# Topic 8 - Hypothesis Testing (Part 1)

## Contents

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
• 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

### Method

1. State Null (Assume to Be True) & Alternative Hypothesis
2. Identify Rejection Region (Graph it!)
3. State the Decision Rule (+Significance Level) - Reject H0 if Z>Zx = 'xx'
4. Calculate Test Statistic + Compare
5. Conclusion
1. Enough evidence to support the alternative hypothesis – large error, more likely that the null is not correct
2. 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

## 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

1. We reject the null hypothesis, we conclude that there is enough statistical evidence to infer that the alternative hypothesis is true
2. 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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