How to Calculate Tolerance Intervals on Ti-84plus

Tolerance intervals provide a range of values within which a population parameter is expected to fall with a certain level of confidence. This guide explains how to calculate tolerance intervals using the TI-84 Plus calculator, including step-by-step instructions, formulas, and practical examples.

What is a Tolerance Interval?

A tolerance interval is a range of values that is expected to contain a specified percentage of a population with a given level of confidence. Unlike confidence intervals, which estimate a population parameter, tolerance intervals provide bounds for a specified proportion of the population.

Tolerance intervals are commonly used in quality control, manufacturing, and statistical process control to ensure product consistency and reliability.

Tolerance Interval Formula

The formula for a tolerance interval for a normal distribution is:

Tolerance Interval = X̄ ± t_{α/2, n-1} × S × √(1 + 1/n)

Where:

X̄ = sample mean
t_{α/2, n-1} = critical t-value from t-distribution
S = sample standard deviation
n = sample size
α = significance level (1 - confidence level)

For non-normal distributions, alternative methods such as the Wilson score interval or Bayesian methods may be used.

Calculating on TI-84 Plus

Step 1: Enter Your Data

Press STAT then select EDIT.
Enter your data points in list L1.
Press STAT again, then select CALC.
Choose option 1:1-Var Stats and press ENTER.
Enter L1 for the list and press ENTER.

Step 2: Calculate Tolerance Interval

Press 2nd then DISTR to access the distribution menu.
Select 7:tcdf.
Enter the lower bound: (-1E99) for a one-sided interval or (-t_{α/2, n-1}) for a two-sided interval.
Enter the upper bound: t_{α/2, n-1}.
Enter degrees of freedom: n-1.
Press ENTER to get the probability.
Use the calculated t-value in the tolerance interval formula.

Note: The TI-84 Plus does not have a built-in tolerance interval function, so you'll need to calculate it manually using the formula and the values from the 1-Var Stats output.

Example Calculation

Suppose you have a sample of 20 measurements with a mean of 50 and a standard deviation of 5. You want a 95% confidence level for a 90% tolerance interval.

Step 1: Find the t-value

Degrees of freedom = n - 1 = 19
Confidence level = 95% → α = 0.05 → α/2 = 0.025
Using tcdf on TI-84 Plus:

Lower bound: -1E99
Upper bound: t_0.025,19 ≈ 2.093
Probability ≈ 0.975

Step 2: Calculate the tolerance interval

Tolerance Interval = 50 ± 2.093 × 5 × √(1 + 1/20)

= 50 ± 2.093 × 5 × 1.06066

= 50 ± 11.15

= (38.85, 61.15)

This means we are 95% confident that 90% of the population falls between 38.85 and 61.15.

Interpreting Results

The tolerance interval provides a range of values that is expected to contain a specified percentage of the population. For example, a 90% tolerance interval with 95% confidence means that 95% of the time, 90% of the population will fall within the calculated range.

Key considerations when interpreting tolerance intervals:

The confidence level (e.g., 95%) refers to the reliability of the interval, not the proportion of the population it covers.
The tolerance level (e.g., 90%) is the proportion of the population expected to fall within the interval.
Larger sample sizes provide more precise tolerance intervals.
Assumes the population is normally distributed. For non-normal data, alternative methods may be needed.

Frequently Asked Questions

What is the difference between a confidence interval and a tolerance interval?: A confidence interval estimates a population parameter, while a tolerance interval provides bounds for a specified proportion of the population.
Can I calculate tolerance intervals on the TI-84 Plus without using the formula?: No, the TI-84 Plus does not have a built-in tolerance interval function, so you'll need to calculate it manually using the formula and the values from the 1-Var Stats output.
What assumptions are made when calculating tolerance intervals?: The primary assumption is that the data is normally distributed. For non-normal data, alternative methods may be needed.
How does sample size affect tolerance intervals?: Larger sample sizes provide more precise tolerance intervals, as they reduce the variability in the estimated parameters.
What is the difference between one-sided and two-sided tolerance intervals?: A one-sided tolerance interval provides bounds in one direction, while a two-sided interval provides bounds in both directions. The formula adjusts accordingly based on the desired interval type.