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Steps to a Hypothesis Test - PowerPoint PPT Presentation

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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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
• 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).”
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.

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.