A spinner that has 5 areas of equal size, blue, red, green, yellow, and purple, is spun 25 times. The results are shown in the table. Are all outcomes equally likely according to the results of the experiment?
Since the spinner has 5 areas of equal size, each outcome has an initial probability of 1/5.
To test whether the outcomes are equally likely according to the results of the experiment, we can use a hypothesis test.
Null hypothesis: The outcomes are equally likely.
Alternative hypothesis: The outcomes are not equally likely.
We can use a chi-square goodness-of-fit test to determine whether the observed frequencies differ significantly from the expected frequencies (under the assumption that the null hypothesis is true). The test statistic is calculated as follows:
χ2 = Σ (observed frequency - expected frequency)2 / expected frequency
where Σ is the sum over all outcomes.
Using the observed frequencies and an expected frequency of 25/5 = 5 for each outcome, we get:
χ2 = (7-5)2/5 + (7-5)2/5 + (4-5)2/5 + (3-5)2/5 + (4-5)2/5 = 4.8
Using a chi-square distribution table with 4 degrees of freedom (5 outcomes - 1), we find that the probability of getting a test statistic of 4.8 or higher is about 0.3. This means that there is insufficient evidence to reject the null hypothesis at the 0.05 significance level.
Therefore, based on the results of the experiment, we cannot conclude that the outcomes are not equally likely.