Ti 84 How to Calculate Confidence Level with Confidence Interval

Introduction

Confidence intervals and confidence levels are fundamental concepts in inferential statistics. A confidence interval provides a range of values that is likely to contain the true population parameter, while the confidence level represents the probability that this interval contains the true value.

For normally distributed populations, the confidence interval for the population mean is calculated using the sample mean, standard deviation, sample size, and the appropriate z-score or t-score from the t-distribution. The TI-84 calculator can efficiently perform these calculations, especially when working with large datasets.

Formula

The formula for the confidence interval for a population mean is:

Where:

• Sample Mean - The average of the sample data
• Critical Value - The z-score or t-score corresponding to the desired confidence level
• Standard Deviation - The measure of data dispersion
• Sample Size - The number of observations in the sample

The confidence level is typically expressed as a percentage (e.g., 95% confidence level) and corresponds to a specific critical value from the standard normal distribution or t-distribution.

Steps to Calculate on TI-84

1. Enter Data: Press STAT, then EDIT to enter your data into a list (e.g., L1).
2. Calculate Sample Mean: Press STAT, then CALC, select 1:1-Var Stats, and enter L1. The sample mean (x̄) will be displayed.
3. Calculate Standard Deviation: From the same 1-Var Stats screen, note the standard deviation (σx).
4. Determine Sample Size: Count the number of data points in your list.
5. Find Critical Value: For a 95% confidence level, the critical value is approximately 1.96. For smaller samples, use the t-distribution by pressing 2ND DISTR and selecting tcdf.
6. Calculate Margin of Error: Multiply the critical value by (Standard Deviation / √Sample Size).
7. Compute Confidence Interval: Add and subtract the margin of error from the sample mean to get the lower and upper bounds of the confidence interval.

Worked Example

Suppose you have a sample of 20 test scores with a mean of 75 and a standard deviation of 10. Calculate the 95% confidence interval for the population mean.

1. Sample Mean (x̄) = 75
2. Standard Deviation (σx) = 10
3. Sample Size (n) = 20
4. Critical Value (for 95% confidence) = 2.093 (from t-table with df=19)
5. Margin of Error = 2.093 × (10 / √20) ≈ 4.62
6. Confidence Interval = 75 ± 4.62 → (70.38, 79.62)

This means we are 95% confident that the true population mean test score is between 70.38 and 79.62.

Interpreting Results

A 95% confidence interval means that if the same sampling process were repeated many times, approximately 95% of the calculated intervals would contain the true population mean. The width of the confidence interval depends on the sample size and the variability of the data.

To increase the confidence level, you can use a higher critical value (e.g., 99% confidence level), but this will result in a wider interval. Conversely, increasing the sample size will narrow the confidence interval for the same confidence level.