How to Calculate Confidence Interval with Ti-84 Plus Ce

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 of a population, you can be 95% confident that the true population mean falls within that range.

Confidence intervals are commonly used in hypothesis testing, quality control, and survey analysis. They provide a measure of the precision of an estimate and help determine whether differences between groups are statistically significant.

Calculating Confidence Interval on TI-84 Plus CE

The TI-84 Plus CE calculator has built-in functions to calculate confidence intervals for means and proportions. Here's how to use them:

For Means (Z-Interval)

To calculate a confidence interval for the mean when the population standard deviation is known, use the Z-Interval function:

1. Press STAT then select TESTS.
2. Scroll down to Z-Interval and press ENTER.
3. Enter the sample size (n), sample mean (x̄), population standard deviation (σ), and confidence level (CLevel).
4. Press ENTER to calculate the interval.

For Means (T-Interval)

When the population standard deviation is unknown, use the T-Interval function:

1. Press STAT then select TESTS.
2. Scroll down to T-Interval and press ENTER.
3. Enter the sample size (n), sample mean (x̄), sample standard deviation (s), and confidence level (CLevel).
4. Press ENTER to calculate the interval.

For Proportions (1-PropZInt)

To calculate a confidence interval for a proportion:

1. Press STAT then select TESTS.
2. Scroll down to 1-PropZInt and press ENTER.
3. Enter the sample size (n), number of successes (x), and confidence level (CLevel).
4. Press ENTER to calculate the interval.

Step-by-Step Guide

Let's go through a complete example of calculating a confidence interval for the mean using the TI-84 Plus CE calculator.

Step 1: Enter Your Data

First, enter your sample data into the calculator's list editor. For this example, let's use the following sample of 10 test scores: 72, 75, 80, 82, 85, 88, 90, 92, 95, 98.

1. Press STAT then select EDIT.
2. Enter the data into List1.
3. Press STAT then select CALC.
4. Scroll down to 1-Var Stats and press ENTER.
5. Enter L1 for the list and press ENTER.

The calculator will display the sample size (n=10), sample mean (x̄≈86.1), and sample standard deviation (s≈8.03).

Step 2: Calculate the Confidence Interval

Now, let's calculate a 95% confidence interval for the population mean.

1. Press STAT then select TESTS.
2. Scroll down to T-Interval and press ENTER.
3. Enter the following values:
   • n: 10
   • x̄: 86.1
   • s: 8.03
   • CLevel: 95
4. Press ENTER to calculate the interval.

The calculator will display the confidence interval: (78.0, 94.2).

Step 3: Interpret the Results

We can be 95% confident that the true population mean test score falls between 78.0 and 94.2. This means if we were to take many samples and calculate a 95% confidence interval for each, approximately 95% of those intervals would contain the true population mean.

Example Calculation

Let's look at another example where we calculate a confidence interval for a proportion.

Suppose a survey of 200 people found that 120 supported a new policy. We want to calculate a 90% confidence interval for the true proportion of people who support the policy.

1. Press STAT then select TESTS.
2. Scroll down to 1-PropZInt and press ENTER.
3. Enter the following values:
   • n: 200
   • x: 120
   • CLevel: 90
4. Press ENTER to calculate the interval.

The calculator will display the confidence interval: (0.53, 0.67).

This means we can be 90% confident that between 53% and 67% of all people support the new policy.

Interpreting Results

When you calculate a confidence interval, it's important to understand what the interval represents and how to interpret it properly.

Understanding the Confidence Level

The confidence level (often 90%, 95%, or 99%) represents the probability that the calculated interval contains the true population parameter. For example, a 95% confidence interval means that if you were to take many samples and calculate a 95% confidence interval for each, approximately 95% of those intervals would contain the true population parameter.

Margin of Error

The margin of error is half the width of the confidence interval. It represents the maximum expected difference between the sample estimate and the true population parameter. For example, if your confidence interval is (78.0, 94.2), the margin of error is (94.2 - 78.0)/2 = 8.1.

Practical Implications

When interpreting confidence intervals, consider the following:

• Narrower intervals indicate more precise estimates.
• Wider intervals indicate less precision, often due to smaller sample sizes.
• Confidence intervals can help determine whether differences between groups are statistically significant.
• They provide a range of plausible values for the population parameter.