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Statistics: Unlocking the Power of Data Lock5
Hypothesis Testing: Conclusions
STAT 250
Dr. Kari Lock Morgan
SECTION 4.3• Significance level (4.3)• Statistical conclusions (4.3)• Statistical significance (4.3)
Statistics: Unlocking the Power of Data Lock5
A formal hypothesis test has only two possible conclusions:
1. The p-value is small: reject the null hypothesis in favor of the alternative
2. The p-value is not small: do not reject the null hypothesis
Formal Decisions
How small?
Statistics: Unlocking the Power of Data Lock5
Significance Level
The significance level, , is the threshold αbelow which the p-value is deemed small enough to reject the null hypothesis
p-value < α => Reject H0
p-value > α => Do not Reject H0
Often = 0.05, unless otherwise specifiedα
Statistics: Unlocking the Power of Data Lock5
p-value < α
Results would be rare, if the null were true
Reject H0
We have evidence that the alternative is true!
p-value ≥ α
Results would not be rare, if the null were true
Do not reject H0
Our test is inconclusive
Statistics: Unlocking the Power of Data Lock5
Statistical Conclusions
In a hypothesis test of
H0: = 10 vs Hμ a: < 10 μ
the p-value is 0.002. With α = 0.05, we conclude:
a) Reject H0
b) Do not reject H0
c) Reject Ha
d) Do not reject Ha
Statistics: Unlocking the Power of Data Lock5
Statistical Conclusions
In a hypothesis test of
H0: = 10 vs Hμ a: < 10 μ
the p-value is 0.002. With α = 0.01, we conclude:
a) There is evidence that = 10 μb) There is evidence that < 10 μ
c) We have insufficient evidence to conclude anything
Statistics: Unlocking the Power of Data Lock5
Statistical Conclusions
In a hypothesis test of
H0: = 10 vs Hμ a: < 10 μ
the p-value is 0.21. With α = 0.01, we conclude:
a) Reject H0
b) Do not reject H0
c) Reject Ha
d) Do not reject Ha
Statistics: Unlocking the Power of Data Lock5
Statistical Conclusions
In a hypothesis test of
H0: = 10 vs Hμ a: < 10 μ
the p-value is 0.21. With α = 0.01, we conclude:
a) There is evidence that = 10 μb) There is evidence that < 10 μ
c) We have insufficient evidence to conclude anything
Statistics: Unlocking the Power of Data Lock5
H0 : X is an elephantHa : X is not an elephant
Would you conclude, if you get the following data?
• X walks on two legs
• X has four legs
Elephant Example
Statistics: Unlocking the Power of Data Lock5
“For the logical fallacy of believing that a hypothesis has been proved to be true, merely because it is not contradicted by the available facts, has no more right to insinuate itself in statistical than in other kinds of scientific reasoning…” -Sir R. A. Fisher
Never Accept H0
•“Do not reject H0” is not the same as “accept H0”!
• Lack of evidence against H0 is NOT the same as evidence for H0!
Statistics: Unlocking the Power of Data Lock5
Statistical Significance
When the p-value is less than , the αresults are statistically significant.
If our sample is statistically significant, we have convincing evidence against H0, in favor of Ha
Statistics: Unlocking the Power of Data Lock5
p-value < α
Results would be rare, if the null were true
Reject H0
We have evidence that the alternative is true!
Results are statistically significant
p-value ≥ α
Results would not be rare, if the null were true
Do not reject H0
Our test is inconclusive
Results are not statistically significant
Statistics: Unlocking the Power of Data Lock5
Question #1 of the Day
Does red wine (resveratrol) promote weight loss?
If so, why?
Statistics: Unlocking the Power of Data Lock5
• Resveratrol, an ingredient in red wine and grapes, has been shown to promote weight loss in rodents. We’ll look at a study on Grey Mouse Lemurs (a primate).
• A sample of lemurs had various measurements taken before and after receiving resveratrol supplementation for 4 weeks:• body mass• resting metabolic rate• locomotor activity• food intake
Red Wine and Weight Loss
BioMed Central (2010, June 22). “Lemurs lose weight with ‘life-extending’ supplement resveratrol. Science Daily.
Statistics: Unlocking the Power of Data Lock5
Study Design
This is a
a) Experimentb) Observational study
Statistics: Unlocking the Power of Data Lock5
Study Design
Can we use this study to make conclusions about causality?
a) Yesb) No
Statistics: Unlocking the Power of Data Lock5
Red Wine and Weight Loss
In the test to see if the mean resting metabolic rate is higher after treatment, the p-value is 0.013.
