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Chapter 8

Steps to a Hypothesis Test

- Hypotheses
- Null Hypothesis (Ho)
- Alternative Hypothesis (Ha)

- Alpha
- Distribution (aka model)
- Test Statistics and P-value
- Decision
- Conclusion

Steps to a Hypothesis Test

- Can remember the steps by the sentence:
“Happy Aunts Make The Darndest Cookies”

Example 1– Hypothesis Testing

- An attorney claims that more than 25% of all lawyers advertise. A sample of 200 lawyers in a certain city showed that 63 had used some form of advertising. At α = 0.05, is there enough evidence to support the attorney’s claim?

Hypotheses (Sets up the two sides of the test)

- Build the Alternative Hypothesis (Ha) first.
- based on the claim you are testing (you get this from the words in the problem)
- Three choices
- Ha: parameter ≠ hypothesized value
- Ha: parameter < hypothesized value
- Ha: parameter > hypothesized value

- Three choices

- based on the claim you are testing (you get this from the words in the problem)
- Build Null Hypothesis (Ho) next.
- opposite of the Ha (i.e. = , ≥ , ≤ )

Example 1– Constructing Hypotheses

- We need to know what parameter we are testing and which of the three choices for alternative hypothesis we are going to use.
- “An attorney claims that more than 25% of all lawyers advertise” tells us that this is a test for proportions so our parameter is p.
- “claims that more than 25%” tells us that
Ha: p > .25 and therefore Ho: p ≤ .25

Alpha

- Alpha = α = significance level
- How much proof we are requiring in order to reject the null hypothesis.
- The complement of the confidence level that we learned in the last chapter
- Usually given to you in the problem, if not, you can choose.
- Most popular alphas: 0.05, 0.01, and 0.10

Example 1 – Alpha

- “At α = 0.05” is given to us in the problem so we just copy α = 0.05

Model

- The model is the distribution used for the parameter that you are testing. These are just the same as we used in the confidence intervals.
- p and μ (n ≥ 30) use the normal distribution
- μ (n < 30) uses the t-distribution
- uses the chi-squared distribution

Example 1 - Model

- The model used for a proportion is the normal.

Test Statistic

- You will have a different test statistic for each of the four different parameters that we have learned about.
- p :
- μ (n ≥ 30) :

Test Statistic

- You will have a different test statistic for each of the four different parameters that we have learned about.
- μ (n < 30) :
- :

p-value

- This is the evidence (probability) that you will get off of your chart and then compare against your criteria (alpha).
- You will need to find the appropriate probability that goes with your Ha.
- > and < Ha’s are called one-tailed tests.
- ≠ Ha’s are called two-tailed tests.
- For z and χ2 you have to take the > probability X2

Example 1 – Test Statistic and p-value

- The formula for a test statistic for proportions is:
- So, from our problem we need a proportion from a sample (p-hat), the proportion from our hypothesis (po), and a sample size (n).

Example 1 – Test Statistic and p-value

- “A sample of 200 lawyers in a certain city showed that 63 had used some form of advertising” tells us that
- p-hat = 63/200 or 0.315

- From our hypothesis we know
- po = 0.25 (which means that qo = 0.75)

- “sample of 200” tells us that
- n = 200

Example 1 – Test Statistic and p-value

- So our test statistic and p-value are

Decision – (always about Ho)

- We have two choices for decision
- Reject Ho
- Do Not Reject Ho

- If our evidence (p-value) is less than α we REJECT Ho.
- If our evidence (p-value) is greater than α we DO NOT REJECT Ho.

Example 1 - Decision

- Our p-value is 0.0170 and our alpha is 0.05
- So, since our p-value is less than our alpha our decision is: REJECT Ho.

Conclusion – (always in terms of Ha)

- Conclusions
- Reject Ho
- “There is enough evidence to suggest (Ha).”

- Do Not Reject
- “There is not enough evidence to suggest (Ha).”

- Reject Ho

Example 1 - Conclusion

- Our decision to was to reject Ho, so our conclusion is:
“There is enough evidence to suggest that p>0.25”

Example 1 - Summary

- Ho: p ≤ 0.25
Ha: p > 0.25

- α = 0.05
- Model: Normal
- z = 2.12 and p-value = 0.0170
- Reject Ho
- There is enough evidence to suggest that p>0.25.

Example 2 – Hypothesis Testing

A researcher reports that the average salary of assistant professors is more than $42,000. A sample of 30 assistant professors has a mean of $43,260. At α = 0.05, test the claim that assistant professors earn more than $42,000 a year. The standard deviation of the population is $5230.

Example 2 (cont.)

- Hypotheses
- Ho: μ ≤ $42,000
- Ha: μ > $42,000 (given claim is “more than”)

- Alpha
- α = 0.05 (given)

- Model
- Normal (n ≥ 30 and it’s a mean)

Example 2 (cont.)

- Test statistic and p-value:

Example 2 (cont.)

- Decision
- 0.0934 > 0.05 (p-value > alpha)
- DO NOT REJECT Ho

- Conclusion
- We do not have evidence to suggest that
μ > $42,000.

- We do not have evidence to suggest that

Example 3 – Hypothesis Testing

A physician claims that joggers’ maximal volume oxygen uptake is greater than the average of all adults. A sample of 15 joggers has a mean of 40.6 milliliters per kilogram (ml/kg) and a standard deviation of 6 ml/kg. If the average of all adults is 36.7 ml/kg, is there enough evidence to support the physicians claim at α = 0.05?

Example 3 (cont.)

- Hypotheses
- Ho: μ ≤ 36.7
- Ha: μ > 36.7

- Alpha
- α = 0.05 (given)

- Model
- t(14)

Example 3 (cont.)

- Test statistic and p-value:

Example 3 (cont.)

- Decision
- (0.01,0.025) < 0.05 (p-value < alpha)
- REJECT Ho

- Conclusion
- There is evidence to suggest that μ > 36.7.

Example 4 – Hypothesis Testing

A researcher knows from past studies that the standard deviation of the time it takes to inspect a car is 16.8 minutes. A sample of 24 cars is selected and inspected. The standard deviation was 12.5 minutes. At α=0.05, can it be concluded that the standard deviation has changed?

Example 4 (cont.)

- Hypotheses
- Ho: σ = 16.8
- Ha: σ≠ 16.8

- Alpha
- α = 0.05 (given)

- Model
- χ2(23)

Example 4 (cont.)

- Test statistic and p-value:

Example 4 (cont.)

- Decision
- (0.05,0.10) > 0.05 (p-value > alpha)
- DO NOT REJECT Ho

- Conclusion
- There is not enough evidence to suggest that
σ≠ 16.8.

- There is not enough evidence to suggest that

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