44 9.4 Rare Events, the Sample, Decision and Conclusion
44 9.4 Rare Events, the Sample, Decision and Conclusion
Establishing the type of distribution, sample size, and known or unknown standard deviation can help you figure out how to go about a hypothesis test. However, there are several other factors you should consider when working out a hypothesis test.
Rare Events
Suppose you make an assumption about a property of the population (this assumption is the null hypothesis). Then you gather sample data randomly. If the sample has properties that would be very unlikely to occur if the assumption is true, then you would conclude that your assumption about the population is probably incorrect. (Remember that your assumption is just an assumption—it is not a fact and it may or may not be true. But your sample data are real and the data are showing you a fact that seems to contradict your assumption.)
Using the Sample to Test the Null Hypothesis
Use the sample data to calculate the actual probability of getting the test result, called the p-value. The p-value is the probability that, if the null hypothesis is true, the results from another randomly selected sample will be as extreme or more extreme as the results obtained from the given sample.
A large p-value calculated from the data indicates that we should not reject the null hypothesis. The smaller the p-value, the more unlikely the outcome, and the stronger the evidence is against the null hypothesis. We would reject the null hypothesis if the evidence is strongly against it.
Draw a graph that shows the p-value. The hypothesis test is easier to perform if you use a graph because you see the problem more clearly.
Example 1
A baker bakes 10 loaves of bread. The mean height of the sample loaves is 17 cm. The baker knows from baking hundreds of loaves of bread that the standard deviation for the height is 0.5 cm. The distribution of heights is normal. He claims that his bread height is more than 15 cm, on average. Several of his customers do not believe him.
To persuade his customers that he is right, the baker decides to do a hypothesis test.
Solution:
Since the baker knows the standard deviation from baking hundreds of loaves of bread, we will run Normal Z-Test.
The null hypothesis could be H0: μ ≤ 15.
The alternate hypothesis is Ha: μ > 15.
The words “is more than” translates as a “>” so “μ > 15″ goes into the alternate hypothesis.
The null hypothesis must contradict the alternate hypothesis.
Since σ is known (σ = 0.5 cm.), the distribution for the population is known to be normal with
- mean μ = 15 and
- standard deviation [latex]\displaystyle\frac{\sigma}{\sqrt{n}}=\frac{0.5}{\sqrt{10}}=0.16[/latex]
The p-value, then, is the probability that a sample mean is the same or greater than 17 cm. when the population mean is, in fact, 15 cm. We can calculate this probability using the normal distribution for means.
p-value
= P([latex]\overline{X}[/latex] > 17)
= P([latex]\frac{\overline{X}-{\mu}}{\frac{\sigma}{\sqrt{n}}}[/latex] > [latex]\frac{17 - {\mu}}{\f