Let the Calculator Handle the Arithmetic
In “The Form of a Two-Sample t Interval,” you learned that an interval for \(\mu_1-\mu_2\) combines the difference in sample means, its standard error, and a t critical value. The 2-SampTInt function performs those calculations for two independent samples. You can enter either summary statistics or the raw observations in lists.
The calculator is a useful arithmetic tool, but it cannot decide whether the samples are independent, whether the data support a t procedure, or what the population difference means in context. Check those parts yourself. The calculator also follows the order of the groups you enter, so label sample 1 and sample 2 before typing anything.
Choose the Correct Input Mode
On a TI-84, open STAT, choose TESTS, and select 2-SampTInt. Menu wording and numbering may vary slightly on other calculator models. The important choice is whether you are entering summary statistics or lists of observations.
| Calculator input | Choose this when | Enter for each sample |
|---|---|---|
| Stats | You are given summary statistics rather than individual observations. | \(\bar{x}\), \(Sx\), and \(n\) |
| Data | You have the individual observations stored in lists. | The list containing each sample and its frequency list, if applicable |
For Stats mode, enter the first group’s sample mean, sample standard deviation, and sample size, followed by the corresponding values for the second group. Enter \(s\), the sample standard deviation, in the \(Sx\) field—not the population standard deviation \(\sigma\). Then enter the confidence level as a decimal, such as 0.95 for 95% confidence.
For Data mode, enter or select a list for each group. On a TI-84, lists such as L1 and L2 are common choices. If each observation appears once, set each frequency to 1. The frequency setting is not a substitute for checking that the observations are recorded correctly. Confirm that the first list is the group you defined as sample 1 and the second list is sample 2.
For the usual AP Statistics two-sample t interval, select Pooled: No. This uses the unpooled procedure and Welch degrees of freedom, which allow the population standard deviations to differ. Do not select pooled simply because the sample sizes are equal or the sample standard deviations look similar. Use a pooled procedure only if the problem specifically asks for it and its assumptions are part of the method being used.
Check Conditions Before Interpreting the Output
A calculator will produce an interval even when the design or data do not justify one. As in “The Form of a Two-Sample t Interval” and the earlier tutorials on independent samples and two-sample degrees of freedom, check the conditions before treating the output as a useful estimate.
- Independent groups: The two samples must be independent, not paired or linked by the design. Random sampling supports generalizing to the population sampled; in a randomized experiment, random assignment supports causal conclusions about the study subjects but does not by itself justify generalizing to a broader population.
- 10% condition: If sampling without replacement, each sample should be no more than 10% of its population, so observations within a sample can reasonably be treated as independent.
- Nearly Normal condition: For small samples, inspect each group’s data or graph for strong skewness and outliers. Larger samples can generally tolerate more departure from Normality, but the data still need to be examined for severe problems.
With raw lists, the calculator can help you find sample statistics, but it does not verify the sampling design or make a careful judgment about the shape of each distribution. With summary statistics alone, you may need information from the problem about how the data were collected and what plots show. If the conditions are not described, do not invent evidence that they are satisfied; state what information is missing.
The output typically includes the lower and upper interval endpoints, the confidence level, the degrees of freedom, and summaries such as the sample means and sample standard deviations. Use the output as a check on the inputs, not as a replacement for explaining the analysis. The endpoints are in the original variable’s units and estimate \(\mu_1-\mu_2\), not either population mean by itself.
Worked Examples
Worked Example: Summary Statistics for Two Delivery Methods
An invented comparison measures delivery time, in minutes, for two independent random samples. Method 1 has \(\bar{x}_1=72.4\), \(s_1=6\), and \(n_1=20\). Method 2 has \(\bar{x}_2=69.1\), \(s_2=6\), and \(n_2=20\). Find a 95% confidence interval for the difference in population mean delivery times, method 1 minus method 2.
Let \(\mu_1-\mu_2\) be the true difference in mean delivery time, in minutes, for method 1 minus method 2. We want a 95% confidence interval for this difference.
Use a two-sample t interval for independent means. The samples are described as independent random samples. Assume each sample is less than 10% of its respective population. Suppose plots show no pronounced skewness or outliers in either group. These facts support the procedure.
Choose 2-SampTInt and Stats. Enter \(72.4,6,20\) for sample 1 and \(69.1,6,20\) for sample 2. Enter C-Level \(=0.95\) and Pooled: No.
Report the calculator interval and interpret it as a plausible range for the true difference in mean delivery times, method 1 minus method 2.
The sample difference and standard error provide useful checks on the calculator’s work:
Because the two variance contributions are equal, Welch’s degrees of freedom are \(38\). For 95% confidence, the critical value is about \(2.0244\), giving a margin of error of \(2.0244(1.8974)\approx3.8410\) minutes. Thus the calculator interval is approximately:
We are 95% confident that the true mean delivery time for method 1 minus the true mean delivery time for method 2 is between about \(-0.541\) and \(7.141\) minutes. The interval includes zero, so values compatible with either ordering of the two population means are plausible.
