Two Independent Means: Add Variances, Keep the Populations Clear
These lessons and problems are newly written for the revised course. Worked examples show complete reasoning; the application questions ask you to make and defend your own choices.
Independent samples estimate a difference in means
Two packaging designs may have different mean opening times. A researcher independently samples 32 packages of design A from a production population of 10,000 and 36 of design B from 12,000, without replacement. Under a standardized opening task, A has sample mean 25 seconds and sample SD 4 seconds; B has mean 23 seconds and SD 5 seconds. Both population SDs are unknown.
Define μ_A and μ_B as the population mean opening times in seconds for packages of designs A and B under the stated task. The parameter μ_A−μ_B is A minus B. A two-sample t-interval for the difference between independent population means estimates this quantity. A two-sample t-test for the difference between population means tests an equality claim. Means describe quantitative time responses; proportions and categorical-count tests answer different questions.
Independence is a design feature, not something guaranteed by two columns of numbers. Here different independently sampled packages produce the two groups. If instead each person opens both designs and observations are matched within person, those responses form dependent pairs: analyze one ordered difference per person with the one-sample paired t method. The number of pairs is its sample size. Choosing the independent method for paired data loses the design’s within-pair information.
Verify both designs and both distributions
The interval and test use the same three condition groups. Randomization requires two independent random samples or a randomized experiment. For sampling without replacement, each n must be at most 10% of its own population: n_A≤0.10N_A and n_B≤0.10N_B. The condition is unnecessary for randomized treatment assignment in an experiment; volunteer selection still limits population generalization.
For the sample-data condition, both samples should have n≥30, or both population distributions should be indicated as approximately Normal. If either sample has fewer than 30 observations, both sample-data distributions should be free from strong skewness and outliers. This rule examines both groups when either is small. A clean sample of 12 does not excuse strong skew and an isolated extreme in the other sample of 40.
| Samples/populations | Course condition assessment |
|---|---|
| n_A=32, n_B=36 | Both sample sizes at least 30 |
| n_A=9, n_B=11; both populations approximately Normal; clean observed samples | Normal-population route with small-sample shape supported |
| n_A=12, n_B=40; both observed samples symmetric without outliers | Either-small rule checks both distributions |
| n_A=12 clean; n_B=40 strongly skewed with isolated outlier | Either-small/both-clean route fails |
The worked packaging samples are independent SRSs. Their fractions satisfy 32≤1,000 and 36≤1,200, and both n values are at least 30. These actual design and count statements justify the course approximation. For a small-sample study, examine a meaningful plot or complete data instead of inventing a population Normality claim. Randomness cannot be supplied by a histogram, and a large n cannot repair a substantial without-replacement sampling fraction.
Each sample contributes its own uncertainty
The point estimate is the difference in sample means, 25−23=2 seconds. Independent variance contributions add. Use the two sample SDs s_A and s_B to estimate variability; they are not known population σs.
For the worked data, s_A²/n_A=16/32=0.50 and s_B²/n_B=25/36≈0.6944444. SE=√1.1944444≈1.0929064 seconds. Technology supplies appropriate Welch degrees of freedom df≈65.315481 and the central 95% t*=1.9969551. The df falls between min(n_A−1,n_B−1)=31 and n_A+n_B−2=66. It need not be a whole number; an appropriate t-distribution technology calculation handles it. If a table uses a conservative smaller df, state that approximation explicitly.
MOE=t*SE≈2.1824850 seconds, so the 95% interval is 2±2.1824850, approximately (−0.1824850,4.1824850) seconds. A central C% t critical value determines the interval’s confidence multiplier. The margin is half the full width. Adding SDs, subtracting variances, using a known-σ z critical, or forcing equal sample SDs would change this uncertainty calculation without justification.
Read the sign, units and zero together
We are 95% confident that design A’s population mean opening time is from about 0.1825 seconds lower to 4.1825 seconds higher than design B’s. The lower endpoint is negative, the upper positive, and both describe A minus B. This is an interval for population means, not an assertion that every A package takes between these amounts longer than every B package.
In repeated independent random sampling with these same sample sizes from these same populations, approximately 95% of intervals constructed this way capture μ_A−μ_B. This one interval may or may not capture that fixed difference. Confidence refers to the procedure’s approximate long-run capture rate, rather than a probability distribution for a changing population parameter.
