Topic 8 - Hypothesis Testing (Part 1)
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This article is a topic within the subject Business & Economic Statistics.
Contents |
Required Reading
Gerald Keller (2011), Statistics for Management and Economics (Abbreviated), 9th Edition, pp. value.
Hypothesis Testing
Basic Inference Problems
- Estimation: Process or obtaining reliable information about population parameters (point/interval)
- Does the data support a hypothesis about the parameter of interest? is there enough evidence to infer the alternative hypothesis is true?
- Typical Example: The Judicial System
- Null Hypothesis = Defendant presumed innocent; Alternative Hypothesis = Defendant is guilty
- Given evidence we will decide –
- Guilty à Rejecting the null hypothesis in favour of the alternative
- “Statistically Significant” or sufficient evidence to reject the null
- Guilty à Rejecting the null hypothesis in favour of the alternative
- Not Guilty à Not rejecting the null hypothesis in favour of the alternative (fail to reject the null)
- Possible errors – guilty person goes free, innocent person found guilty
- Types of Errors – Trade-off
- Type I - Rejecting a true null hypothesis = α = Significance Level
- Sets the rejection region
- Type II - Don’t reject a false null hypothesis = β
- Type I - Rejecting a true null hypothesis = α = Significance Level
Method
- State Null (Assume to Be True) & Alternative Hypothesis
- Identify Rejection Region (Graph it!)
- State the Decision Rule (+Significance Level) - Reject H0 if Z>Zx = 'xx'
- Calculate Test Statistic + Compare
- Conclusion
- Enough evidence to support the alternative hypothesis – large error, more likely that the null is not correct
- Not enough evidence to support the alternative hypothesis – small error, due to random sampling errors
Example: McDonalds Quality Control
- Quarter pounder is presumed to have 0.25 pounds of pre-cooked meat, is this correct or incorrect?
- Maintained or Null Hypothesis
- Let X = weight of precooked meat with mean µ
- H0: μ = 0.25
- H1: μ ≠ 0.25
- ≠ 2 tailed hypothesis (truth in advertising)
- < one (lower) tailed hypothesis
- Consider H0: μ = 0.25, H1:μ < 0.25,
- A sample of 25 is taken 4 times = 0.24, 0.22, 0.28, 0.21
- 0.25, 0.22, 021 represent evidence against the null
- Assume X ~ N (0.25, 0.05^2)
- Let α = 0.05, n =25, σ = 0.05, the sample mean = 23.75
- Find i.e. the 5th percentile (Z.05) (0.05 probability area) Z = -1.645
- Standardise the sample mean --> the test statistic = (0.2375-0.25)/(0.05/rt(25)) = -1.25 which is not within the rejection region
- Conclusion
- Let α = 0.05, n =25, σ = 0.05, the sample mean = 23.75
- A sample of 25 is taken 4 times = 0.24, 0.22, 0.28, 0.21
Example: Skills Test Alternative Method: X = 7.09
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P Values
- The p-value test is the probability of observing a test statistic at least as extreme as the one computed given the null hypothesis is true
- The minimum level of significance needed to reject the null
- In “Skills Test” it would be the probability of getting a score of 7.09 or larger
- If p-value > α, fail to reject the null
- If p-value < α, reject the null
Conclusions of a Test of Hypothesis
- We reject the null hypothesis, we conclude that there is enough statistical evidence to infer that the alternative hypothesis is true
- We do not reject the null hypothesis, we conclude that there is not enough statistical evidence to infer that the alternative hypothesis is true
End
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References
Textbook refers to Gerald Keller (2011), Statistics for Management and Economics (Abbreviated), 9th Edition,.
- ↑ ASB Lecture Slides