Worked Example: Raw Data Stored in Lists
An invented school project compares the number of minutes two types of study activity take to complete. The independent random samples are:
| Activity 1 (L1) | Activity 2 (L2) |
|---|---|
| 12 | 8 |
| 14 | 10 |
| 15 | 11 |
| 16 | 12 |
| 18 | 14 |
Construct a 90% confidence interval for \(\mu_1-\mu_2\), activity 1 minus activity 2.
First check the conditions. The samples are described as independent random samples. Assume each group is less than 10% of its target population. Each list has only five observations, so inspect the values: both lists are roughly symmetric around their centers, with no obvious outlier. This supports using a two-sample t interval. The observations are from different students in the two groups, not paired measurements.
On the calculator, choose 2-SampTInt and Data. Select L1 for sample 1 and L2 for sample 2, set both frequencies to 1, enter C-Level \(=0.90\), and choose Pooled: No. The calculator finds \(\bar{x}_1=15\), \(\bar{x}_2=11\), \(s_1=s_2=\sqrt{5}\approx2.2361\), and \(n_1=n_2=5\).
Check the center and standard error from the interval formula:
The equal variance contributions give Welch \(df=8\). For a 90% interval, \(t^*\approx1.8595\), so the margin of error is \(1.8595(1.4142)\approx2.6298\) minutes. The calculator’s interval, rounded to three decimals, is:
We are 90% confident that the true mean time for activity 1 is about 1.370 to 6.630 minutes greater than the true mean time for activity 2. The positive endpoints describe the difference in the stated order, activity 1 minus activity 2.
Worked Example: Interpreting an Interval That Crosses Zero
An invented environmental comparison records the amount of time, in hours, that two independent random samples of water filters operate before replacement. Filter 1 has \(\bar{x}_1=28.6\), \(s_1=4\), and \(n_1=10\). Filter 2 has \(\bar{x}_2=30.1\), \(s_2=4\), and \(n_2=10\). Use 2-SampTInt in Stats mode to construct a 95% confidence interval for \(\mu_1-\mu_2\).
The samples are independent random samples, and we assume each is less than 10% of its respective population. Suppose plots show no pronounced skewness or outliers in either sample. These checks support a two-sample t interval. Enter \(28.6,4,10\) for sample 1 and \(30.1,4,10\) for sample 2; set C-Level to 0.95 and Pooled to No.
The calculator’s center should equal the difference in the sample means, \(\bar{x}_1-\bar{x}_2=28.6-30.1=-1.5\) hours. The standard error is:
Welch’s degrees of freedom are \(18\), since both variance contributions are \(1.6\). The 95% critical value is about \(2.1009\), so the margin of error is \(2.1009(1.7889)\approx3.7582\) hours. The interval is approximately:
We are 95% confident that the true mean operating time for filter 1 minus the true mean operating time for filter 2 is between about \(-5.258\) and \(2.258\) hours. Because the interval includes zero, it is compatible with filter 1 having a lower mean, equal mean, or higher mean operating time. The negative sample difference is not an error: filter 1’s sample mean was 1.5 hours below filter 2’s.
Common Mistakes and AP Exam Tips
- Switching the sample order: The calculator subtracts sample 2 from sample 1. If your parameter is \(\mu_1-\mu_2\), enter group 1 first and group 2 second. Reversing them changes the signs of the center and endpoints.
- Putting the wrong statistic in the \(Sx\) field: Enter the sample standard deviation \(s\), not the variance \(s^2\), the standard error, or a population standard deviation.
- Using the wrong mode: Choose Stats when you have \(\bar{x}\), \(s\), and \(n\); choose Data when you have individual observations in lists. Do not enter a sample mean as though it were a data list.
- Leaving pooling on: For the usual AP Statistics two-sample t interval, set Pooled: No. The unpooled method uses the groups’ separate standard deviations and Welch degrees of freedom.
- Treating calculator output as proof that conditions hold: The calculator does not establish random sampling, independence, the 10% condition, or acceptable distribution shapes. Explain those checks separately.
- Describing the endpoints as separate group means: The interval estimates a difference in population means. Name both groups, the variable, the units, and the subtraction order in your interpretation.
For full credit, identify \(\mu_1-\mu_2\) in context, state that you used a two-sample t interval, check the conditions, show or report the calculator settings, and interpret both endpoints in the original units. Include the confidence level in the interpretation. Keep enough digits during calculations and round the reported interval consistently.
Check Your Understanding
Answer each question using the calculator procedure and interpretation ideas from this tutorial.
- You are given \(\bar{x}_1\), \(s_1\), and \(n_1\), along with the corresponding summary statistics for sample 2. Which 2-SampTInt input mode should you use?
- For the usual AP Statistics two-sample t interval, what should the Pooled setting be, and what degrees-of-freedom approach does that use?
- When using raw observations, what should you check about the two lists before running the interval?
- An interval for \(\mu_1-\mu_2\) is \((-2.4,\ 1.1)\) seconds. What does including zero tell you about plausible differences?
- Write a contextual interpretation for a 90% interval of \((3.2,\ 8.6)\) kilograms for \(\mu_1-\mu_2\), where population 1 is the first group named in your study.