How to Calculate Confidence Interval for Proportion in Ti84

Calculating a confidence interval for a proportion is essential in statistics to estimate the true proportion of a population based on a sample. This guide explains how to perform this calculation using the TI-84 calculator, including step-by-step instructions and practical examples.

Introduction

A confidence interval for a proportion provides a range of values that is likely to contain the true population proportion. The TI-84 calculator can efficiently compute this interval using the normal approximation method or exact methods when appropriate.

Key concepts include:

Sample proportion (p̂)
Sample size (n)
Confidence level (typically 90%, 95%, or 99%)
Standard error of the proportion
Critical value from the standard normal distribution

Formula

The confidence interval for a proportion is calculated using the following formula:

Confidence Interval = p̂ ± z*(√(p̂*(1-p̂)/n))

Where:

p̂ = sample proportion
z = critical value from standard normal distribution
n = sample size

The critical value z depends on the desired confidence level. Common values include:

90% confidence: z ≈ 1.645
95% confidence: z ≈ 1.960
99% confidence: z ≈ 2.576

Note: For small samples (n*p̂ < 5 or n*(1-p̂) < 5), use the exact binomial method instead of normal approximation.

TI-84 Calculator Steps

Step 1: Enter Data

Press STAT and select EDIT to enter your data.
Enter your sample proportions in list L1.
Note the sample size (n) and sample proportion (p̂).

Step 2: Calculate Standard Error

Press 2ND and select DISTR to access the distribution menu.
Select A:normalcdf(.
Enter the values: normalcdf(-1E99, (p̂ - p̂)/√(p̂*(1-p̂)/n), 0).
This calculates the standard error.

Step 3: Find Critical Value

Press 2ND and select DISTR.
Select 2:InvNorm(.
Enter the confidence level (e.g., 0.95 for 95% confidence).
This gives you the z-value.

Step 4: Calculate Confidence Interval

Multiply the z-value by the standard error.
Add and subtract this value from the sample proportion to get the interval.

Worked Example

Suppose you conducted a survey and found that 60 out of 100 people support a new policy. Calculate a 95% confidence interval for the true proportion.

Step 1: Calculate Sample Proportion

p̂ = 60/100 = 0.60

Step 2: Find Critical Value

For 95% confidence, z ≈ 1.960

Step 3: Calculate Standard Error

SE = √(0.60 * 0.40 / 100) ≈ 0.049

Step 4: Compute Confidence Interval

Margin of error = 1.960 * 0.049 ≈ 0.096

Lower bound = 0.60 - 0.096 = 0.504

Upper bound = 0.60 + 0.096 = 0.696

The 95% confidence interval is approximately (0.504, 0.696).

Interpreting Results

Interpreting a confidence interval for a proportion involves understanding what the interval represents and its limitations.

The interval provides a range of plausible values for the true population proportion.
A 95% confidence interval means that if we took many samples and calculated intervals, 95% of them would contain the true proportion.
The width of the interval depends on the sample size and confidence level.

Tip: Larger sample sizes produce narrower intervals, providing more precise estimates.

FAQ

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

A confidence level (e.g., 95%) is the probability that the interval contains the true proportion. The confidence interval is the actual range of values calculated from the sample data.

When should I use the exact binomial method instead of normal approximation?

Use exact methods when the sample size is small (n*p̂ < 5 or n*(1-p̂) < 5) or when the confidence level is extreme (e.g., 99%).

How does sample size affect the confidence interval?

Larger sample sizes result in narrower confidence intervals, providing more precise estimates of the population proportion.