Since zero is included, the interval provides insufficient evidence of a mean difference at this confidence level. It does not prove equal means. A separate study yielding a valid interval (0.8,3.0) seconds would exclude zero and supply convincing evidence of a population mean difference with A higher. Reversing the worked parameter to B minus A changes the interval to (−4.1824850,0.1824850), reversing the direction of interpretation while retaining zero. No change in subtraction order can create stronger evidence.
Standardize a difference under the equality null
For a question asking whether the mean opening times differ, write H₀:μ_A=μ_B or H₀:μ_A−μ_B=0. Write Hₐ:μ_A≠μ_B or Hₐ:μ_A−μ_B≠0. A higher-A investigative question instead uses μ_A>μ_B or μ_A−μ_B>0; a lower-A question uses μ_A<μ_B or μ_A−μ_B<0. Each form refers to population means, not the observed 25 and 23.
The worked test gives t=2/1.0929064≈1.8299828. Under equal population means and the verified conditions, use the appropriate t reference distribution with technology df≈65.315481. Its allowable df range is again 31 to 66. The test retains separate estimated variance terms; equality of means does not assert equality of population variances.
A suitable t table or technology gives upper-tail probability about 0.0359081. Since “differ” is two-sided, p≈2(0.0359081)=0.0718162. A prespecified higher-A alternative would use the upper tail alone; a lower-A alternative uses the lower tail. Choose the alternative from the research question before the observed sign. Neither a single shaded upper tail nor a standard normal probability is the full two-sided t p-value here.
Keep the null assumption in the probability
The worked p-value means: assuming the population mean opening times for A and B are equal, the probability of an independent-sample standardized mean difference at least this far from zero in either direction is about 0.0718162. The same statistic, response, design and equality assumption should be identifiable in the interpretation. This probability is not the chance that equality is true.
At prespecified α=0.05, 0.0718162>0.05 gives fail to reject H₀. There is not convincing statistical evidence that the population mean opening times differ between designs under the task conditions. The central 95% interval includes zero, agreeing with this matched two-sided test. Failure to reject is not proof of equal means; a real difference can remain difficult to distinguish from sampling variation.
For a different valid test with p=0.008 at the same α, p≤α gives reject H₀, with convincing evidence of a population mean difference. Both decisions need an explicit comparison, an answer to the original question, the parameter/populations and qualified language consistent with the alternative. Random samples support population inference; random treatment assignment is a different design feature needed for causal comparison.
Independent Application
Work through each investigation before opening the separate solutions. State parameters, show calculations and connect your reasoning to the actual study.
Original generated practice for the Fall 2026 course.
Glossary and deeper reading · 24 refreshers
These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.
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Mean
The mean is a measure of center often called an average. For a finite data set, the arithmetic mean is the sum of the values divided by their number; for a random variable, the mean is its probability-weighted expected value when that value exists. Means can be pulled toward extreme values and are not generally resistant. For example, the arithmetic mean of 2, 4, and 9 is 5.
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Sample
A sample is the subset of a population actually studied. How it is selected matters for the conclusions that can be drawn about the population. For example, 60 selected students may be a sample from a school’s full enrollment.
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Population parameter
A population parameter is a numerical feature of the population or probability model being studied. It is fixed for that defined population, although its value may be unknown. For example, the true proportion of all enrolled students who walk to school is a parameter.
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Statistical independence
Statistical independence means that learning one outcome does not change the probability distribution of another. For two events, it implies that the probability of both occurring equals the product of their probabilities. For example, independent draws with replacement keep the second draw’s color probabilities unchanged by the first.
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Observation
An observation is a recorded value or set of values for an individual or occasion. The individual may be a person, object, or event rather than a human subject. For example, one plant’s measured height is an observation in a plant-growth study.
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Random sampling
Random sampling uses a specified chance process to select individuals from a population. It supports population inference when the sampling frame and data collection are appropriate, but does not guarantee perfect balance or eliminate every source of bias. For example, a random-number generator can select student identifiers from an enrollment list.
