Confidence Intervals on TI-84 Calculator

Confidence Intervals on TI-84 Calculator, A confidence interval (CI) is a statistical interval used to estimate an unknown population parameter from sample data. It provides a range of plausible values for the population parameter at a specified confidence level, such as 90%, 95%, or 99%.

The TI-84 calculator provides several built-in functions for calculating confidence intervals. The appropriate procedure depends on the type of parameter being estimated and the information available from the sample.

This guide covers three common confidence intervals:

1. Confidence interval for a population mean when the population standard deviation (σ) is known
2. Confidence interval for a population mean when σ is unknown
3. Confidence interval for a population proportion

What Is a Confidence Interval?

A confidence interval is commonly expressed as:

Point Estimate ± Margin of Error

For example, suppose a 95% confidence interval for a population mean is:

(12.5, 17.5)

The sample estimate is the midpoint:

(12.5 + 17.5) / 2 = 15

The margin of error is:

(17.5 − 12.5) / 2 = 2.5

Therefore, the result can be written as:

15 ± 2.5

A 95% confidence interval does not mean that there is a 95% probability that the fixed population parameter lies inside this particular interval. Rather, the 95% confidence level describes the long-run performance of the interval-generating procedure.

Choosing the Correct TI-84 Confidence Interval

Before calculating a confidence interval, identify the parameter you are estimating.

Choosing the correct procedure is important because the formulas and sampling distributions differ.

Example 1: Confidence Interval for a Population Mean When σ Is Known

Suppose we want to calculate a 95% confidence interval for a population mean with:

• Sample mean (x̄) = 14
• Sample size (n) = 35
• Population standard deviation (σ) = 4
• Confidence level = 95%

Because the population standard deviation is known, we use a Z interval.

Step 1: Select ZInterval

On the TI-84, press:

STAT → TESTS → ZInterval

Select:

Stats

Step 2: Enter the Summary Statistics

Enter:

• σ = 4
• x̄ = 14
• n = 35
• C-Level = 0.95

Then select:

Calculate

Step 3: Interpret the Result

The TI-84 will produce a confidence interval of approximately:

(12.674, 15.326)

The small difference from other reported values can occur because of rounding.

Therefore, the 95% confidence interval for the population mean is approximately:

12.67 < μ < 15.33

We can also express this as:

14 ± 1.326

This means that the sample provides an interval estimate extending approximately 1.326 units on either side of the sample mean.

Example 2: Confidence Interval for a Population Mean When σ Is Unknown

Now suppose:

• Sample mean (x̄) = 12
• Sample standard deviation (Sx) = 6.3
• Sample size (n) = 19
• Confidence level = 95%

Here, the population standard deviation is unknown. We therefore use a t interval.

Step 1: Select TInterval

On the TI-84, press:

STAT → TESTS → TInterval

Select:

Stats

Step 2: Enter the Summary Statistics

Enter:

• x̄ = 12
• Sx = 6.3
• n = 19
• C-Level = 0.95

Then select:

Calculate

Step 3: Interpret the Result

The TI-84 will produce a confidence interval of approximately:

(8.963, 15.037)

Therefore, the 95% confidence interval is approximately:

8.963 < μ < 15.037

The margin of error is approximately:

(15.037 − 8.963) / 2 ≈ 3.037

So the result can be written approximately as:

12 ± 3.037

Why Use TInterval?

When the population standard deviation is unknown, the sample standard deviation is used to estimate it. The t-distribution accounts for the additional uncertainty introduced by estimating the population standard deviation.

The t distribution is particularly important for smaller samples.

Example 3: Confidence Interval for a Population Proportion

Suppose a survey produces:

• Number of successes (x) = 12
• Number of trials (n) = 19
• Confidence level = 95%

The sample proportion is:

p̂ = x / n

Therefore:

p̂ = 12 / 19 ≈ 0.6316

We can use the TI-84’s 1-PropZInt function to calculate the confidence interval.

Step 1: Select 1-PropZInt

On the TI-84, press:

STAT → TESTS → 1-PropZInt

Step 2: Enter the Information

Enter:

• x = 12
• n = 19
• C-Level = 0.95

Then select:

Calculate

Step 3: Interpret the Result

The calculator will produce a confidence interval of approximately:

(0.4147, 0.8485)

Therefore, the 95% confidence interval for the population proportion is approximately:

0.4147 < p < 0.8485

Expressed as percentages:

41.47% < p < 84.85%

The corresponding margin of error is approximately:

(0.8485 − 0.4147) / 2 ≈ 0.2169

or approximately:

21.69 percentage points

How to Interpret a Confidence Interval

Suppose your TI-84 produces:

95% CI = (8.963, 15.037)

A suitable interpretation is:

We are 95% confident that the population mean lies between 8.963 and 15.037, based on the procedure used to construct the interval.

For a population proportion, suppose the result is:

95% CI = (0.4147, 0.8485)

You could report:

We are 95% confident that the population proportion is between approximately 41.47% and 84.85%.

