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Hypothesis Test. 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:

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Hypothesis test

Hypothesis Test

Chapter 8


Steps to a hypothesis test
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 test1
Steps to a Hypothesis Test

  • Can remember the steps by the sentence:

    “Happy Aunts Make The Darndest Cookies”


Example 1 hypothesis testing
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
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

  • Build Null Hypothesis (Ho) next.

    • opposite of the Ha (i.e. = , ≥ , ≤ )


Example 1 constructing hypotheses
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

  • 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
Example 1 – Alpha

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


Model
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
Example 1 - Model

  • The model used for a proportion is the normal.


Test statistic
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 statistic1
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
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
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 value1
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 value2
Example 1 – Test Statistic and p-value

  • So our test statistic and p-value are


Decision always about ho
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
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
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).”


Example 1 conclusion
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
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
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
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 cont1
Example 2 (cont.)

  • Test statistic and p-value:


Example 2 cont2
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.


Example 3 hypothesis testing
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
Example 3 (cont.)

  • Hypotheses

    • Ho: μ ≤ 36.7

    • Ha: μ > 36.7

  • Alpha

    • α = 0.05 (given)

  • Model

    • t(14)


Example 3 cont1
Example 3 (cont.)

  • Test statistic and p-value:


Example 3 cont2
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
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
Example 4 (cont.)

  • Hypotheses

    • Ho: σ = 16.8

    • Ha: σ≠ 16.8

  • Alpha

    • α = 0.05 (given)

  • Model

    • χ2(23)


Example 4 cont1
Example 4 (cont.)

  • Test statistic and p-value:


Example 4 cont2
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.