Writing the Full Two-Sample t Interval Solution

From a Calculated Interval to a Complete Response

A calculator can produce a two-sample t interval, but a complete AP Statistics response must do more than report its endpoints. You need to identify the population difference being estimated, justify the procedure, show how the interval was constructed, and interpret the result in context.

The interval is for \(\mu_1-\mu_2\): the true mean for population 1 minus the true mean for population 2. As emphasized in “Order of Subtraction and Interval Interpretation,” the order matters. A negative estimate or interval endpoint means the mean for population 1 is lower than the mean for population 2; it does not mean that both means are negative.

Definition: A complete two-sample t interval response describes an interval for \(\mu_1-\mu_2\), checks whether the study design and data support the procedure, shows the interval calculation, and interprets the interval’s endpoints in context.

The four steps—State, Plan, Do, and Conclude—are a way to make that reasoning easy to follow. They build on the interval formula from “The Form of a Two-Sample t Interval” and the conditions discussed in “Conditions for a Two-Sample t Interval.” The goal here is to connect those parts into one coherent response.

State.
Define \(\mu_1\) and \(\mu_2\) in context, identify the parameter \(\mu_1-\mu_2\), and include its units.

Plan.
Name the unpooled two-sample t interval and check the design, independence, the 10% condition when relevant, and whether the sample distributions support a t procedure.

Do.
Show the sample mean difference, standard error, degrees of freedom, critical value, margin of error, and interval. Keep units visible and rounding consistent.

Conclude.
State the confidence level and interpret the range of plausible differences in context, preserving the order \(\mu_1-\mu_2\).

What to Include in Each Step

In State, name the populations—not just “group 1” and “group 2”—so the parameter has a clear meaning. For example, “Let \(\mu_1\) be the true mean weekly practice time for students at School A, and let \(\mu_2\) be the true mean weekly practice time for students at School B.” Then specify that the parameter is \(\mu_1-\mu_2\), measured in minutes per week.

In Plan, identify an unpooled two-sample t interval. The two samples should be independent, not paired, as described in “Identifying Two Independent Samples.” Check the specific design: were the groups formed by appropriate random samples or random assignment? If sampling without replacement, check that each sample is no more than 10% of its population. Finally, judge whether the sample sizes and distributions make a t procedure reasonable. For small samples, look for distributions that are roughly symmetric without strong outliers; larger samples can better support the procedure when the population shapes are not known to be normal.

Conditions: For a two-sample t interval, explain why the groups are independent and why the design supports inference. Check the 10% condition for each sample drawn without replacement. Assess the sample distributions or sample sizes to justify using a t procedure.

In Do, use the unpooled standard error, which estimates uncertainty in the difference between the sample means:

SE_{\bar{x}_1-\bar{x}_2} = \sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}

Then use the two-sample t interval formula. Use the Welch degrees of freedom provided by the calculator or an assigned conservative method, as covered in “Computing Two-Sample Degrees of Freedom With the Welch Formula.” Do not silently use \(n_1+n_2-2\) as the degrees of freedom for the standard unpooled interval.

(\bar{x}_1-\bar{x}_2) \ \pm\ t^* \sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}

In Conclude, translate the lower and upper endpoints into a range for the true population mean difference. If the interval is for \(\mu_1-\mu_2\), describe population 1 relative to population 2. A confidence interval describes plausible values for the parameter; it is not a range containing a stated percentage of individual observations.

Worked Examples

Worked Example: A Full 95% Interval Response

In an invented comparison, independent random samples of 15 students from each of two large schools record weekly minutes spent on mathematics practice. School A’s sample mean is 34.6 minutes and its sample standard deviation is 6 minutes. School B’s sample mean is 31.2 minutes and its sample standard deviation is also 6 minutes. Both schools have more than 150 students, and each sample distribution is described as roughly symmetric with no strong outliers. Construct and interpret a 95% interval.

State: Let \(\mu_A\) be the true mean weekly mathematics practice time for students at School A, and let \(\mu_B\) be the true mean for students at School B. The parameter of interest is \(\mu_A-\mu_B\), measured in minutes per week.

Plan: Use an unpooled two-sample t interval for \(\mu_A-\mu_B\). The students were selected in separate random samples, and a student can appear in only one school’s sample, so the groups are independent rather than paired. Each sample of 15 is less than 10% of its school’s more-than-150 students, satisfying the 10% condition. The sample distributions are roughly symmetric with no strong outliers, which supports using a t procedure for these small samples.

Do: The sample mean difference is \(34.6-31.2=3.4\) minutes per week. The standard error is:

SE = \sqrt{\frac{6^2}{15}+\frac{6^2}{15}} = \sqrt{2.4+2.4} = \sqrt{4.8} \approx 2.191

The Welch degrees of freedom are 28. For a 95% interval, \(t^*\approx2.048\). The margin of error is \(2.048(2.191)\approx4.487\) minutes per week. Therefore:

3.4\pm4.487 \quad\Longrightarrow\quad (-1.087,\ 7.887)

Conclude: We are 95% confident that the true mean weekly mathematics practice time at School A is between about 1.087 minutes less and 7.887 minutes more than at School B. The interval includes zero, so it does not provide convincing evidence of a difference between the two population means at the corresponding two-sided 5% significance level, since \(1-0.95=0.05\). The interval also includes both negative and positive differences, so the data are compatible with School A’s mean being somewhat lower or higher.

Worked Example: A 90% Interval for Plant Growth

In an invented greenhouse comparison, independent random samples of 10 plots using Soil Mix 1 and 10 plots using Soil Mix 2 are used to measure plant growth after six weeks, in centimeters. Mix 1 has a sample mean of 18.2 cm and a sample standard deviation of 4 cm. Mix 2 has a sample mean of 16.0 cm and a sample standard deviation of 4 cm. Each mix is used in more than 100 plots, and the sample distributions are roughly symmetric without strong outliers. Construct and interpret a 90% interval for the difference in mean growth.