Using = 0.05, is this difference statistically αsignificant? (should we reject H0: no difference?)
a) Yesb) No
Statistics: Unlocking the Power of Data Lock5
Red Wine and Weight Loss
In the test to see if the mean resting metabolic rate is higher after treatment, the p-value is 0.013.
Using = 0.05, what can we conclude?α
a) Mean resting metabolic rate is higher after resveratrol supplementation in lemursb) Mean resting metabolic rate is not higher after resveratrol supplementation in lemursc) Nothing
Statistics: Unlocking the Power of Data Lock5
Red Wine and Weight Loss
In the test to see if the mean body mass is lower after treatment, the p-value is 0.007.
Using = 0.05, is this difference statistically αsignificant? (should we reject H0: no difference?)
a) Yesb) No
Statistics: Unlocking the Power of Data Lock5
Red Wine and Weight Loss
In the test to see if the mean body mass is lower after treatment, the p-value is 0.007.
Using = 0.05, what can we conclude?α
a) Mean body mass is higher after resveratrol supplementation in lemursb) Mean body mass is not higher after resveratrol supplementation in lemursc) Nothing
Statistics: Unlocking the Power of Data Lock5
Red Wine and Weight Loss
In the test to see if locomotor activity changes after treatment, the p-value is 0.980.
Using = 0.05, is this difference statistically αsignificant? (should we reject H0: no difference?)
a) Yesb) No
Statistics: Unlocking the Power of Data Lock5
Red Wine and Weight Loss
In the test to see if locomotor activity changes after treatment, the p-value is 0.980.
Using = 0.05, what can we conclude?α
a) Locomotor activity changes after resveratrol supplementation in lemursb) Locomotor activity does not change after resveratrol supplementation in lemursc) Nothing
Statistics: Unlocking the Power of Data Lock5
Red Wine and Weight Loss
In the test to see if mean food intake changes after treatment, the p-value is 0.035.
Using = 0.10, is this difference statistically αsignificant? (should we reject H0: no difference?)
a) Yesb) No
Statistics: Unlocking the Power of Data Lock5
Red Wine and Weight Loss
In the test to see if mean food intake changes after treatment, the p-value is 0.035.
Using = 0.01, is this difference statistically αsignificant? (should we reject H0: no difference?)
a) Yesb) No
Statistics: Unlocking the Power of Data Lock5
Informal strength of evidence against H0:
Formal decision of hypothesis test, based on = 0.05 :
statistically significant
not statistically significant
Statistical Conclusions
Statistics: Unlocking the Power of Data Lock5
Question #2 of the Day
What aspect of sunlight might help protect against multiple
sclerosis?
Statistics: Unlocking the Power of Data Lock5
Multiple Sclerosis and Sunlight• It is believed that sunlight offers some protection
against multiple sclerosis, but the reason is unknown
• Researchers randomly assigned mice to one of:• Control (nothing)• Vitamin D Supplements• UV Light
• All mice were injected with proteins known to induce a mouse form of MS, and they observed which mice got MS
Seppa, Nathan. “Sunlight may cut MS risk by itself”, Science News, April 24, 2010 pg 9, reporting on a study appearing March 22, 2010 in the Proceedings of the National Academy of Science.
Statistics: Unlocking the Power of Data Lock5
Study Design
Can this study be used to make conclusions about causality, at least in mice?
a) Yesb) No
Statistics: Unlocking the Power of Data Lock5
Multiple Sclerosis and SunlightFor each situation below, write down
Null and alternative hypotheses Informal description of the strength of evidence against H0
Formal decision about H0, using α = 0.05 Conclusion in the context of the question
In testing whether UV light provides protection against MS (UV light vs control group), the p-value is 0.002.
In testing whether Vitamin D provides protection against MS (Vitamin D vs control group), the p-value is 0.47.
Statistics: Unlocking the Power of Data Lock5
Multiple Sclerosis and SunlightIn testing whether UV light provides
protection against MS (UV light vs control group), the p-value is 0.002.
Statistics: Unlocking the Power of Data Lock5
Multiple Sclerosis and SunlightIn testing whether Vitamin D provides
protection against MS (Vitamin D vs control group), the p-value is 0.47.
Statistics: Unlocking the Power of Data Lock5
Conclusions
p-value < α
Generic conclusion:Reject H0
Conclusion in context:We have (strong?) evidence
that [fill in alternative hypothesis]
p-value ≥ α
Generic conclusion:Do not reject H0
Conclusion in context:We do not have enough
evidence to conclude that [fill in alternative hypothesis]
H0
Ha
Statistics: Unlocking the Power of Data Lock5
To DoRead Section 4.3
Do HW 4.2, 4.3 (due Friday, 10/23)