Constructing a Confidence Interval for a Population Mean or Population Mean Difference

Point estimate plus or minus the margin of error, and the margin of error is the critical value times the standard error.

• Correct: statistic $\pm$ (critical value)(standard error):
• statistic $\pm$ standard error:
• parameter $\pm$ margin of error:
• critical value $\pm$ statistic:

Means vary less than single values, by a factor of $\sqrt{n}$.

• Correct: $\frac{\sigma}{\sqrt{n}}$:
• $\frac{\sigma}{n}$:
• $\sigma$:
• $\sigma\sqrt{n}$:

Matched pairs compare two treatments within each pair, so the two responses belong together.

• Correct: each subject (or pair of similar subjects) gets both treatments:
• subjects are split into two unrelated groups:
• no treatment is imposed:
• every subject gets a placebo:

With $\sigma$ unknown, $\frac{\bar{x} - \mu}{s/\sqrt{n}}$ follows a t distribution with $n - 1$ degrees of freedom, which is wider than the standard normal.

• The data come from a skewed population: The shape condition is separate. Even for a normal population, using $s$ calls for t.
• The sample is smaller than 30: Large samples use $t^*$ too, whenever $\sigma$ is estimated by $s$.
• Correct: The interval uses $s$ in place of the unknown $\sigma$: Replacing $\sigma$ with $s$ adds variability, because $s$ changes from sample to sample. The t distribution, with heavier tails, accounts for it.
• By definition, intervals for means use t and intervals for proportions use z: There is a reason: for means the standard error estimates $\sigma$ with $s$, which adds extra uncertainty.

The confidence is about the method capturing the parameter, the true mean distance of all the city's bike commuters, not about individual commuters or the sample mean.

• A full-credit answer: We are 95% confident that the interval from 5.60 to 8.00 miles captures the true mean one-way distance of all the city's bike commuters.

Pairs: compute husband minus wife for each couple, then $\bar{x}_d \pm t^*\frac{s_d}{\sqrt{30}}$ with 29 df.

• Two separate t intervals, one for the husbands' ages and one for the wives' ages, then compare: Two intervals do not estimate the mean of the paired difference, and they ignore the pairing.
• A two-sample z interval for the difference of two proportions: Ages are quantitative, so this is about means, not proportions.
• A one-sample z interval using 1.96: The population standard deviation of the differences is unknown, so use $t^*$.
• Correct: A one-sample t interval for the mean of the 30 differences, husband minus wife: The two ages in a couple belong together, so the data are paired: one difference per couple, then a t interval for $\mu_d$.