Simple Random Sample Calculate Confidence Interval with Ti 84

Calculating confidence intervals for simple random samples is a fundamental statistical task. This guide explains how to perform this calculation using the TI-84 calculator, including step-by-step instructions, the underlying formula, and practical examples.

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 simple random samples, we typically calculate confidence intervals for the population mean when the population standard deviation is unknown.

The TI-84 calculator is a powerful tool for performing these calculations quickly and accurately. This guide will walk you through the process of calculating confidence intervals using your TI-84, including the necessary formulas and practical examples.

How to Use the TI-84 Calculator

Step 1: Enter Your Data

First, you need to enter your sample data into the TI-84 calculator. Follow these steps:

Press the STAT button to access the statistics menu.
Select 1:Edit to enter your data.
Enter your sample values into the list editor. You can enter up to 999 data points.
Press STAT again to exit the list editor.

Step 2: Calculate the Sample Statistics

Next, you'll calculate the sample mean and standard deviation:

Press STAT and select CALC.
Choose 1:1-Var Stats to calculate the one-variable statistics.
Enter the list name where your data is stored (e.g., L1).
Press ENTER to see the sample statistics, including the sample mean (x̄) and sample standard deviation (s).

Step 3: Calculate the Confidence Interval

Now, you can calculate the confidence interval using the t-distribution:

Press STAT and select TESTS.
Choose A:1-PropZInt if you're calculating a confidence interval for a proportion, or D:2-SampTInt if you're comparing two means.
For a confidence interval for the mean, select 8:TInterval.
Enter the sample size (n), sample mean (x̄), and sample standard deviation (s).
Enter the confidence level (e.g., 95 for 95% confidence).
Press ENTER to see the confidence interval.

Note

The TI-84 uses the t-distribution for calculating confidence intervals when the population standard deviation is unknown. This accounts for the additional uncertainty in estimating the standard deviation from the sample.

Formula

The formula for calculating a confidence interval for the population mean when the population standard deviation is unknown is:

Confidence Interval Formula

Confidence Interval = x̄ ± t*(s/√n)

Where:

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

The critical t-value depends on the confidence level and the degrees of freedom (df = n - 1). You can find this value using the TI-84's t-distribution function or a t-table.

Worked Example

Let's walk through a practical example to calculate a 95% confidence interval for the mean height of a sample of 20 students.

Example Data

Sample size (n): 20

Sample mean (x̄): 165 cm

Sample standard deviation (s): 8 cm

Confidence level: 95%

Step 1: Find the Critical t-Value

For a 95% confidence level and degrees of freedom (df) = 20 - 1 = 19, the critical t-value is approximately 2.093.

Step 2: Calculate the Margin of Error

Margin of Error = t* × (s/√n) = 2.093 × (8/√20) ≈ 2.093 × 1.581 ≈ 3.33

Step 3: Calculate the Confidence Interval

Lower Bound = x̄ - Margin of Error = 165 - 3.33 ≈ 161.67 cm

Upper Bound = x̄ + Margin of Error = 165 + 3.33 ≈ 168.33 cm

Result

We are 95% confident that the true population mean height is between 161.67 cm and 168.33 cm.

Interpreting Results

When you calculate a confidence interval, it's important to understand what the result means. 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.

In our example, we can be 95% confident that the true mean height of all students is between 161.67 cm and 168.33 cm. This provides a range of plausible values for the population mean based on our sample data.

FAQ

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. It provides a measure of the uncertainty associated with a sample estimate.
Why do we use the t-distribution instead of the normal distribution?: We use the t-distribution when the population standard deviation is unknown and must be estimated from the sample. The t-distribution accounts for the additional uncertainty in estimating the standard deviation.
How do I choose the confidence level?: The confidence level is typically chosen based on the desired level of certainty. Common choices are 90%, 95%, and 99%. A higher confidence level results in a wider interval, providing more certainty but less precision.
What if my sample size is small?: For small sample sizes, the t-distribution will be wider than the normal distribution, resulting in a wider confidence interval. This accounts for the greater uncertainty in estimating the population parameters with a small sample.
How do I interpret a confidence interval?: A confidence interval provides a range of values that is likely to contain the true population parameter. For example, a 95% confidence interval means that if you were to take 100 different samples, you would expect approximately 95 of those intervals to contain the true population mean.