TIP: Compare the observed probability of success to the expected value in a binomial experiment.
Now we rolled a die 248 times and number 3 comes up 57 times. We want to test whether the probability is 1/6 of rolling a 3. And is the proportion of 3's higher than 1/6 (Alternative = greater)?
- In this case, we should denote Number of success = 57, Number of trials = 248, Hypothesized probability of success = 1/6 and Alternative = greater.
- We can fill in the data according to the following picture.
The test result is as follows:
Exact binomial test
data: 57 and 248
number of successes = 57, number of trials = 248, p-value = 0.006334
alternative hypothesis: true probability of success is greater than 0.1666667
95 percent confidence interval:
probability of success
- In this case, under the hypothesis that the true probability of rolling a 3 on each trial is 1/6, the probability of getting number 3 for 57 or more times on is 0.006334 (p-value). If we were looking for significance at the 5% level, this result indicates that the probability of getting a number 3 is greater than 1/6. What's more, there is 95% chance that the true probability of success falls in [0.1864715 1.0000000] and we estimate it at 0.2298387.