In performing two-tailed hypothesis test with v=20, you obtain a t-statistic of 2.41. The p-value is:
Between 2% and 5%.
I need help understanding how this range was gotten.
I know that the t-statistic is between t.025 and t.01 and since this is a 2-tailed test these t-values correspond to 5% and 2% tests.
Please help
To understand how the range for the p-value was obtained, let's break down the steps:
1. The given t-statistic is 2.41, which represents the distance of the sample mean from the null hypothesis mean, measured in standard errors.
2. The next step is to determine the critical values of the t-distribution for a given significance level (alpha) and degrees of freedom (v) to define the rejection region.
3. Since this is a two-tailed test, we need to split the alpha level into two equal parts. If the desired significance level is 5%, we divide it by 2 to get 2.5% for each tail.
4. The degrees of freedom for the t-distribution are v = 20. Using a t-table or statistical software, you can find the critical t-values for a two-tailed test at the 2.5% level of significance and v = 20 degrees of freedom.
5. The critical t-value at the 2.5% level of significance for this two-tailed test (t.025) will correspond to the upper and lower bounds of the rejection region. The size of the rejection region on each tail is equal to 2.5%.
6. By comparing the t-statistic (2.41) to the critical values obtained in step 4, we find that it falls between the critical t.025 and t.01 values.
7. From this, we conclude that the p-value associated with the t-statistic of 2.41 in a two-tailed test is between the 2% (t.025) and 5% (t.01) significance levels.
Therefore, the range for the p-value is between 2% and 5% for this particular hypothesis test.
To understand how the range of the p-value was determined, you need to consider the significance level or alpha value chosen for the hypothesis test. The significance level determines the probability level at which you reject the null hypothesis.
For a two-tailed hypothesis test, the significance level is usually divided equally between the two tails. This means that both the upper and lower tails of the t-distribution are considered for determining the p-value.
Typically, the 2% and 5% values you mentioned are associated with a 95% confidence level, which is a common choice for hypothesis testing. This means that you are willing to accept a 5% chance of making a Type I error (rejecting the null hypothesis when it is actually true) and splitting this 5% equally between the two tails of the distribution.
In a t-distribution table (also available in most statistical software), you can find critical values corresponding to different confidence levels and degrees of freedom. Since you mentioned a degree of freedom (v) of 20, you would look up the t-values for v=20 and a confidence level of 95%.
For a two-tailed test, you divide the desired significance level (in this case, 5%) by 2, giving you a 2.5% significance level for each tail. The critical t-value for a 2.5% significance level with v=20 is t.025. Similarly, the critical t-value for a 5% significance level with v=20 is t.05.
Now, the t-statistic you obtained in your hypothesis test is 2.41. To determine the p-value, you need to check where this t-statistic falls in the distribution. If it falls between t.025 and t.05, then the p-value would be between 2% and 5%, as you mentioned.
However, it's important to note that the exact p-value can be obtained using statistical software or t-distribution tables. These tools provide more precise p-values based on the specific t-statistic and degrees of freedom.