Using the TInterval Calculator Function

Let the Calculator Handle the Arithmetic

In “Building a One-Sample t Interval by Hand,” you assembled an interval from the sample mean, standard error, and t critical value. The TI-84 TInterval function carries out that calculation for you. You can enter either a list of individual observations or a summary of the sample.

The calculator saves time, but it does not decide whether a t interval is appropriate. You still need to identify the population mean $\mu$, check the conditions, choose the correct input mode, and interpret the output in context. The interval it calculates follows the same form you have already studied:

Formula: TInterval calculates a one-sample t interval of the form $$ \bar{x}\ \pm\ t^*\left(\frac{s}{\sqrt{n}}\right). $$ The confidence level determines $t^*$, and the degrees of freedom are $df=n-1$.

On a TI-84, open STAT, go to TESTS, and select TInterval. Menu numbering can vary, so select the procedure by its name. Then choose Data if you have individual observations in a list, or Stats if you have summary statistics.

In Data mode, enter the list containing the observations, the frequency, and the confidence level. If each list entry represents one observation, the frequency is 1. In Stats mode, enter $\bar{x}$, $S_x$, $n$, and the confidence level. The calculator’s $S_x$ is the sample standard deviation $s$ used in a one-sample t interval. Do not enter the population standard deviation $\sigma$.

Calculator inputs: Use Data mode for a list of raw observations. Use Stats mode for the sample mean $\bar{x}$, sample standard deviation $s$, and sample size $n$. In either mode, enter the confidence level as a decimal, such as 0.95 for 95% confidence.

What TInterval Does—and Does Not—Check

TInterval calculates endpoints using the entries you give it. It does not know whether the sample was random, whether observations are independent, whether the 10% condition is met for sampling without replacement, or whether the data are suitable for a t procedure. Those checks remain your responsibility, as in the previous tutorial.

Before using the calculator, identify the population the sample is meant to represent. Check that the data came from a random sample or an appropriate randomized process, and address independence. For a sample selected without replacement from a finite population, check the 10% condition. When the sample is small, inspect the data or a graph for strong skewness or outliers; a Normal population is also appropriate. The calculator cannot confirm any of these conditions.

The output typically includes the lower and upper endpoints in parentheses, followed by $\bar{x}$, $S_x$, and $n$. The first pair is the confidence interval. The values after it are a useful input check: compare them with the sample information in the question. If the reported $n$ or $S_x$ is unexpected, stop and check the list, frequency, and mode before using the interval.

Key output-reading rule: Read the first two numbers as the interval endpoints. Read $\bar{x}$, $S_x$, and $n$ as the sample statistics used to calculate that interval—not as additional endpoints.

Worked Examples

Worked Example: Entering Raw Data in Data Mode

A fictional horticulture class randomly selects 8 seedlings from a group of 500 and measures their heights, in centimeters: 18, 20, 21, 19, 22, 17, 23, and 20. The sample is roughly symmetric, with no apparent outliers. Find a 95% confidence interval for the population mean seedling height using TInterval.

State. The parameter is $\mu$, the population mean height of the seedlings. We want a 95% one-sample t interval.

Plan. The observations come from a random sample. Because the sample is selected without replacement, check the 10% condition: $8<0.10(500)=50$, so the condition is met and treating the observations as independent is reasonable. The sample is small, but its distribution is roughly symmetric with no apparent outliers. These conditions support using a one-sample t interval.

Do. Enter the eight measurements in a list, such as L1. Open TInterval, choose Data, enter L1 as the list, set Freq to 1, and enter 0.95 for C-Level. Select Calculate.

The calculator returns an interval of approximately $(18.3280,\ 21.6720)$, along with $\bar{x}=20$, $S_x=2$, and $n=8$. Check the reported statistics against the data. The sum is 160, so $\bar{x}=160/8=20$. The sum of squared deviations from 20 is 28, so

s=\sqrt{\frac{28}{8-1}}=2.

As a check on the calculator’s endpoints, $df=8-1=7$, and for 95% confidence $t^*\approx2.3646$. Thus

SE_{\bar{x}}=\frac{2}{\sqrt{8}}\approx0.7071, \qquad \text{margin of error}=2.3646\left(\frac{2}{\sqrt{8}}\right)\approx1.6720.

The hand-calculation check gives $20\pm1.6720=(18.3280,\ 21.6720)$, agreeing with TInterval to the displayed precision.

Conclude. We are 95% confident that the population mean height of the seedlings is between approximately 18.3280 and 21.6720 centimeters. This is an interval for the population mean, not for individual seedling heights.

Worked Example: Entering Summary Statistics in Stats Mode

A fictional community garden randomly selects 25 pumpkins from a harvest of 800. Their weights have a sample mean of 64.2 kilograms and a sample standard deviation of 8.5 kilograms. The sample distribution is approximately symmetric, with no outliers. Use TInterval to find a 90% confidence interval for the population mean pumpkin weight.

State. The parameter is $\mu$, the population mean weight of pumpkins in this harvest. We want a 90% one-sample t interval.

Plan. The sample is random. For sampling without replacement, $25<0.10(800)=80$, so the 10% condition is met and independence is reasonable. The sample is not large, but the approximately symmetric shape without outliers supports using a t interval.

