Question

how to find the percentiles of the t - distribution
when you want to find percentiles for a t - distribution, you can use the t - table. a percentile is a number on a statistical distribution who is less - than the probability in the given percentage; for example, the 95th percentile of the t - distribution with n - 1 degrees of freedom is that value of whose left - tail (less than) probability is 0.05.
the t - table shows right - tail probabilities for selected t - distributions. you can use it to solve the following problems.
example no. 2
suppose you have a sample size 10 and you want to find the 95th percentile of its corresponding t - distribution. you have n - 1 = 9. the 95th percentile is the number where 95% of the values lie below it, and 5% lie above it, so you want the right - tail area to be 0.05. move across the row, find the column for 0.05, and you get 1.8331. this is the 95th percentile of the t - distribution with 9 degrees of freedom.
now, if you increase the sample size to n = 20, the value of the 95th percentile decreases; look at the row for 20 - 1 = 19 degrees of freedom, and in the column for 0.05 (a right - tail probability of 0.05) you find 1.7291.
what’s more
activity 1. supply the missing piece!
find the missing value. use the t - distribution table.

1. given: 90th percentile, sample size n = 10

find the following:
a) α
b) t - value
c) df

1. given: t - value = 2.492

find the following:
a) sample size n
b) df
c) α
d) percentile

Explanation:

We know that the t - value of 2.492 with a certain \( df \) and \( \alpha \). Looking at t - distribution tables, for \( \alpha = 0.01 \) (two - tailed or one - tailed? Let's assume one - tailed for percentile calculation, but for a common case, when \( df = 24 \), the t - value for \( \alpha=0.01 \) (right - tail) is 2.492? Wait, no, let's think again. Wait, for a two - tailed test, but here we can use the fact that \( df=n - 1 \). Let's check the t - table: the t - value 2.492 corresponds to \( df = 24 \) (since for \( df = 24 \) and \( \alpha = 0.01 \) (right - tail), the t - value is 2.492? Wait, no, actually, for a one - tailed test, when \( df = 24 \), the t - value for \( \alpha=0.01 \) is 2.492? Wait, no, let's correct. The t - value of 2.492: looking at t - tables, for \( df = 24 \), the two - tailed \( \alpha = 0.02 \) (so one - tailed \( \alpha = 0.01 \)) gives t - value 2.492? Wait, no, let's find the \( df \) first. If \( t = 2.492 \), from t - tables, when \( df = 24 \), the t - value for \( \alpha=0.01 \) (one - tailed) is 2.492. Then \( df=n - 1\), so \( n=df + 1=24 + 1 = 25 \).

Step 1: From part (a)

We found that \( df=n - 1 \), and \( n = 25 \), so \( df=25 - 1 = 24 \). Also, since \( t = 2.492 \) corresponds to \( df = 24 \) (from t - table), \( df = 24 \).

Step 1: Recall t - table

With \( df = 24 \) and \( t = 2.492 \), from the t - table, the right - tail probability \( \alpha \) is 0.01 (since for \( df = 24 \), the t - value for \( \alpha=0.01 \) (one - tailed) is 2.492).

Step 1: Understand percentile

The percentile is \( 1-\alpha \) (if \( \alpha \) is right - tail probability). Since \( \alpha = 0.01 \), the percentile is \( 1 - 0.01=0.99 \) or 99th percentile.

Answer:

The sample size \( n = 25 \)

Part (b): Find \( df \)