1. Warm Up Check your understanding p. 563
Construct a 95% confidence interval of the data in the CYU.

2. The POWER of a hypothesis test and the probability of a Type II error
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3. Today’s Objectives
Students will be able to define what is meant by the “power” of a hypothesis test
Students will be able to calculate the power of a test against a given alternative
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4. So what is power?
Power is the probability that a level, α, significance test will properly reject Ho when a given alternative value is true.
In other words, power is the probability that you get it right! (When Ho false)
To calculate power, find 1 minus the probability of a Type II error for the given alternative hypothesis.
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5. Example
An Agronomist tests sugar content in plants to see if the sugar had increased. A previous study claimed that the mean cellulose content was 140 mg/g and has a known pop. standard deviation of 8 mg/g., but the agronomist believes that the mean is higher than 140. A sample of 15 cuttings is taken.
σ=8 n = 15 What are the Ho and Ha?
Ho: μ = 140
Ha: μ > 140
Part 1: For which sample means will you fail to reject the null at α = .05? Remember: “When P is high, let it fly”
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6. Example Recall the agronomist testing sugar content in plants
Ho: μ = 140
Ha: μ > 140
Part 1: For which sample means will you fail to reject the null at α = .05?
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7. Example
Why?
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8. Example continued
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9. Example concluded
Now re-do part 2 twice, first assuming the true mean is 147 and then 150
normalcdf(-999,143.4,147, 2.066)4%
So power against µa = 147 is 96%
normalcdf(-999,143.4,150,2.066)0.0007
So power against µa = 150 is virtually 100%
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10. Power Summary
So power is the probability we draw the right conclusion when the null is not true
The farther from the null a particular alternative is, the greater the power
How could we change our sampling strategy to increase power against a close alternative?
Increase sample size
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11. Why does increasing sample size help?
Increasing sample size decreases the spread of the sampling distribution.
So for a given alternative mean, there is less chance that a particular sample mean will be that far (or farther) from the assumed mean.
Also: Increased sample size reduces confidence interval width (why?), so the “acceptance region” is reduced.
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