Assume that I run a test with p-value 0.021. Which statements are true? (Choose all that apply.)
There is strong evidence in favor of H_0
There is strong evidence against H_0
The probability of type I error of the test is equal to 2.1%
The test rejects H_0 at level 5\%
The test rejects H_0 at level 2\%
The test fails to reject H_0 at level 1\%
The statements that are true are:
- The test rejects H_0 at level 5%: Given that the p-value is 0.021, which is less than 0.05 (5%), we reject the null hypothesis (H_0) at the 5% significance level.
- The test fails to reject H_0 at level 1%: Although the p-value is 0.021, which is less than 0.01 (1%), we do not have enough evidence to reject the null hypothesis (H_0) at the 1% significance level.
not correct
I apologize for the incorrect response. Let me correct it:
- The test rejects H_0 at level 5%: Given that the p-value is 0.021, which is less than 0.05 (5%), we reject the null hypothesis (H_0) at the 5% significance level. Therefore, this statement is true.
- The probability of type I error of the test is equal to 2.1%: The p-value of 0.021 represents the probability of obtaining the observed data or more extreme results under the assumption that the null hypothesis (H_0) is true. Therefore, the p-value is equal to the probability of type I error, which is 2.1%. Therefore, this statement is also true.
- There is strong evidence against H_0: Since the p-value is less than the chosen level of significance (0.05), we can conclude that there is strong evidence against the null hypothesis (H_0). Therefore, this statement is true.
- The test rejects H_0 at level 2%: Given that the p-value is 0.021, which is greater than 0.02 (2%), we do not have enough evidence to reject the null hypothesis (H_0) at the 2% significance level. Therefore, this statement is false.
Please let me know if you have any further questions.
this is also incorrect
I apologize for the confusion. Let me correct my response:
- There is strong evidence against H_0: Since the p-value is less than the chosen level of significance (0.05), we can conclude that there is strong evidence against the null hypothesis (H_0). Therefore, this statement is true.
- The test rejects H_0 at level 5%: Given that the p-value is 0.021, which is less than 0.05 (5%), we reject the null hypothesis (H_0) at the 5% significance level. Therefore, this statement is also true.
- The probability of type I error of the test is equal to 2.1%: The p-value is not directly equal to the probability of type I error. The p-value represents the probability of obtaining the observed data or more extreme results under the assumption that the null hypothesis (H_0) is true. The type I error rate is the probability of rejecting the null hypothesis when it is true, and it is equal to the chosen level of significance. In this case, the type I error rate is 5% since it is the chosen level of significance. Therefore, this statement is false.
- The test fails to reject H_0 at level 1%: Since the p-value is 0.021, which is greater than 0.01 (1%), we do not have enough evidence to reject the null hypothesis (H_0) at the 1% significance level. Therefore, this statement is true.
- The test rejects H_0 at level 2%: Given that the p-value is 0.021, which is less than 0.02 (2%), we reject the null hypothesis (H_0) at the 2% significance level. Therefore, this statement is true.
I apologize for the earlier incorrect responses. Thank you for pointing it out. Please let me know if you have any further questions.
The correct statements are:
- There is strong evidence against H_0
- The test rejects H_0 at level 5%
- The test fails to reject H_0 at level 1%
Explanation:
- A p-value of 0.021 is less than the commonly used significance level of 0.05, which means there is strong evidence against the null hypothesis (H_0). Therefore, the statement "There is strong evidence against H_0" is true.
- The test rejects H_0 at level 5% because the p-value is less than the significance level. Therefore, the statement "The test rejects H_0 at level 5%" is true.
- The test fails to reject H_0 at level 1% because the p-value is greater than the significance level. Therefore, the statement "The test fails to reject H_0 at level 1%" is true.
To determine which statements are true, we need to understand what the p-value represents and how it relates to the hypothesis test.
The p-value is the probability, assuming the null hypothesis (H_0) is true, of observing a test statistic as extreme as, or more extreme than, the one obtained from the sample data. It helps us evaluate the strength of evidence against the null hypothesis.
Now, let's evaluate each statement:
1. There is strong evidence in favor of H_0: False
The p-value is 0.021, which is less than the significance level commonly used (such as 0.05 or 0.01). Therefore, there is evidence against the null hypothesis, not in favor of it.
2. There is strong evidence against H_0: True
As mentioned above, the p-value of 0.021 indicates evidence against the null hypothesis. Consequently, there is strong evidence to reject the null hypothesis based on this p-value.
3. The probability of type I error of the test is equal to 2.1%: False
The p-value does not represent the probability of a type I error. Type I error refers to incorrectly rejecting the null hypothesis when it is true. The p-value is instead the probability of observing test results as extreme or more extreme than the ones obtained, assuming the null hypothesis is true. In this case, it is 0.021 or 2.1%.
4. The test rejects H_0 at a level of 5%: True
When performing a hypothesis test at a given level of significance (such as 5%), we compare the p-value to that significance level. In this case, the p-value (0.021) is less than 0.05, which leads us to reject the null hypothesis at the 5% level.
5. The test rejects H_0 at a level of 2%: False
Since the p-value is greater than the significance level of 0.02, the test does not reject the null hypothesis at the 2% level.
6. The test fails to reject H_0 at a level of 1%: True
Similarly, since the p-value is greater than the significance level of 0.01, the test fails to reject the null hypothesis at the 1% level.
So, the true statements are:
- There is strong evidence against H_0.
- The test rejects H_0 at a level of 5%.
- The test fails to reject H_0 at a level of 1%.