A confidence interval for a difference estimates how far apart two population values may be, such as two means or two proportions. Instead of giving only one number from a sample, it gives a range of plausible differences. This matters because real samples vary, and two groups can look different just by chance.
The key reference point is difference = 0, which means no difference between the two populations.
Understanding Statistics: Confidence Interval for a Difference
The order of subtraction controls the meaning of every result. If a study defines the difference as new method minus old method, a positive value favors the new method. Reversing the order changes every sign but does not change the evidence.
Students should write the subtraction order in words before calculating. For example, a difference in average test scores of five points can mean the first group scored higher by five, or the second group scored higher by five. The number alone is incomplete without a clear direction.
The size of the interval depends on how much information the samples provide. Larger samples usually produce narrower intervals because random variation has less effect on the estimates. More spread within each group makes an interval wider when comparing means.
For proportions, results near one half tend to have more sampling variation than results near zero or one when sample sizes are the same. A higher confidence level needs a wider interval. This is a tradeoff.
Greater confidence gives a range that is less precise. A narrow interval is useful only when the data collection was sound.
The method must match the study design. Independent groups contain measurements from different people or objects, such as comparing heights from two separate classes. Paired data come from linked measurements, such as a student's score before tutoring and that same student's score afterward.
For paired data, the important values are the individual changes. Treating paired data as independent throws away useful information and can give an interval that is too wide or misleading. Random samples, or random assignment in an experiment, matter too.
A confidence interval describes uncertainty from sampling. It cannot fix a biased survey, missing responses, unfair group selection, or inaccurate measurements.
Students often meet these comparisons in health reports, school surveys, product testing, and sports statistics. A report might compare the proportion of students who prefer two lunch options. A medical experiment might compare average recovery times for two treatments.
The interval gives more than a decision about whether a difference seems present. Its endpoints show which difference sizes remain believable based on the sample. An interval entirely above zero may support a difference, yet that difference could still be too small to matter in real life.
Consider the units and context. A one point score change, a one day recovery change, and a one percent change in safety risk have very different practical meanings.
Confidence describes the long run performance of the method across many similar samples. It does not mean there is a fixed chance that the already calculated interval contains the population difference.
Key Facts
- For two independent means, point estimate = x1bar - x2bar.
- For two independent proportions, point estimate = p1hat - p2hat.
- General form: confidence interval = estimate ± critical value × standard error.
- Two means, large sample or t method: (x1bar - x2bar) ± t* sqrt(s1^2/n1 + s2^2/n2).
- Two proportions: (p1hat - p2hat) ± z* sqrt(p1hat(1 - p1hat)/n1 + p2hat(1 - p2hat)/n2).
- If 0 is inside the interval, the data do not show a clear difference at that confidence level; if 0 is outside, the data support a difference.
Vocabulary
- Confidence interval
- A range of values calculated from sample data that is likely to contain the true population parameter.
- Difference of means
- The subtraction of one population or sample mean from another, often written as μ1 - μ2 or x1bar - x2bar.
- Difference of proportions
- The subtraction of one population or sample proportion from another, often written as p1 - p2 or p1hat - p2hat.
- Standard error
- An estimate of how much a sample statistic would vary from sample to sample.
- Critical value
- A multiplier such as z* or t* that sets how wide the confidence interval must be for a chosen confidence level.
Common Mistakes to Avoid
- Interpreting a 95% confidence interval as a 95% chance that the fixed true difference is in this one interval. The correct idea is that the method captures the true difference in about 95% of repeated samples.
- Forgetting to check whether 0 is in the interval. If 0 is included, a true difference of zero is still plausible at that confidence level.
- Using the wrong standard error formula for means versus proportions. Means use sample standard deviations, while proportions use p-hat values and binomial variation.
- Reversing the subtraction order without changing the interpretation. An interval for group 1 minus group 2 has the opposite sign of an interval for group 2 minus group 1.
Practice Questions
- 1 Two independent samples have x1bar = 84, s1 = 10, n1 = 40 and x2bar = 78, s2 = 12, n2 = 35. Using t* = 2.00, find a 95% confidence interval for μ1 - μ2.
- 2 In group 1, 64 out of 200 students passed a test. In group 2, 45 out of 180 students passed. Using z* = 1.96, find a 95% confidence interval for p1 - p2.
- 3 A confidence interval for the difference in average reaction time, μ1 - μ2, is from -0.12 seconds to 0.05 seconds. Explain what the inclusion of 0 means and whether this interval supports a clear difference between the groups.