Two Ways to Calculate the Same t Interval
A one-sample t interval for a population mean can be calculated from the individual observations or from a summary of those observations. The raw-data approach starts with a list of values. The summary-statistics approach starts with the sample size \(n\), sample mean \(\bar{x}\), and sample standard deviation \(s\). When both approaches use the same unrounded sample information and confidence level, they produce the same interval.
In “Using the TInterval Calculator Function,” we saw that the calculator’s TInterval function has a Data mode for a list of observations and a Stats mode for summary statistics. This tutorial focuses on choosing between those modes and, when given raw data, finding \(s\) with 1-Var Stats. As in “Building a One-Sample t Interval by Hand,” the interval is centered at \(\bar{x}\), uses estimated standard error \(s/\sqrt{n}\), and has \(df=n-1\).
The two input methods answer different practical needs. Use Data mode when the individual observations are available. The calculator finds the sample statistics from the list and calculates the interval. Use Stats mode when you have \(n\), \(\bar{x}\), and \(s\), but do not have or need to enter every observation. Neither mode changes the statistical procedure.
When you use 1-Var Stats to summarize a list, pay close attention to the standard deviation labels. On a TI-84, Sx is the sample standard deviation \(s\), the one needed for a one-sample t interval. \(\sigma x\) is the population standard deviation calculated by treating the entered list as the entire population. Do not use \(\sigma x\) in place of \(s\) for this t interval.
Choosing and Checking the Input Method
Before calculating, identify what information the problem gives you. If it provides a list, enter the observations as data and use Data mode; you can also run 1-Var Stats to inspect the list’s summaries. If it provides \(n\), \(\bar{x}\), and \(s\), enter those in Stats mode. If the problem gives a list and you calculate the interval in both ways, the comparison is a useful check that the list was entered and summarized correctly.
Use Data mode for individual observations. Use Stats mode when \(n\), \(\bar{x}\), and \(s\) are supplied or have already been found.
For a raw list, use 1-Var Stats to check the number of observations, \(\bar{x}\), and Sx. Confirm that the sample size matches the number of data values.
Enter the requested confidence level as a decimal, such as 0.95 for 95%. The confidence level determines the critical value \(t^*\).
Confirm that the degrees of freedom are \(n-1\), the interval is centered at \(\bar{x}\), and the endpoints are in the response variable’s units.
The calculator can carry out the arithmetic, but it cannot decide whether a t interval is appropriate. As in “Building a One-Sample t Interval by Hand,” check that the data come from a random sample or an appropriate randomized process, that observations are independent, and that the sample distribution is sufficiently well behaved for the sample size. For sampling without replacement, check the 10% condition. With a small sample, a Normal population or data without strong skewness or outliers supports using the t procedure.
Worked Examples
Worked Example: From a Raw List to a 95% Interval
A fictional laboratory takes a random sample of five independent batches from a much larger production process. The process is known to have a Normal distribution for the measurement of interest. The measurements, in milliliters, are 12, 14, 15, 17, and 17. Find a 95% confidence interval for the population mean measurement.
State. Let \(\mu\) be the true mean measurement, in milliliters, for all batches from this process. We want a 95% confidence interval for \(\mu\).
Plan and conditions. Use a one-sample t interval for a population mean because the population standard deviation is unknown and we have a sample standard deviation. The five batches were randomly sampled, and the observations are stated to be independent. The process has far more than 50 batches, so \(n=5\) is less than 10% of the population size. The population is Normal, which supports a t interval even with this small sample.
Do: find the summary statistics. Enter the five measurements in a list and run 1-Var Stats. The list contains \(n=5\) observations and has \(\bar{x}=15\) milliliters. The sum of squared deviations from the mean is
Thus the sample standard deviation is \(s=\sqrt{18/(5-1)}=\sqrt{4.5}\approx2.1213\) milliliters. This is the TI-84’s Sx value. In contrast, \(\sigma x=\sqrt{18/5}\approx1.8974\) milliliters; do not use that population-standard-deviation calculation for this t interval.
The degrees of freedom are \(df=5-1=4\). For a 95% interval with 4 degrees of freedom, \(t^*\approx2.7764\). The estimated standard error and margin of error are:
So the interval is:
Conclude. We are 95% confident that the true mean measurement for all batches from this process is between approximately 12.366 and 17.634 milliliters. The interval is centered at the sample mean of 15 milliliters.
Worked Example: Checking Data Mode Against Stats Mode
Use the five batch measurements from the previous example to compare the calculator’s Data and Stats approaches. The purpose is to check that the two routes use the same sample information and return the same interval.
