How to Calculate Confidence Interval on A Ti 84

Calculating a confidence interval on a TI-84 calculator is a straightforward process that helps you estimate the range within which a population parameter (like a mean) is likely to fall. This guide will walk you through the steps, explain the formula, and provide a practical example.

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 students, you can be 95% confident that the true mean height falls within that range.

The confidence level is typically expressed as a percentage, such as 90%, 95%, or 99%. A higher confidence level means a wider interval, while a lower confidence level means a narrower interval.

Steps to Calculate on TI-84

Enter your data: Press STAT, then EDIT to enter your data into a list (e.g., L1).
Calculate the sample mean: Press 2ND then STAT to access the STAT menu. Choose 1-Var Stats and select your list (e.g., L1). The sample mean (x̄) will be displayed.
Calculate the sample standard deviation: The standard deviation (s) is also displayed in the 1-Var Stats output.
Determine the sample size: Count the number of data points in your list.
Calculate the critical value: Use the t-distribution table (2ND DISTR) for the appropriate degrees of freedom (n-1) and confidence level.
Calculate the margin of error: Multiply the critical value by the standard error (s/√n).
Calculate the confidence interval: Subtract and add the margin of error to the sample mean to get the lower and upper bounds.

Note

For large sample sizes (typically n > 30), you can use the z-distribution instead of the t-distribution. The TI-84 will automatically use the appropriate distribution based on your sample size.

Confidence Interval Formula

For a population mean (μ) with unknown standard deviation:

Confidence Interval = x̄ ± t*(s/√n)

Where:

x̄ = sample mean
t* = critical value from t-distribution
s = sample standard deviation
n = sample size

For large samples (n > 30), you can use the z-distribution instead of the t-distribution, replacing t* with z*.

Worked Example

Suppose you have a sample of 20 students with an average height of 68 inches and a standard deviation of 3 inches. Calculate a 95% confidence interval for the mean height of all students.

Sample mean (x̄): 68 inches
Sample standard deviation (s): 3 inches
Sample size (n): 20
Degrees of freedom: 19 (n-1)
Critical value (t*): 2.093 (from t-distribution table for 95% confidence)
Standard error: 3/√20 ≈ 0.424
Margin of error: 2.093 × 0.424 ≈ 0.895
Confidence interval: 68 ± 0.895 → (67.105, 68.895)

You can be 95% confident that the true mean height of all students falls between 67.11 and 68.89 inches.

Interpreting Results

A 95% confidence interval means that if you were to take 100 different samples and calculate a 95% confidence interval for each, you would expect approximately 95 of those intervals to contain the true population mean.

If your confidence interval is wide, it suggests that your sample size is small or the variability in your data is high. A narrow confidence interval indicates that your estimate is precise.

FAQ

What is the difference between a confidence interval and a confidence level?

The confidence level is the percentage that represents how confident you are that the interval contains the true population parameter. The confidence interval is the actual range of values calculated from your sample data.

When should I use a t-distribution instead of a z-distribution?

Use the t-distribution when your sample size is small (n < 30) and the population standard deviation is unknown. For larger samples (n > 30), the z-distribution is appropriate.

How does sample size affect the confidence interval?

A larger sample size results in a narrower confidence interval because the standard error decreases as the sample size increases. This means your estimate is more precise.