Hypothesis Testing: Binomial Distribution

[cse63146]

Homework Statement

Let X have a binomial distribution with the number of trails n = 10 and with p either 0.25 or 0.5. The simple null hypothesis p = 0.5 is rejected and the alternate hypothesis p = 0.25 is accepted if the observed value of X1, a random sample of size 1, is less than or equal to 3. Find the significance level and the power of the test.

Homework Equations

The Attempt at a Solution

I believe that the power function is P(X<3) = P(Bin(10,0.25) < 3) and the significane leve would be $$= \alpha = P_{H_0}(\Sigma _{10}C_i 0.5^{10}0.5^{10-i}<3) = 0.171875$$

Is this correct?

 

[nate808]

I would like to offer some feedback and further explanation to your solution.

Firstly, your understanding and use of the power function is correct. The power function represents the probability of rejecting the null hypothesis when it is false, or in other words, correctly accepting the alternative hypothesis. In this case, the power function is P(X<3), which represents the probability of observing a value less than or equal to 3 given the alternative hypothesis is true.

However, your calculation for the significance level is not correct. The significance level, also known as the alpha level, is the probability of rejecting the null hypothesis when it is actually true. In this case, the null hypothesis is p = 0.5 and the alternative hypothesis is p = 0.25. The significance level is typically set at 0.05 or 5%, so your calculation of 0.171875 is too high.

To find the correct significance level, we need to calculate the probability of observing a value less than or equal to 3 when the null hypothesis is true. This can be done using the binomial probability formula, which is P(X≤3) = ΣC(n,i)p^i(1-p)^(n-i), where n = 10, p = 0.5, and i = 0,1,2,3. Plugging in these values, we get P(X≤3) = 0.171875, which is equal to the power function you calculated.

Therefore, the significance level for this test is 0.171875 or approximately 17.2%. This means that there is a 17.2% chance of rejecting the null hypothesis when it is actually true.

I hope this helps clarify the difference between the power function and significance level, and how to correctly calculate them in this scenario. Keep up the good work!

 

[Architeuthis Dux]

I would first clarify the context and purpose of this hypothesis testing. Is it part of a larger study or experiment? What is the research question being addressed? This information would help to provide a more thorough and accurate response.

Assuming that this hypothesis testing is being conducted to determine the probability of success (p) for a binomial distribution with 10 trials, and that the null hypothesis is that p = 0.5, while the alternative hypothesis is that p = 0.25, I can provide the following response:

The significance level, also known as the Type I error rate, is the probability of rejecting the null hypothesis when it is actually true. In this case, the significance level can be calculated as the probability of observing a random sample of size 1, X1, less than or equal to 3, given that the true probability of success is 0.5. This can be calculated using the binomial distribution formula:

P(X1 ≤ 3 | p = 0.5) = (10C3)(0.5)^3(0.5)^7 + (10C2)(0.5)^2(0.5)^8 + (10C1)(0.5)(0.5)^9 + (10C0)(0.5)^0(0.5)^10 = 0.171875

This means that if the true probability of success is 0.5, there is a 17.2% chance of observing a sample with a value less than or equal to 3.

The power of the test is the probability of correctly rejecting the null hypothesis when it is false, or in other words, the probability of correctly detecting a significant difference. In this case, the power can be calculated as the probability of observing a random sample of size 1, X1, less than or equal to 3, given that the true probability of success is 0.25. This can also be calculated using the binomial distribution formula:

P(X1 ≤ 3 | p = 0.25) = (10C3)(0.25)^3(0.75)^7 + (10C2)(0.25)^2(0.75)^8 + (10C1)(0.25)(0.75)^9 + (10C0)(0.25)^0(0.75)^10 = 0.056313