Another example

The following example shows again how the binomial distribution can be used to obtain the p-value for a test about a population probability.

Design of mustard jar

In the trial of three new designs for a mustard jar that was described at the start of this section, all three were offered for sale at the same price in a supermarket. Out of the first 90 jars that were purchased, 36 chose the design that was more expensive to manufacture than the other two. Since more than a third chose this design for the mustard jar, is there strong evidence that customers prefer it?

The null and alternative hypotheses are...

H0:  π = 1/3       (no preference)

HA:  π > 1/3       (preference for the expensive design)

The p-value is the probability of 36 or more expensive designs being purchased when π = 1/3. This can be obtained directly from a binomial distribution with π = 1/3 and n = 90.

Use the slider below to obtain the p-value for this test.

The p-value for the test is 0.1103, meaning that there is a probability of 0.1103 of the expensive design being purchased 36 of more times even if there is no real preference for it. We therefore conclude that there is no evidence of any preference from the data.

Interpretation of p-values

If the p-value for a test is very small, the data are 'inconsistent' with the null hypothesis. (The observed data may still be possible, but are at least extremely unlikely.)

From a very small p-value, we can conclude that the null hypothesis is probably wrong.

However a high p-value cannot allow us to conclude that the null hypothesis is correct — only that the observed data are consistent with it. For example, if exactly 30 expensive cases (a third) were purchased in the CD-case example above, it would be wrong to conclude that there was no preference for it. The data are also consistent with other values of π near 1/3, so we cannot conclude that π is not 0.32 or 0.34.

A hypothesis test can never conclude that the null hypothesis is correct.

The correct interpretation of p-values for the CD-case experiment would be...

p-value Interpretation Conclusion
p >  0.1 x is not unusually high. It would be as high in more than 10% of samples if π = 1/3. There is no evidence against π = 1/3.
0.05 < p < 0.1 We would find x as high in only 5% to 10% of samples if π = 1/3. There is only slight evidence against π = 1/3.
0.01 < p < 0.05 We would find x this high in only 1% to 5% of samples if π = 1/3. There is moderately strong evidence against π = 1/3.
p < 0.01 We would find x this high in under 1% of samples if π = 1/3. There is strong evidence against π = 1/3.