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How to Calculate Confidence Intervals on A Ti 84 Plus

Reviewed by Calculator Editorial Team

Confidence intervals are a fundamental concept in statistics that help quantify the uncertainty around an estimated parameter. This guide explains how to calculate confidence intervals for a population mean using a TI-84 Plus calculator, including step-by-step instructions, formulas, and practical examples.

What is a Confidence Interval?

A confidence interval is a range of values that is likely to contain the true population parameter with a certain level of confidence. For example, if you calculate a 95% confidence interval for the mean height of adults in a country, you can be 95% confident that the true mean height falls within that range.

Confidence Interval Formula:

For a population mean: CI = x̄ ± z*(σ/√n)

For a sample mean: CI = x̄ ± t*(s/√n)

Where:

  • x̄ = sample mean
  • σ = population standard deviation (known)
  • s = sample standard deviation (unknown)
  • n = sample size
  • z = z-score from standard normal distribution
  • t = t-score from t-distribution

The width of the confidence interval depends on:

  • The desired confidence level (common levels are 90%, 95%, and 99%)
  • The sample size (larger samples produce narrower intervals)
  • The variability in the data (higher variability produces wider intervals)

Calculating Confidence Intervals on TI-84 Plus

The TI-84 Plus calculator can compute confidence intervals for both population and sample means. Here's how to do it:

For Population Mean (σ Known)

  1. Enter your data into the calculator's list editor (STAT → EDIT)
  2. Press STAT → CALC → 1:1-VarStats to get the sample mean (x̄) and standard deviation (σ)
  3. Press 2ND DISTR to access the distribution menu
  4. Select A:normalcdf and enter the parameters for your confidence level
  5. Calculate the margin of error: z*(σ/√n)
  6. Add and subtract the margin of error from the sample mean to get the confidence interval

For Sample Mean (σ Unknown)

  1. Enter your data into the calculator's list editor
  2. Press STAT → CALC → 1:1-VarStats to get the sample mean (x̄), sample standard deviation (s), and sample size (n)
  3. Press 2ND DISTR to access the distribution menu
  4. Select D:tcdf and enter the parameters for your confidence level and degrees of freedom (n-1)
  5. Calculate the margin of error: t*(s/√n)
  6. Add and subtract the margin of error from the sample mean to get the confidence interval

Note: The TI-84 Plus uses the t-distribution for sample means because the population standard deviation is unknown. For large samples (n > 30), the t-distribution approaches the normal distribution.

Worked Example

Let's calculate a 95% confidence interval for the mean height of a sample of 25 adults, with a sample mean of 68 inches and a sample standard deviation of 2.5 inches.

Step-by-Step Calculation

  1. Determine the degrees of freedom: df = n - 1 = 25 - 1 = 24
  2. Find the t-score for 95% confidence and 24 degrees of freedom using the TI-84 Plus:
    • Press 2ND DISTR → D:tcdf
    • Enter: tcdf(-1E99,1.711,24) ≈ 0.975
    • This gives the t-score of approximately 1.711
  3. Calculate the margin of error: 1.711 * (2.5/√25) = 1.711 * 0.5 = 0.8555
  4. Calculate the confidence interval:
    • Lower bound: 68 - 0.8555 ≈ 67.1445
    • Upper bound: 68 + 0.8555 ≈ 68.8555

The 95% confidence interval for the mean height is approximately 67.14 to 68.86 inches.

Interpreting Results

When you calculate a confidence interval, you're making a statement about the range of values that is likely to contain the true population parameter. For the example above:

We can be 95% confident that the true mean height of all adults in the population falls between approximately 67.14 and 68.86 inches. This means if we were to take many samples and calculate 95% confidence intervals for each, about 95% of those intervals would contain the true population mean.

Important Notes:

  • The confidence level (95% in this case) refers to the long-run success rate of the method, not a probability about a specific interval.
  • A 95% confidence interval does not mean there's a 95% probability that the true mean falls within the interval.
  • Wider intervals indicate more uncertainty about the true parameter value.

FAQ

What is the difference between a confidence interval and a confidence level?
The confidence level is the percentage of confidence you have in your method (e.g., 95%). The confidence interval is the range of values calculated using that confidence level.
How do I choose the right confidence level?
Common choices are 90%, 95%, and 99%. Higher confidence levels result in wider intervals. Choose based on your tolerance for error and the importance of the decision.
Can I calculate a confidence interval for proportions?
Yes, the TI-84 Plus can calculate confidence intervals for proportions using the normal approximation or exact methods for small samples.
What if my sample size is small?
For small samples (n < 30), use the t-distribution instead of the normal distribution. The TI-84 Plus automatically handles this when using the appropriate statistical functions.
How do I know if my confidence interval is narrow enough?
A narrow interval indicates more precise estimation. You can make the interval narrower by increasing the sample size or reducing the confidence level.