In Data mode, enter 12, 14, 15, 17, and 17 as the observations and select a 95% confidence level. The calculator obtains the sample size, mean, and sample standard deviation from the list. In Stats mode, enter those same summaries directly: \(n=5\), \(\bar{x}=15\), and \(s\approx2.1213\), again with confidence level 0.95.
Both modes use \(df=5-1=4\) and \(t^*\approx2.7764\). The estimated standard error is \(2.1213/\sqrt{5}\approx0.9487\), giving a margin of error of approximately \(2.7764(0.9487)=2.6340\) milliliters. Therefore each mode returns the interval \((12.3660,\ 17.6340)\), rounded to four decimal places.
This agreement is expected: Data mode first calculates the summaries and then forms the interval, while Stats mode uses the summaries you provide. If the endpoints differ noticeably, check for a mistyped observation, an incorrect sample standard deviation, a different confidence level, or a mistaken sample size. Small differences in the last displayed digit can result from rounding.
The raw list contains more detail than \(n\), \(\bar{x}\), and \(s\). Those three summaries are enough to calculate the one-sample t interval, but they do not let you reconstruct the individual measurements. For example, the list is still useful for inspecting the data’s shape and looking for unusual observations before deciding whether the t procedure is suitable.
Worked Example: Calculating From Summary Statistics
A fictional community program takes a random sample of 16 participants to estimate the mean time, in minutes, needed to complete a particular activity. The sample summary is \(n=16\), \(\bar{x}=42.6\) minutes, and \(s=5.2\) minutes. Assume the population distribution is approximately Normal, the sample is independent, and the population contains at least 160 participants. Find a 90% confidence interval for the population mean activity time.
Because the summary statistics are given, use Stats mode. Let \(\mu\) be the population mean activity time. For the plan, a one-sample t interval is appropriate: the sample was randomly selected, observations are independent, and the sample size of 16 is less than 10% of the stated population size. The population is approximately Normal, supporting the procedure.
The degrees of freedom are \(df=16-1=15\). For a central 90% interval with 15 degrees of freedom, \(t^*\approx1.7531\). The estimated standard error is:
The margin of error is \(1.7531(1.3)\approx2.2790\) minutes. The interval is:
We are 90% confident that the true mean activity time for participants in this population is between approximately 40.321 and 44.879 minutes. In Stats mode, the calculator uses the same \(n\), \(\bar{x}\), and \(s\) shown in the calculations above.
Common Mistakes and AP Exam Tips
- Using \(\sigma x\) instead of Sx. For a one-sample t interval, use the sample standard deviation \(s\). A full-credit calculation identifies Sx as \(s\), not the population-standard-deviation output.
- Confusing Data and Stats mode. Data mode expects the observations; Stats mode expects \(n\), \(\bar{x}\), and \(s\). Entering a raw list as if it were a summary, or entering \(\bar{x}\) where \(s\) belongs, does not calculate the intended interval.
- Using the wrong sample size or degrees of freedom. Check that \(n\) is the number of observations and use \(df=n-1\), not \(df=n\).
- Rounding too early. Keep the full Sx and calculator precision while finding the standard error and endpoints. Round the final interval consistently with the context and the precision of the data.
- Assuming a calculator verifies the conditions. TInterval performs calculations, not a check of random sampling, independence, the 10% condition, or sample shape. State and assess relevant conditions in the written response.
- Claiming that summary statistics preserve the whole data set. The interval can be calculated from \(n\), \(\bar{x}\), and \(s\), but those values do not reveal the individual observations or their shape.
For a clear AP response, name the parameter in context, identify the one-sample t interval, and show the relevant conditions. If the observations are given, report how you obtained \(n\), \(\bar{x}\), and \(s\); if summaries are given, state them. Then include \(df\), the interval, units, and a confidence statement about the population mean. “Recovering \(\bar{x}\) and Margin of Error From an Interval” used the interval’s endpoints to find its center and half-width; here, we go in the forward direction and use sample information to calculate the interval.
Check Your Understanding
Show your calculations where appropriate, identify the relevant calculator input, and include units when provided.
- A data list has 12 observations. Which TInterval mode uses the list, and which 1-Var Stats output gives the sample standard deviation?
- A sample has \(n=9\), \(\bar{x}=31.4\) seconds, and \(s=4.5\) seconds. Which TInterval mode is appropriate, and what are the degrees of freedom?
- Why should you not use \(\sigma x\) instead of Sx when making a one-sample t interval from a sample list?
- If Data mode and Stats mode give noticeably different intervals for the same sample, name two things you would check.
- A random sample of 20 values is taken without replacement from a population of 150. Does the 10% condition hold? Explain.