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.
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.