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Experiment
An experiment deliberately imposes treatments and measures responses. Random assignment, comparison, and replication help distinguish treatment effects from other variation. For example, assigning plants to fertilizer treatments is an experiment.
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Population
A population is the full set of individuals or measurements a study aims to describe. It must be defined clearly before deciding whether a sample supports generalization. For example, all students enrolled in a particular school this year form a population.
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Skewness
Skewness describes asymmetry in a quantitative distribution’s tails. A longer right tail indicates right skew, while a longer left tail indicates left skew. For example, many short waits with a few very long waits produce right skew.
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Histogram
A histogram groups quantitative observations into numerical intervals and displays their frequencies or proportions. Neighboring bars represent adjacent intervals, so the graph preserves numerical order. For example, a histogram can group travel times into ten-minute intervals.
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Point estimate
A point estimate is the specific numerical value obtained from an estimator for the observed sample. It gives one estimate without expressing uncertainty by itself. For example, a sample mean of 14 minutes is a point estimate of the population’s mean travel time.
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Variance
Variance is the square of the standard deviation. It measures spread in squared units, with the precise averaging rule depending on whether it describes a sample or a probability distribution. For example, a standard deviation of 3 centimeters corresponds to a variance of 9 square centimeters.
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Degrees of freedom
Degrees of freedom describe the independent information remaining after constraints or estimated quantities are accounted for. The value depends on the procedure, rather than always being the sample size minus 1. For example, a one-sample t procedure uses n − 1 degrees of freedom, while simple regression inference uses n − 2.
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Student’s t-distribution
A Student’s t-distribution is a symmetric reference distribution with heavier tails than the standard Normal distribution. Its shape depends on degrees of freedom and approaches the standard Normal shape as those degrees of freedom grow. For example, inference about a Normal population mean with an unknown standard deviation uses a t reference with one fewer degree of freedom than the sample size.
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Critical value
A critical value is a cutoff from a reference distribution chosen for a confidence level or testing rule. The relevant distribution, degrees of freedom, and one-sided or two-sided procedure determine the cutoff. For example, a symmetric 95% Normal interval uses a cutoff near 1.96 standard-error units.
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Probability distribution
A probability distribution describes a random variable’s possible values and their probabilities. A discrete distribution lists probability masses, while a continuous distribution describes probabilities through areas under a density. For example, a distribution can list the probabilities of 0, 1, 2, or 3 red cards in three draws.
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Confidence level
A confidence level is the long-run fraction of intervals from a procedure that capture the true parameter under its assumptions. Holding the data and procedure fixed, increasing the confidence level generally widens the interval. For example, a 99% interval is usually wider than a 90% interval calculated from the same sample.
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T-table
A t-table lists critical values for t-distributions at specified degrees of freedom and tail probabilities. Its column convention must be checked before using it for a one-sided test or a two-sided interval. For example, a two-sided 95% interval leaves 2.5% in each tail when selecting its critical value.
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Probability
Probability quantifies how likely an event is under a specified chance model. It lies between 0 and 1, with larger values representing greater likelihood. For example, drawing a red card from a well-shuffled standard deck has probability one half.
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P-value
A p-value is the probability, under the null model and its assumptions, of results at least as extreme as those observed in the direction or directions specified by the test. It does not give the probability that the null hypothesis is true or measure the practical size of an effect. For example, a p-value of 0.03 describes a 3% tail probability under the null model, rather than a 3% probability that the null is correct.
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Sample statistic
A statistic is a numerical summary calculated from sample data. Its value can change when a different sample is selected from the same population. For example, the proportion of walkers among 60 sampled students is a statistic.
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Two-sided test
A two-sided test seeks evidence for departures in either direction from the null claim. For symmetric z or t tests, extreme evidence appears in both tails of the reference distribution. For example, a not-equal alternative asks whether a population mean is either lower or higher than a stated value.
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Statistical inference
Statistical inference uses data and a study design to draw conclusions beyond the observed values. Random sampling supports population generalization, while well-conducted random assignment supports treatment-effect conclusions. For example, a random sample can estimate a school-wide preference without surveying every student.
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Treatment
A treatment is a specific condition imposed on an experimental unit. In a multifactor experiment, a treatment is a combination of factor levels. For example, low fertilizer with eight hours of light is one treatment combination.