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.

Telepathy experiment

In the telepathy experiment that was described at the start of this section, one subject selects cards with a random shape (circle, square or cross) and attempts to 'send' this shape to another subject who is seated behind a screen; this second subject reports the shape imagined for the card.

Out of 90 cards, 36 are correctly guessed. Since more than a third are correct, does this provide strong evidence that information is being telepathically transmitted?

The null and alternative hypotheses are...

H0:   π = 1/3       (guessing)

HA:   π > 1/3       (telepathy)

The p-value is the probability of getting 36 or more cards 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 or more correct cards if there is no telepathy. We therefore conclude that there is no evidence of telepathy 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 cards (a third) were correctly picked in the telepathy example above, it would be wrong to conclude that there was no telepathy. 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 telepathy 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.