In a simple random sample of 50 plain M&M candies, it is found that none of them are blue. We want to use a 0.01 significance level to test the claim of Mars, Inc., that the proportion of M&M candies that are blue is equal to 0.10.
Based on data from the Federal Bureau of Investigation, violent crimes in a recent year occurred with the
distribution given in the accompanying table. The listed percentages are based on a total of 1,424,287 cases of
violent crime. Use a 0.01 significance level to test the claim that violent crimes are distributed equally among
the 12 months.
To test this claim, we can use a hypothesis test for proportions. Let's outline the steps involved in conducting this hypothesis test:
Step 1: Identify the null and alternative hypotheses
- Null hypothesis (H0): The proportion of M&M candies that are blue is equal to 0.10.
- Alternative hypothesis (Ha): The proportion of M&M candies that are blue is not equal to 0.10.
Step 2: Determine the significance level
In this case, the significance level (α) is given as 0.01. This value represents the probability of rejecting the null hypothesis when it is actually true.
Step 3: Conduct the test statistic
To conduct the test statistic, we need to calculate the sample proportion and the standard error.
Sample proportion (p̂): The sample proportion is the number of successes (blue M&M candies) divided by the sample size (50 plain M&M candies). In this case, since none of the 50 candies are blue, the sample proportion is 0/50 = 0.
Standard error (SE): The standard error measures the variability of the sample proportion. It is calculated as the square root of (p̂*(1-p̂)/n), where n is the sample size. In this case, since the sample proportion is 0 and 1-p̂ is also 0, the standard error is undefined.
Step 4: Calculate the test statistic
Since the standard error is undefined, we cannot calculate the test statistic. However, we can still conduct a hypothesis test using an alternative approach called the exact test.
Step 5: Conduct the exact test
Using software or statistical tables, we can find the p-value associated with the exact test. The p-value represents the probability of obtaining a sample proportion as extreme as or more extreme than what we observed (0) under the assumption that the null hypothesis is true.
Step 6: Make a decision
Compare the p-value to the significance level (α) to make a decision regarding the null hypothesis. If the p-value is less than or equal to α, we reject the null hypothesis. If the p-value is greater than α, we fail to reject the null hypothesis.
It's important to note that in this particular case, the test statistic is undefined, and it becomes impossible to conduct a hypothesis test using the standard approach.