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.

Rat recognition of symbols

In the experiment that was described at the start of this section, a rat was placed in a cage containing three boxes marked with a circle, square and cross. To train the rat, 100 cards were shown to the rat, with the corresponding box unlocked allowing access to food.

After this training period, 90 cards were presented to the rat and the box with the same symbol was picked 36 times. Since more than a third are correct, does this provide strong evidence that the rat is partially recognising the symbols?

The null and alternative hypotheses are...

H0:   π = 1/3       (guessing)

HA:   π > 1/3       (learned)

The p-value is the probability of getting 36 or more symbols correct 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 getting 36 of more correct symbols if the rat has not learned. We therefore conclude that there is no evidence of learning 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 the rat chose correctly exactly 30 times (a third) in the rat-learning example above, it would be wrong to conclude that the rats had not learned at all. 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 rat-learning test 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.