Utk yg mo Bantu2 Keuangan saya..monggo ke Bank Central Asia BCA 5520166779 a.n. Andreas Tparlaungan Manurung (Indonesia) For those who would like to help support my finances ..please feel free to send it to Bank Central Asia (BCA) account number 5520166779 under the name Andreas Tparlaungan Manurung (Indonesia)

How to Calculate a T-Distribution Critical Value Using Basic Arithmetic

To approximate a t-distribution critical value using only ordinary mathematical operators (+, -, *, /), mathematicians use an asymptotic approximation formula known as the Cornish-Fisher expansion. This method adjusts a standard Z-score based on the degrees of freedom using basic arithmetic operations.

---

The Approximation Formula

To find the estimated t-value for a standard 95% confidence level, we start with the base Z-score (1.960) and apply a correction term:

t = z + ( (z * z * z + z) / (4 * nu) )

• z = The base Z-score for 95% confidence (1.960)
• nu = Degrees of freedom ($df$)

---

Step-by-Step Calculation Example

Let's calculate the estimated t-value for a small sample size where $df = 10$ ($\nu = 10$).

Step 1: Cube the Z-score (z * z * z)

• 1.960 * 1.960 = 3.8416
• 3.8416 * 1.960 = 7.529536

Step 2: Add the Z-score to the cubed value (... + z)

• 7.529536 + 1.960 = 9.489536

Step 3: Multiply 4 by the degrees of freedom (4 * nu)

• 4 * 10 = 40

Step 4: Divide the numerator by the denominator (... / ...)

• 9.489536 / 40 = 0.2372384

Step 5: Add the adjustment to the original Z-score (z + ...)

• 1.960 + 0.2372384 = 2.1972384

---

Final Comparison

Method	Result Value
Arithmetic Approximation	2.197
True T-Table Value (Exact via integration)	2.228

By using just basic arithmetic steps (+, -, *, /), the formula successfully adjusts the flat Z-score (1.96) outward to account for small-sample uncertainty, landing very close to the true statistical table value.