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.
a) 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.]
49% Right?
b) 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.]
The power is not 100-49=51?
Both answer you listed are incorrect.
ohhh.. Why? the significance is not 49%?
L of significance is 2.33
a) The significance level of the test is determined by the chosen threshold for determining whether to reject the null hypothesis. In this case, the threshold is set at 5,000 or more tickets marked "1" out of a sample of 10,000 tickets.
To calculate the significance level, we need to determine the probability of observing 5,000 or more tickets marked "1" if the null hypothesis is true (i.e., if the true proportion of tickets marked "1" is 49%). We can use a binomial probability calculation to find this probability.
Using the binomial probability formula, the probability of getting k successes (k tickets marked "1") in n trials (total sample size of 10,000 tickets) with a success probability of p (49%) is:
P(X ≥ 5,000) = 1 - P(X < 5,000)
where P(X < 5,000) represents the cumulative probability of getting fewer than 5,000 tickets marked "1".
To calculate this probability, we can use a binomial probability calculator or software. For instance, using a statistical software or calculator, the probability can be determined using a cumulative binomial probability distribution. Given the large sample size, an approximation can be made using a normal approximation to the binomial distribution.
After calculating the probability, the significance level is equal to 1 minus this probability, expressed as a percentage.
b) The power of the test is the probability of correctly rejecting the null hypothesis when the alternative hypothesis is true. In this case, the alternative hypothesis assumes a true proportion of 50.5% of tickets marked "1".
To calculate the power, we need to determine the probability of observing 5,000 or more tickets marked "1" when the alternative hypothesis is true (i.e., when the true proportion is 50.5%). Similar to calculating the significance level, we can use the binomial probability formula to calculate this probability.
Using a binomial probability calculator or software, we can calculate the probability of getting 5,000 or more tickets marked "1" out of a sample of 10,000 tickets when the true proportion is 50.5%.
The power of the test is equal to this probability, expressed as a percentage.
Therefore, the significance level and power cannot be determined as 49% and 51% respectively without performing the necessary calculations.