Remember that the confidence level describes the statistical procedure rather than the probability that the parameter moves into or out of the calculated interval.

Confidence Level and Interval Width

The confidence level affects the width of the confidence interval.

For the same dataset:

90% confidence interval → narrower interval

95% confidence interval → wider interval

99% confidence interval → even wider interval

Therefore, increasing the confidence level generally increases the margin of error.

This creates a trade-off between confidence and precision.

Sample Size and Confidence Intervals

Sample size also affects the width of a confidence interval.

Generally, as sample size increases, the standard error decreases and the confidence interval becomes narrower.

For example, if two studies estimate the same population parameter and have similar variability, the study with the larger sample size will generally produce a more precise estimate.

However, increasing the sample size does not necessarily eliminate other sources of bias. A very large biased sample can still produce a misleading estimate.

ZInterval vs TInterval

One of the most common questions when using a TI-84 is whether to select ZInterval or TInterval.

Use ZInterval when:

• Estimating a population mean
• The population standard deviation σ is known

Use TInterval when:

• Estimating a population mean
• The population standard deviation σ is unknown
• The sample standard deviation Sx is used

For example:

σ = 4 is known → ZInterval

σ unknown and Sx = 6.3 → TInterval

Do not choose the procedure simply because the sample size is large or small. The availability of the population standard deviation is a key consideration for these two procedures.

Confidence Interval for a Mean From Raw Data

The examples above use summary statistics. The TI-84 can also calculate confidence intervals directly from raw data stored in a list.

Suppose your observations are stored in L1.

For a T interval:

STAT → TESTS → TInterval

Select:

Data

Then enter:

• List = L1
• Freq = 1
• C-Level = 0.95

Select:

Calculate

The calculator will calculate the sample mean and sample standard deviation from the data and then construct the confidence interval.

This is particularly useful when you have the individual observations rather than only the summary statistics.

Confidence Interval for a Proportion

For a population proportion, the TI-84 uses the 1-PropZInt procedure.

The basic inputs are:

x = number of successes

n = total number of trials

C-Level = desired confidence level

The sample proportion is:

p̂ = x/n

The confidence interval then provides an estimate of the plausible range for the population proportion.

Common Mistakes When Calculating Confidence Intervals

Mistake 1: Using ZInterval when σ is unknown

If the population standard deviation is unknown and you are using the sample standard deviation, TInterval is generally the appropriate procedure for estimating a population mean.

Mistake 2: Entering the wrong sample size

Make sure n represents the number of observations in the sample.

Mistake 3: Entering 95 instead of 0.95

The TI-84 expects the confidence level as a decimal.

For example:

90% → 0.90

95% → 0.95

99% → 0.99

Mistake 4: Confusing x with n for a proportion

For 1-PropZInt:

x = number of successes

n = total number of observations

Mistake 5: Interpreting a confidence interval as a probability statement about a fixed parameter

The confidence level refers to the performance of the interval procedure over repeated samples. It is not technically correct to say that there is a 95% probability that the fixed population parameter is inside the already calculated interval.

Frequently Asked Questions

What is a confidence interval?

A confidence interval is a range of plausible values for an unknown population parameter based on sample data and a specified confidence level.

How do you calculate a confidence interval on a TI-84?

Choose the appropriate procedure under:

STAT → TESTS

Common options include ZInterval, TInterval, and 1-PropZInt.

What is the difference between ZInterval and TInterval?

ZInterval is used for a population mean when the population standard deviation is known. TInterval is generally used when the population standard deviation is unknown and the sample standard deviation is used.

How do you calculate a 95% confidence interval?

Enter the appropriate sample information into the corresponding TI-84 interval function and set:

C-Level = 0.95

What happens when the confidence level increases?

The confidence interval generally becomes wider, resulting in a larger margin of error.

What happens when sample size increases?

Holding other factors constant, a larger sample generally produces a narrower confidence interval.

What does a 95% confidence interval mean?

It means the interval-generating procedure has a 95% long-run coverage rate under the assumptions of the method. It does not mean there is a 95% probability that the fixed population parameter is inside this particular calculated interval.

Online Confidence Interval Calculator

If you do not have access to a TI-84 calculator, an online confidence interval calculator can be used to calculate confidence intervals for means, proportions, and other statistical parameters.

An online calculator can also be useful for checking TI-84 calculations and experimenting with different sample sizes and confidence levels.

Conclusion

Confidence intervals are an essential part of statistical inference because they provide a range of plausible values for an unknown population parameter rather than relying only on a single sample estimate.

On a TI-84 calculator, the basic workflow is:

Identify the parameter → choose the appropriate interval procedure → enter the sample information → select the confidence level → calculate → interpret the interval

For the three common situations covered in this tutorial:

• Known population standard deviation → ZInterval
• Unknown population standard deviation → TInterval
• Population proportion → 1-PropZInt

The confidence level and sample size both affect the precision of the resulting estimate. A higher confidence level generally produces a wider interval, while a larger sample generally produces a narrower interval.

For more calculator-based statistics tutorials, explore the TI-84 Archives » FINNSTATS.