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]
Suppose the null hypothesis is true (the mean height of the loaves is no more than 15 cm). Then is the mean height (17 cm) calculated from the sample unexpectedly large? The hypothesis test works by asking the question how unlikely the sample mean would be if the null hypothesis were true. The graph shows how far out the sample mean is on the normal curve. The p-value is the probability that, if we were to take other samples, any other sample mean would fall at least as far out as 17 cm.

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.

Normal distribution curve on average bread heights with values 15, as the population mean, and 17, as the point to determine the p-value, on the x-axis.

Figure 1

   p-value
= P([latex]\overline{X}[/latex] > 17)
= P([latex]\frac{\overline{X}-{\mu}}{\frac{\sigma}{\sqrt{n}}}[/latex] > [latex]\frac{17 - {\mu}}{\frac{\sigma}{\sqrt{n}}}[/latex])
= P([latex]\frac{\overline{X}-{\mu}}{\frac{\sigma}{\sqrt{n}}}[/latex] > [latex]\frac{17-15}{\frac{0.5}{\sqrt{10}}}[/latex])
= P( Z > 12.64911 ) which is approximately zero.

A p-value of approximately zero tells us that it is highly unlikely that a loaf of bread rises no more than 15 cm, on average.
That is, almost 100% of all loaves of bread would be at least as high as 17 cm. purely by CHANCE had the population mean height really been 15 cm.
Because the outcome of 17 cm is so unlikely to happen (meaning it is happening NOT by chance alone), we conclude that the evidence is strongly against the null hypothesis (the mean height is at most 15 cm.).
There is sufficient evidence that the true mean height for the population of the baker’s loaves of bread is greater than 15 cm.

Using TI-83/84:

  1. Press STAT.
  2. Arrow right to TESTS.
  3. Choose 1: Z-Test.
  4. For Inpt, arrow right to STATS and press ENTER.
  5. Input the following information.
    [latex]{\mu}_{0}[/latex] = 15
    [latex]\beta[/latex] = 0.5
    [latex]\overline{X}[/latex] = 17
    n = 10
    [latex]{\mu}[/latex] : > [latex]{\mu}_{0}[/latex]
  6. Then arrow down to CALCULATE.

The result:
[latex]{\mu}[/latex] > 15
z = 12.64911064
p = 5.854831 * 10-37
[latex]\overline{X}[/latex] = 17
n = 10

Interpretation of the result: 

The z-score of height 17cm is 12.64911. The blue shaded area of Figure 1, also known as p-value, is 5.854831 * 10-37.


Try It

A normal distribution has a standard deviation of 1. We want to verify a claim that the mean is greater than 12. A sample of 36 is taken with a sample mean of 12.5.

H0: μ ≤ 12

Ha: μ > 12

The p-value is 0.0013

Draw a graph that shows the p-value.



Decision and Conclusion

A systematic way to make a decision of whether to reject or not reject the null hypothesis is to compare the p-value and a preset or preconceived α (also called a “significance level”). A preset α is the probability of a Type I error (rejecting the null hypothesis when the null hypothesis is true). It may or may not be given to you at the beginning of the problem.

When you make a decision to reject or not reject H0, do as follows:

  • If α > p-value, reject H0.
    The results of the sample data are significant.
    There is sufficient evidence to conclude that H0 is an incorrect belief and that the alternative hypothesis, Ha, may be correct.
    p-value < α , then reject H0. Hence we have sufficient evidence to conclude Ha. )
  • If αp-value, do not reject H0. The results of the sample data are not significant.
    There is not sufficient evidence to conclude that the alternative hypothesis,Ha, may be correct.
    p-value > α , then not reject H0. Hence we have no sufficient evidence to conclude Ha. )
When you “do not reject H0,” it does not mean that you should believe that H0 is true. It simply means that the sample data have failed to provide sufficient evidence to cast serious doubt about the truthfulness of Ho.

Conclusion: After you make your decision, write a thoughtful conclusion about the hypotheses in terms of the given problem.

 



Example 2

When using the p-value to evaluate a hypothesis test, it is sometimes useful to use the following memory device

If the p-value is low, the null must go.

If the p-value is high, the null must fly.

This memory aid relates a p-value less than the established alpha (the p is low) as rejecting the null hypothesis and, likewise, relates a p-value higher than the established alpha (the p is high) as not rejecting the null hypothesis.

Solution:

Reject the null hypothesis when ______________________________________.

 

The results of the sample data _____________________________________.

 

Do not reject the null when hypothesis when __________________________________________.

 

The results of the sample data ____________________________________________.


Try It

CuteBaby Genetics Labs claim their procedures improve the chances of a boy being born. The results for a test of a single population proportion are as follows:

H0: p = 0.50, Ha: p > 0.50

α = 0.01

p-value = 0.025

Interpret the results and state a conclusion in simple, non-technical terms.