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Chapter 9: Chapter 9: PowerPowerTarget Goal:
I can make really good decisions I can make really good decisions ..
9.1d
h.w: pg 548: 23, 25
• The probability of a Type II Error tells us the probability of accepting the null hypothesis when it is actually false.
• The complement of this would be the probability of not accepting (in other words rejecting) the null hypothesis when it is actually false. (good decision!)
PowerPower
• To calculate the probability of rejecting the null hypothesis when it is actually false, compute
Power = 1 – P(Type II Error),
Or, (1 – b).
This is called the power of a significance test.
• A P-value describes what would happen supposing that the null hypothesis is true.
• Power describes what would happen supposing that a particular alternative hypothesis is true.
Ex: Exercise is goodEx: Exercise is good
Can a six month exercise program increase the total body mineral content (TBBMC) of young women? A team of researchers is planning a study to examine this question.
Power = 1 – β = 0.80
Interpret results:
This significance test correctly rejects the null hypothesis that exercise has no effect on TBBMC 80% of the time if the true effect of exercise is a 1% increase in TBBMC.
Type II error calculated to be = .20.
Power:Power: the probability of rejecting the null hypothesis when it is actually false.
We must use the shifted curve!We must use the shifted curve!
How Does Power Change?Suppose
H0: µ = 500
Ha: μ > 500
What is the power of the test if µ = 520?
Power = 1 - .239 = .761
Rejection Region
Suppose that = $85 and n = 100.
We would reject H0 for x > 513.98.
= .239
Power is the probability of correctly rejecting H0.
Notice that power is in the SAME curve as Power = 1 –
H0: μ = 500
Ha: µ > 500
Find b and power.
b = .03 Power = .97 vs, Rejection Region
Suppose that = $85 and n = 100.
We would reject H0 for x > 513.98.
If we reject H0, then > 500. What if = 530?
Notice that, as the distance between the null hypothesized value for and our alternative value for increases, decreases AND power
increases.
530 520
.
H0: = 500
Ha: > 500
Find and power.
= .03 power = .97 vs.
= .68 power = .32
Rejection Region
Suppose that = $85 and n = 100.
We would reject H0 for x > 513.98.
If the null hypothesis is false, then > 500.
What if = 510?
Notice that, as the distance between the null hypothesized value for μ and our alternative
value for μ decreases, β increases AND power decreases.
H0: = 500
Ha: > 500
What happens if we use = .01?
Rejection Region
Suppose that = $85 and n = 100.
will increase and power will decrease.
Rejection Region
Power
Power
What happens to What happens to , , , & power , & power when the sample size is when the sample size is
increased?increased?Reject H0Fail to Reject H0
0
a
The standard deviation will decrease making the curve taller and
skinnier.
The significance level () remains the same – so the value where the rejection region begins must move.
β decreases and power
increases!
Power
If power is too small, If power is too small, increase the increase the PowerPower
1. Increase α.
2. Consider an alternative that is further away from μo.
3. Increase the sample size.
4. Decrease σ.
What does this look like?What does this look like?Increase α.
Slides reference point to the left.
Consider an alternative that is further away from μo.
Shift “new” curve to the right.
Increase the sample size.Less spread and narrower curve.
Decrease σ. Less spread and narrower curve.