Question 1183940: Question 5
For each of the following tobs values, determine the highest level of significance associated with the decision taken.
tobs df Decision
(i) 4.000 17 Reject Ha for a two-tailed test
(ii) 1.200 120 Reject Ha for a one-tailed test
(iii) -2.660 16 Reject Ha for both one-tailed and two-tailed test
(iv) -1.586 60 Reject Ha for a one-tailed test
NB. I have seen already answered answers of this same question but it came with only the answers. I want to see step by step process to how the answer was arrived at. And Excel formulas doesn't help too. Thank you.
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! Here's how to determine the highest level of significance for each t-value:
The level of significance (alpha, α) is the probability of rejecting the null hypothesis when it is actually true (Type I error). We find it by looking up the *area in the tails* of the t-distribution that corresponds to our calculated t-statistic (tobs).
**Understanding the Process**
1. **Degrees of Freedom (df):** This value is crucial for looking up the correct probability in a t-table or using a statistical calculator.
2. **One-tailed vs. Two-tailed Test:** This determines how we interpret the t-value and where we look for the probability.
* **Two-tailed:** We are interested in differences in *either* direction (greater than or less than). We split the alpha in half and look at both tails.
* **One-tailed:** We are interested in a difference in *only one* direction (either greater than *or* less than). We look at only one tail.
3. **Finding Alpha:** We are given a t-value and a decision (reject Hₐ). This means the t-value is *in the rejection region*. We want to find the smallest alpha where this would be true.
**Calculations**
Here's the breakdown for each case:
**(i) tobs = 4.000, df = 17, Two-tailed**
* Look up 4.000 in a t-table with 17 df. Since it is two-tailed, we are looking for the area in *both* tails.
* You'll find that 4.000 is beyond most t-table values, indicating a very small alpha.
* Using a calculator, the p-value (area in both tails) is approximately 0.0008.
* Therefore, the highest level of significance is approximately **0.0008** or **0.08%**.
**(ii) tobs = 1.200, df = 120, One-tailed**
* Look up 1.200 in a t-table with 120 df. Since it is one-tailed, we are looking at only one tail.
* Because this is a one-tailed test and we are rejecting Hₐ, it means the test is for greater than.
* Using a t-table or calculator, the p-value will be approximately 0.1151.
* The highest level of significance is approximately **0.1151** or **11.51%**.
**(iii) tobs = -2.660, df = 16, Both One-tailed and Two-tailed**
* **Two-tailed:** Look up 2.660 (the absolute value) in a t-table with 16 df. Double the value you find because it is two-tailed. The p-value will be approximately 0.0166.
* **One-tailed:** Look up 2.660 in a t-table with 16 df. The p-value will be approximately 0.0083.
* Since the decision is to reject Hₐ for both, we take the smaller p-value.
* The highest level of significance is approximately **0.0083** or **0.83%**.
**(iv) tobs = -1.586, df = 60, One-tailed**
* Look up 1.586 (the absolute value) in a t-table with 60 df. Because the test is one-tailed and tobs is negative, it means the test is for less than.
* The p-value will be approximately 0.0588.
* The highest level of significance is approximately **0.0588** or **5.88%**.
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