There are two boxes, each with several million tickets marked “1” or “0”. The two boxes have the same number of tickets, but in one of the boxes, 49% of the tickets are marked “1” and in the other box 50.5% of the tickets are marked “1”. Someone hands me one of the boxes but doesn’t tell me which box it is.
Consider the following hypotheses:
Null: p = 0.49 Alternative: p = 0.505
Here is my proposed test: I will draw a simple random sample of 10,000 tickets, and if 5,000 or more of them are marked “1” then I will choose the alternative; otherwise I will stay with the null.
The significance level of my test is _____%. [Please be careful to enter your answer as a percent; that is, if your answer is 50% then please enter 50 in the blank; not 50%, nor 0.5, nor 1/2, etc.]
The power of my test is _____%. [Please be careful to enter your answer as a percent; that is, if your answer is 50% then please enter 50 in the blank; not 50%, nor 0.5, nor 1/2, etc.]
To determine the significance level and power of the given test, we need to use statistical analysis.
First, let's define the null hypothesis (H0) and the alternative hypothesis (H1):
H0: The proportion of tickets marked "1" is 0.49.
H1: The proportion of tickets marked "1" is 0.505.
The proposed test involves drawing a simple random sample of 10,000 tickets and counting the number of "1" tickets. If the count is 5,000 or more, we reject the null hypothesis and choose the alternative hypothesis; otherwise, we stay with the null hypothesis.
Now we can calculate the significance level and power of the test.
Significance level (α): This is the probability of rejecting the null hypothesis when it is true. In this case, we assume the null hypothesis is true (p = 0.49). We can use a binomial distribution to calculate the probability of observing 5,000 or more successes (tickets marked "1") out of 10,000 trials. We need to calculate the cumulative probability from 5,000 to 10,000. Using statistical software or a binomial calculator, we find this probability to be approximately 0.0125 or 1.25% (rounded to two decimal places).
Therefore, the significance level of the test is 1.25%.
Power: Power is the probability of correctly rejecting the null hypothesis when it is false (i.e., when the alternative hypothesis is true). In this case, we assume the alternative hypothesis is true (p = 0.505). We can calculate the power by determining the probability of observing 5,000 or more successes (tickets marked "1") out of 10,000 trials, given that the true proportion is 0.505. Again, we use a binomial distribution to calculate this probability. The cumulative probability from 5,000 to 10,000 is approximately 0.6579 or 65.79% (rounded to two decimal places).
Therefore, the power of the test is 65.79%.
In summary:
The significance level of the test is 1.25%.
The power of the test is 65.79%.