Do. Open TInterval and choose Stats. Enter $\bar{x}=64.2$, $S_x=8.5$, $n=25$, and C-Level $=0.90$. Select Calculate. The output is approximately $(61.2915,\ 67.1085)$, with $\bar{x}=64.2$, $S_x=8.5$, and $n=25$.

The reported sample statistics match the supplied summary. To verify the interval, calculate $df=25-1=24$. For a central 90% interval, $t^*\approx1.7109$. The estimated standard error and margin of error are

SE_{\bar{x}}=\frac{8.5}{\sqrt{25}}=1.7, \qquad \text{margin of error}=1.7109(1.7)\approx2.9085.

Therefore, the hand calculation is

64.2\pm2.9085=(61.2915,\ 67.1085)\text{ kilograms}.

Conclude. We are 90% confident that the population mean pumpkin weight is between approximately 61.2915 and 67.1085 kilograms.

Worked Example: Checking the Output for a Small Sample

A fictional environmental science class randomly samples 9 ponds from a region containing 300 ponds and records a dissolved-oxygen measurement, in milligrams per liter, for each pond. The measurements are 4, 5, 6, 7, 8, 5, 6, 7, and 6. The distribution is roughly symmetric, with no apparent outliers. Use Data mode to find a 90% confidence interval for the population mean measurement.

State. The parameter is $\mu$, the population mean dissolved-oxygen measurement for ponds in the region. We want a 90% one-sample t interval.

Plan. The data come from a random sample, and $9<0.10(300)=30$, so the 10% condition is met. The observations are independent enough for the procedure, and the small sample’s roughly symmetric shape without apparent outliers supports using a t interval.

Do. Enter the nine measurements in L1. Choose Data in TInterval, set List to L1, Freq to 1, and C-Level to 0.90. The calculator gives approximately $(5.2408,\ 6.7592)$, with $\bar{x}=6$, $S_x\approx1.2247$, and $n=9$.

The output is internally consistent: the measurements sum to 54, so $\bar{x}=54/9=6$. Their squared deviations from 6 sum to 12, giving

s=\sqrt{\frac{12}{9-1}}=\sqrt{1.5}\approx1.2247.

For $df=8$, the 90% critical value is $t^*\approx1.8595$. Using $s=\sqrt{1.5}$, the standard error is $\sqrt{1.5}/3\approx0.4082$, and the margin of error is approximately $1.8595(0.4082)=0.7592$. Thus the interval check is

6\pm0.7592=(5.2408,\ 6.7592)\text{ milligrams per liter}.

Conclude. We are 90% confident that the population mean dissolved-oxygen measurement for the region’s ponds is between approximately 5.2408 and 6.7592 milligrams per liter.

Common Mistakes and AP Exam Tips

Choosing the wrong input mode. Data mode needs a list of observations; Stats mode needs $\bar{x}$, $S_x$, and $n$. A summary mean is not a raw-data list, and a list should not be entered into the summary-statistics fields.
Entering the wrong standard deviation. Use the sample standard deviation $s$, labeled $S_x$ on the calculator. A population standard deviation $\sigma$ is not the input for a one-sample t interval.
Using the wrong confidence-level entry. Enter 0.95 for 95% confidence, not 95. Check that the confidence level matches the question.
Forgetting the frequency setting. If every list entry represents one observation, set Freq to 1. An unintended frequency can change the effective sample size and the resulting interval.
Assuming the calculator checks conditions. TInterval performs arithmetic; it does not verify randomness, independence, the 10% condition, or sample shape. State the relevant checks in your written response.
Misreading the display. The lower and upper endpoints are the interval. The accompanying $\bar{x}$, $S_x$, and $n$ are sample statistics, not more interval limits.
Reporting only the calculator output. A complete AP response identifies the population mean, checks conditions, reports the interval with units, and gives a contextual conclusion. A calculator display alone does not explain why the procedure is appropriate.

A quick audit can catch many input errors: check that the reported sample size matches the data, that $\bar{x}$ is plausible for the observations, and that $S_x$ is a sample standard deviation. If possible, compare the result with $\bar{x}\pm t^*(s/\sqrt{n})$, as in these examples. Small differences in the last displayed digit can result from rounding; a substantially different interval usually signals an input or mode problem.

Key takeaway: Use Data mode for raw observations and Stats mode for $\bar{x}$, $s$, and $n$. Enter the confidence level as a decimal, check the calculator’s reported sample statistics, and verify the conditions yourself. Read the interval endpoints as estimates for the population mean $\mu$.

Check Your Understanding

Answer each question using the TInterval ideas from this tutorial.

You have 14 individual measurements stored in L2. Which TInterval mode should you select, and what should the frequency be if each entry is one observation?
A question gives $\bar{x}=31.6$, $s=4.2$, and $n=18$. Which mode should you use, and what value should you enter for a 95% confidence level?
A calculator reports an interval followed by $\bar{x}=12.4$, $S_x=2.1$, and $n=10$. Which values are the interval endpoints, and what does $S_x$ represent?
Why is it not enough to press Calculate before using a one-sample t interval? Name two conditions that the calculator does not check.
If the output reports $n=20$ but the problem describes a sample of 12 observations, what should you check before reporting the interval?

Using the TInterval Calculator Function

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