State: Let \(\mu_1\) be the true mean six-week plant growth for plots using Mix 1, and let \(\mu_2\) be the true mean for plots using Mix 2. The parameter \(\mu_1-\mu_2\) is measured in centimeters.

Plan: Use an unpooled two-sample t interval. The samples are random samples from separate plots, so observations are independent and not paired. Each sample of 10 is less than 10% of the more-than-100 plots using its mix. The sample distributions are roughly symmetric and have no strong outliers, supporting the t procedure despite the small sample sizes.

Do: The observed difference is \(18.2-16.0=2.2\) cm. The standard error is:

SE = \sqrt{\frac{4^2}{10}+\frac{4^2}{10}} = \sqrt{1.6+1.6} = \sqrt{3.2} \approx1.789

The Welch degrees of freedom are 18. For a 90% interval, \(t^*\approx1.734\). The margin of error is \(1.734(1.789)\approx3.102\) cm. Thus:

2.2\pm3.102 \quad\Longrightarrow\quad (-0.902,\ 5.302)

Conclude: We are 90% confident that the true mean growth for plots using Mix 1 is between about 0.902 cm less and 5.302 cm more than the true mean growth for plots using Mix 2. Because the interval includes zero, it does not provide convincing evidence of a difference between these population means at the corresponding two-sided 10% significance level. This interval does not establish that the two means are equal.

Worked Example: Unequal Sample Sizes and Welch Degrees of Freedom

In an invented quality-control comparison, independent random samples of 12 irrigation emitters of Type A and 18 of Type B are tested for water flow, measured in liters per minute. Type A has a sample mean of 73.4 liters per minute and a sample standard deviation of 5 liters per minute. Type B has a sample mean of 68.1 liters per minute and a sample standard deviation of 6 liters per minute. Each type has more than ten times the number of emitters sampled, and both sample distributions are roughly symmetric without strong outliers. Construct a 95% interval.

State: Let \(\mu_A\) be the true mean flow rate for Type A emitters and \(\mu_B\) the true mean flow rate for Type B emitters. The parameter \(\mu_A-\mu_B\) is measured in liters per minute.

Plan: Use an unpooled two-sample t interval. The emitters are selected independently from the two types, with no pairing between units. Both samples are less than 10% of their respective populations, and the stated sample shapes support the t procedure. Because the sample sizes and standard deviations differ, use Welch’s degrees of freedom rather than a pooled calculation.

Do: The difference in sample means is \(73.4-68.1=5.3\) liters per minute. The standard error is:

SE = \sqrt{\frac{5^2}{12}+\frac{6^2}{18}} = \sqrt{\frac{25}{12}+2} = \sqrt{4.0833} \approx2.021

The Welch degrees of freedom are approximately 26.47, and the 95% critical value is \(t^*\approx2.054\). The margin of error is \(2.054(2.021)\approx4.151\) liters per minute. The interval is:

5.3\pm4.151 \quad\Longrightarrow\quad (1.149,\ 9.451)

Conclude: We are 95% confident that the true mean flow rate for Type A emitters is about 1.149 to 9.451 liters per minute greater than the true mean flow rate for Type B emitters. The interval is entirely above zero, supporting a positive difference \(\mu_A-\mu_B\) between the population means.

Common Mistakes and Full-Credit Communication

Leaving the populations undefined: “The difference is 3.4” is incomplete. Define each population mean and say which mean is subtracted from which.
Calling the samples paired just because there are two groups: Pairing requires a meaningful link between observations. If the individuals or units are different and unlinked, describe the samples as independent.
Listing conditions without applying them: “Conditions are met” does not explain why. State how the random design, independence, 10% condition, and sample shape apply to the situation.
Using the wrong standard error: Show \(\sqrt{s_1^2/n_1+s_2^2/n_2}\). Do not add the two standard errors or combine the standard deviations into a pooled value for the standard unpooled interval.
Dropping the subtraction order in the conclusion: For an interval for \(\mu_1-\mu_2\), explain what its signs say about population 1 compared with population 2.
Interpreting confidence as a probability about the fixed parameter: Do not say there is a 95% probability that this particular interval contains the true difference. Say, “We are 95% confident,” then state the interval and context.
Overstating an interval that includes zero: Say that it does not provide convincing evidence of a difference at the corresponding two-sided significance level. Do not claim that the means are equal or that there is no difference.

For a polished response, keep the four steps connected: define the parameter, justify the procedure with specific evidence, show the calculation, and interpret the interval in the same units and subtraction order. In a calculator-based solution, report the degrees of freedom and critical value along with the interval so the method is transparent.

Key takeaway: A complete two-sample t interval response identifies \(\mu_1-\mu_2\), checks the conditions in context, shows the unpooled calculation, and interprets both endpoints as plausible differences between the population means.

Check Your Understanding

For each question, focus on writing or interpreting a complete interval response.

Two independent random samples compare mean daily screen time at two schools. What should the State step define before reporting an interval?
For independent samples with \(s_1=7,\ n_1=14,\ s_2=5,\ n_2=10\), write the standard error expression for \(\bar{x}_1-\bar{x}_2\).
A researcher takes random samples without replacement from populations of 240 and 180 units, using sample sizes of 20 and 18. Check the 10% condition for both samples.
A 95% interval for \(\mu_1-\mu_2\), measured in minutes, is \((-2.5,\ 6.1)\). What does the interval say about the plausible direction and size of the difference?
Why is “the two population means are equal” not an appropriate conclusion when a confidence interval for their difference contains zero?