Find the values of the following percentiles: t0.025,15, t0.05,10, t0.10,20, t0.005,25, and t0.001,30.
The Correct Answer and Explanation is :
To find the values of the given percentiles, we need to use the t-distribution. The notation ( t_{\alpha, n} ) represents the t-score (critical value) corresponding to a given percentile (\alpha) with ( n ) degrees of freedom.
Let’s break down each of the values you’ve asked for:
1. ( t_{0.025, 15} )
This represents the t-value that corresponds to the 0.025 percentile in a t-distribution with 15 degrees of freedom. In other words, it is the t-value where the area to the right of it is 0.025, which leaves 2.5% in the tail.
2. ( t_{0.05, 10} )
This represents the 0.05 percentile in a t-distribution with 10 degrees of freedom. It is the t-value that leaves 5% in the tail.
3. ( t_{0.10, 20} )
This represents the 0.10 percentile in a t-distribution with 20 degrees of freedom. It leaves 10% in the tail.
4. ( t_{0.005, 25} )
This represents the 0.005 percentile in a t-distribution with 25 degrees of freedom. It leaves 0.5% in the tail.
5. ( t_{0.001, 30} )
This represents the 0.001 percentile in a t-distribution with 30 degrees of freedom. It leaves 0.1% in the tail.
Finding the Percentiles:
The values can be found using a t-distribution table or a statistical calculator. For each percentile, the t-distribution critical values are determined by the degrees of freedom and the given tail probability (usually denoted as (\alpha)).
You can look these values up in a statistical table or use software like Excel, R, or a calculator. For instance, using a statistical software or a t-distribution table, you would input the desired (\alpha) and the degrees of freedom to find the corresponding critical values.
Here are the approximate values:
- ( t_{0.025, 15} ) ≈ 2.131
- ( t_{0.05, 10} ) ≈ 1.812
- ( t_{0.10, 20} ) ≈ 1.325
- ( t_{0.005, 25} ) ≈ 2.787
- ( t_{0.001, 30} ) ≈ 3.646
Explanation:
The t-distribution is used when the sample size is small, and the population standard deviation is unknown. The percentile values are obtained by finding the value ( t ) that corresponds to a given cumulative probability from the left, with the specified degrees of freedom.
Each of these t-values marks a specific position on the t-distribution curve, corresponding to the given tail probabilities. The more degrees of freedom you have, the closer the t-distribution will resemble the standard normal distribution (Z-distribution), and the values will become similar to those for a Z-score.