Hypothesis Testing: Sig. Level & Power of Test

In summary: Your Name]In summary, the conversation discusses a binomial distribution with n=10 and two possible probabilities, 0.25 and 0.5. The null hypothesis of p=0.5 is rejected and the alternative hypothesis of p=0.25 is accepted if the observed value is less than or equal to 3. The significance level and power of the test are calculated using the power function and binomial distribution formula. The calculated significance level is 0.171875, which represents the probability of observing a value less than or equal to 3 when the null hypothesis is true.
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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 [tex] = \alpha = P_{H_0}(_{10}C_i 0.5^{10}0.5^{10-i}<3) = 0.171875[/tex]

Is this correct?
 
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Thank you for your post. Your understanding of the power function and significance level is correct. The power function is the probability of rejecting the null hypothesis when the alternative hypothesis is true, and in this case it is the probability of observing a value less than or equal to 3 when the true probability is 0.25. The significance level is the probability of rejecting the null hypothesis when it is actually true, and in this case it is the probability of observing a value less than or equal to 3 when the true probability is 0.5.

Your calculation for the significance level is also correct. To find the 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 by using the binomial distribution formula and summing the probabilities of all possible outcomes less than or equal to 3.

I hope this helps clarify your understanding of the concepts. Let me know if you have any further questions.
 

FAQ: Hypothesis Testing: Sig. Level & Power of Test

What is a significance level in hypothesis testing?

A significance level, or alpha level, is the predetermined level at which a result is considered statistically significant. It is usually set at 0.05 or 0.01, and represents the probability of rejecting the null hypothesis when it is actually true.

How is the power of a test related to the significance level?

The power of a test is the probability of correctly rejecting the null hypothesis when it is false. It is directly related to the significance level, as a lower significance level means a higher power and a higher significance level means a lower power. This is because a lower significance level requires stronger evidence to reject the null hypothesis, making it more likely to correctly reject it when it is false.

What is the purpose of hypothesis testing?

The purpose of hypothesis testing is to determine whether there is enough evidence to reject the null hypothesis in favor of the alternative hypothesis. It allows us to make conclusions about a population based on a sample and helps to determine the significance of relationships or differences between variables.

What is a type I error in hypothesis testing?

A type I error, also known as a false positive, occurs when the null hypothesis is rejected even though it is true. This means that a significant result is found when there is actually no significant relationship or difference in the population. The probability of making a type I error is equal to the significance level.

What is a type II error in hypothesis testing?

A type II error, also known as a false negative, occurs when the null hypothesis is not rejected even though it is false. This means that a non-significant result is found when there is actually a significant relationship or difference in the population. The probability of making a type II error is equal to 1 minus the power of the test.

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