400 Statistics Tests of Statistical Hypotheses
Simple hypothesis
Composite Hypothesis
Types of Error Type I Error: If we reject Ho when Ho is true Type II Error: If we fail to reject Ho when Ho is false
Ho True
Ha True
Reject Ho
Type I error
Correct Decision
Fail to reject Ho
Correct Decision
Type II error
= P(Type I error)
= P(Type II error)
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Example: You are inspecting light bulbs to determine whether they are defective or not. Ho: The bulbs are not defective Ha: The bulbs are defective Reject Ho if bulbs are defective Fail to reject Ho if bulbs are not defective Draw a table like above to characterize the Type I and Type II errors associated with this example. Ho: Bulbs are not defective Reject Bulbs Ha: Bulbs are defective
Fail to reject Bulbs
The significance level is the probability of a Type I error. It is the probability that the test will reject the null hypothesis when Ho is true.
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Example An environmentalist collects a liter of water from 45 different locations along the banks of a stream. He measures the amount of dissolved oxygen in each specimen. The mean oxygen level is 4.62 mg, with a standard deviation of 0.92. A water purifying company claims that the mean level of oxygen in the water is 5 mg. Conduct a hypothesis test at a 99% confidence level to determine whether the mean oxygen level is less than 5 mg. Step 1. Null Hypothesis:
Step 2: Alternative Hypothesis:
Step 3: Test Statistic:
Step 4: critical region
Step 5: P-value/Probability
Step 6: Decision Rule:
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Step 7: Conclusion
Confidence Intervals and Two-Sided Tests: When conducting a two-sided test it is possible to use a confidence interval to determine whether or not to reject the null hypothesis. Steps: Null Hypothesis: H o : = 0 Alternative Hypothesis: H a : 0
n Reject H o if 0 falls outside the confidence interval range
Confidence Interval: z x *
Example: For the previous problem, use a 95% confidence interval to determine if the mean oxygen level in the water is different from 5 mg.
Step 1: Null Hypothesis
Step 2: Alternative Hypothesis
Step 3: Confidence Interval
Step 4: Conclusions
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Heads-Up about Confidence Intervals and Hypothesis Testing:
Power and Inference In determining the usefulness of a confidence interval, we need to look at both the level of confidence and the margin of error. Level of confidence: how reliable is our estimate
Margin of Error: how sensitive is our method; how closely we pin our estimate down High confidence is of little value if the interval is so wide that few values of the parameter are excluded A test with a small level of is of little value if it almost never rejects Ho even when the true value is far from the hypothesized value.
Power: The probability that a fixed level significance test rejects Ho when Ha is true. This probability is called the power of the test against the alternative.
**Don't worry about learning HOW to calculate power*** Instead focus on: How to increase Power: Increase , consequently d...