • Alternative Hypothesis: The new bonus plan increases sales.
  • Null Hypothesis: The new bonus plan does not increase sales.
  • Example Question: How would you formulate the alternative hypothesis to test if a bonus plan increases sales?
• Example 3: A new drug aims to lower blood pressure more than the existing drug.
  • Alternative Hypothesis: The new drug lowers blood pressure more than the existing drug.
  • Null Hypothesis: The new drug does not lower blood pressure more than the existing drug.
  • Example Question: What null hypothesis would you use to test if a new drug lowers blood pressure more effectively?

Null Hypothesis as an Assumption to be Challenged

• Start with a belief or assumption about a population parameter's value.
• Use a hypothesis test to challenge the assumption and determine if there's statistical evidence to conclude the assumption is incorrect.
• In these situations, it's helpful to develop the null hypothesis first.
• Example: The label on a soft drink bottle states that it contains 67.6 fluid ounces.
  • Null Hypothesis: The label is correct ($\mu = 67.6$ ounces).
  • Alternative Hypothesis: The label is incorrect ($\mu \neq 67.6$ ounces).
  • Example Question: If you want to challenge the accuracy of a soft drink bottle's label, what initial hypothesis would you set?

Forms for Null and Alternative Hypotheses about a Population Mean

• The equality part of the hypotheses always appears in the null hypothesis.
• A hypothesis test about a population mean $\mu$ must take one of the following three forms (where $\mu_0$ is the hypothesized value of the population mean):
  • One-tailed (lower-tail):
  • $H<em>0: \mu \ge \mu</em>0$
  • H_a: \mu - One-tailed (upper-tail):
  • H0: \mu \le \mu0
  • Ha: \mu > \mu0
  • Example Question: Give an example of when you will use one tailed upper test.
  • Two-tailed:
  • H0: \mu = \mu0
  • Ha: \mu \neq \mu0
  • Example Question: Give an example of when you will use two tailed test.

Example: Metro EMS

• A major west coast city's emergency medical service aims to respond to medical emergencies with a mean time of 12 minutes or less.
• The director wants to formulate a hypothesis test to determine whether the service goal of 12 minutes or less is being achieved, using a sample of emergency response times.
• Null and Alternative Hypotheses:
  • H_0: \mu \le 12 (The emergency service is meeting the response goal; no follow-up action is necessary.)
  • H_a: \mu > 12 (The emergency service is not meeting the response goal; appropriate follow-up action is necessary.)
  • Where \mu = mean response time for the population of medical emergency requests.
  • Example Question: If the EMS wants to ensure they are meeting their response time goals, how should they set up their null and alternative hypotheses?

Type I and Type II Errors

• Hypothesis tests are based on sample data, so errors are possible.
• Type I Error: Rejecting H0 when it is true.
  • The probability of making a Type I error when the null hypothesis is true as an equality is called the level of significance, denoted by \alpha.
  • Hypothesis testing that only controls the Type I error are called significance tests.
  • Example Question: What is the consequence of rejecting a true null hypothesis?
• Type II Error: Accepting H0 when it is false.
  • It is difficult to control for the probability of making a Type II error.
  • Statisticians avoid the risk of making a Type II error by using "do not reject H0" and not "accept H0".
  • Example Question: Why do statisticians prefer to say "do not reject H0" instead of "accept H0"?
• Decision Table:
  | Conclusion (Action) | H0$True ($\mu \le 12$) |$H0$False ($\mu > 12) |
  | :------------------------- | :----------------------------- | :------------------------------ |
  | Reject H0$(Conclude$\mu > 12) | Type I Error | Correct Decision | | Accept H0$(Conclude$\mu \le 12) | Correct Decision | Type II Error |

p-Value Approach to One-Tailed Hypothesis Testing

• Reject H_0$$ if the p-value