Calculating Confidence Interval with Ti-84 with Sample N

Calculating confidence intervals with the TI-84 calculator is a fundamental statistical skill. This guide explains how to use your calculator to determine confidence intervals for sample means when you know the sample size (n).

Introduction

A confidence interval provides a range of values that is likely to contain the true population parameter with a certain level of confidence. For sample means, the confidence interval formula depends on whether you know the population standard deviation (σ) or must estimate it using the sample standard deviation (s).

When you know the population standard deviation (σ), you use the z-distribution. When you don't know σ, you use the t-distribution with n-1 degrees of freedom. This guide focuses on the t-distribution scenario, which is more common when working with sample data.

Confidence Interval Formula

The formula for a confidence interval for the population mean (μ) when using the t-distribution is:

Confidence Interval = x̄ ± t*(s/√n) where: x̄ = sample mean t* = critical t-value from t-distribution table s = sample standard deviation n = sample size

The critical t-value depends on your desired confidence level and degrees of freedom (n-1). Common confidence levels are 90%, 95%, and 99%.

Steps to Calculate with TI-84

Step 1: Enter Your Data

1. Press STAT and then clear any existing data by pressing CLEAR EDIT. 2. Enter your sample data into list L1 by pressing STAT, then 1:Edit, and entering your values.

Step 2: Calculate Sample Statistics

1. Press STAT and select CALC. 2. Choose 1:1-Var Stats and press ENTER. 3. Enter L1 for the list and press ENTER. 4. Note the values for x̄ (sample mean) and s (sample standard deviation).

Step 3: Find the Critical t-Value

1. Press 2ND DISTR to access the DISTR menu. 2. Select 0:tcdf and press ENTER. 3. For a 95% confidence interval: - Lower bound: -1E99 - Upper bound: t*(n-1) - Degrees of freedom: n-1 4. The calculator will display the area to the left of t*. Subtract this from 1 to get the area to the right.

Step 4: Calculate the Margin of Error

1. Calculate the margin of error using: t* × (s/√n) 2. The confidence interval is then x̄ ± margin of error.

Worked Example

Suppose you have a sample of 15 students with an average score of 72 and a standard deviation of 8. Calculate a 95% confidence interval for the population mean.

Step 1: Identify Values

Sample size (n): 15
Sample mean (x̄): 72
Sample standard deviation (s): 8
Degrees of freedom: 14
Confidence level: 95%

Step 2: Find Critical t-Value

Using the TI-84: 1. Press 2ND DISTR, select 0:tcdf 2. Enter -1E99, 2.145, 14 3. The calculator shows 0.975, so t* ≈ 2.145

Step 3: Calculate Margin of Error

Margin of error = 2.145 × (8/√15) ≈ 2.145 × 1.633 ≈ 3.53

Step 4: Determine Confidence Interval

Confidence interval = 72 ± 3.53 → (68.47, 75.53)

Interpretation: We are 95% confident that the true population mean score is between 68.47 and 75.53.

Interpreting Results

A 95% confidence interval means that if you took 100 different samples and calculated the interval for each, about 95 of those intervals would contain the true population mean.

Common confidence levels and their interpretations:

Confidence Level	Interpretation	Critical t-Value (df=14)
90%	We are 90% confident the true mean is in this range	±1.345
95%	We are 95% confident the true mean is in this range	±2.145
99%	We are 99% confident the true mean is in this range	±2.977

FAQ

What if my sample size is small?

For small sample sizes (typically n < 30), you should use the t-distribution rather than the normal distribution. The TI-84 automatically handles this by using the t-distribution when you select 1:1-Var Stats.

How do I know if I can use the z-distribution?

You can use the z-distribution when you know the population standard deviation (σ) and your sample size is large (typically n > 30). For most real-world applications with sample data, you'll use the t-distribution.

What does a 95% confidence interval mean?

A 95% confidence interval means that if you took 100 different samples and calculated 95% confidence intervals for each, approximately 95 of those intervals would contain the true population mean.

How does sample size affect the confidence interval?

Larger sample sizes result in narrower confidence intervals because the standard error (s/√n) decreases as n increases. This means you can be more precise about estimating the population mean with larger samples.