How To Find A Critical Value On A Ti 84

How to Find a Critical Value on a TI-84: A Step-by-Step Guide

Understanding how to find a critical value on a TI-84 calculator is an essential skill for anyone working with statistics, from high school students to college researchers and professionals. That said, this guide will demystify the process, providing clear, actionable steps for the most common scenarios using the TI-84’s built-in statistical functions. When your test statistic crosses this boundary, you have sufficient evidence to reject the null hypothesis. Critical values are the gatekeepers of hypothesis testing; they define the boundary between the rejection region and the non-rejection region. Mastering these keystrokes will save you time, reduce errors from statistical tables, and deepen your conceptual understanding of significance testing.

The Foundation: What is a Critical Value?

Before pressing any buttons, it’s crucial to grasp the concept. , standard normal z, Student’s t, chi-square). The significance level (α), commonly 0.g.05, 0.The type of test: Is it left-tailed, right-tailed, or two-tailed? It marks the cutoff point beyond which we deem our sample result too unlikely to have occurred by random chance if the null hypothesis were true. The probability distribution of your test statistic (e.01, or 0.The location of this value depends entirely on three factors:

1. 2. A critical value is a point on the scale of the test statistic (like a z-score or t-score) that is derived from the chosen significance level (alpha, α). Practically speaking, 3. 10.

The TI-84 calculator excels at computing these values through its inverse probability functions, which essentially answer the question: “What score corresponds to a given cumulative probability from the left?”

Finding Critical Values for the Standard Normal (z) Distribution

The invNorm function is your tool for finding z-critical values. It is used when the population standard deviation is known, or for large samples (n > 30) by the Central Limit Theorem.

Step-by-Step for invNorm:

1. Press 2ND then VARS to access the DISTR (distribution) menu.
2. Scroll down to option 3:invNorm( and press ENTER.
3. Enter the area (probability) to the left of the desired critical value. This is the key step that requires careful thought based on your test’s tail(s):
   • For a right-tailed test: The rejection region is in the upper tail. The area to the left of the critical value is 1 - α. For α=0.05, enter 0.95.
   • For a left-tailed test: The rejection region is in the lower tail. The area to the left of the critical value is α. For α=0.05, enter 0.05.
   • For a two-tailed test: The rejection region is split between both tails. Each tail has an area of α/2. The area to the left of the positive critical value is 1 - α/2. For α=0.05, enter 0.975.
4. Set the parameters for the normal distribution. After the area, enter a comma, then the mean (μ, typically 0 for the standard normal), and a comma, then the standard deviation (σ, typically 1). The syntax is invNorm(area, μ, σ).
5. Close the parenthesis and press ENTER. The calculator will display the z-critical value.

Example (Two-tailed, α=0.05): invNorm(0.975,0,1) returns 1.95996, which rounds to 1.96.

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Finding Critical Values for the Student’s t-Distribution

When the population standard deviation is unknown and the sample size is small (n ≤ 30), you use the t-distribution. The critical value now also depends on the degrees of freedom (df), calculated as df = n - 1. The TI-84 uses the invT function.

Step-by-Step for invT:

1. Press 2ND then VARS to access the DISTR menu.
2. Scroll down to option 4:invT( and press ENTER.
3. Enter the area to the left of the desired critical value. The logic for tails is identical to the z-distribution:
   • Right-tailed: Area = 1 - α
   • Left-tailed: Area = α
   • Two-tailed: Area = 1 - α/2 (for the positive t-value)
4. Enter the degrees of freedom (df) after a comma.
5. Close the parenthesis and press ENTER.

Example (Two-tailed, α=0.05, n=15): df = 14. invT(0.975,14) returns 2.14479, which rounds to 2.145.

Important Note: The t-distribution is symmetric. For a two-tailed test, you will get the positive critical value. The negative critical value is simply its opposite (e.g., ±2.145). For a left-tailed test, you would directly enter the small left-tail area (e.g., invT(0.05,14) gives -1.761).

Handling Other Distributions: Chi-Square and F

For tests involving variance or analysis of variance (ANOVA), you may need critical values from the **chi-square (χ²