AP Statistics · Lesson 1

Confidence intervals for a mean

Know Know it

To estimate a population mean μ\mu, start from the sample mean xˉ\bar{x}. Its spread from sample to sample depends on the population standard deviation, which you almost never know. So you use the sample standard deviation s and the standard error sn\frac{s}{\sqrt{n}}.

Using s instead of the true value adds extra uncertainty, so the standardized statistic follows a t distribution with n−1n - 1 degrees of freedom, not the standard Normal curve. The t curves have heavier tails, so the critical value t∗t^* is larger than z∗z^*, most of all for small samples.

The one-sample t interval is xˉ±t∗sn\bar{x} \pm t^* \frac{s}{\sqrt{n}}. The part after the plus-or-minus sign is the margin of error.

Check three conditions first. Random: the data come from a random sample or a randomized experiment. 10%: when sampling without replacement, n is at most 10% of the population. Normal/Large Sample: the population is roughly Normal, or n≥30n \ge 30; for a small sample, a graph of the data shows no strong skew or outliers.

95% critical values: t gets closer to z = 1.960 as df grows
Degrees of freedomt* for 95% confidence
42.776
92.262
242.064
991.984

Calculator On the exam

A graphing calculator with statistical capabilities is expected on both sections of the exam, and one is built into the testing app. The exam also gives a reference booklet of formulas and tables, including t critical values.

Use the calculator's t interval function to get the interval, but still write the formula, the df, t∗t^* and the numbers you used, so a reader can follow your work.

Trap The interval is about the mean, not about individuals

A 95% interval of 6.4 to 7.0 hours for mean sleep does not say that 95% of students sleep 6.4 to 7.0 hours. Individual values vary far more than the mean does. The interval gives plausible values for the population mean only.

Worked example Easy

A 90% confidence interval for the mean daily screen time of students at a fictional school is 3.4 to 4.6 hours. What are the sample mean and the margin of error?

  1. Sample mean 4.6 hours, margin of error 0.6 hours
  2. Sample mean 4.0 hours, margin of error 0.6 hoursAnswer
  3. Sample mean 4.0 hours, margin of error 1.2 hours
  4. Sample mean 3.4 hours, margin of error 1.2 hours

How to solve it

  1. A t interval is xˉ±\bar{x} \pm margin of error, so the sample mean is the midpoint.
  2. Midpoint: 3.4+4.62=4.0\frac{3.4 + 4.6}{2} = 4.0 hours.
  3. Margin of error: half the width, 4.6−3.42=0.6\frac{4.6 - 3.4}{2} = 0.6 hours.

Why each choice is right or wrong

  • A. Incorrect. 4.6 hours is the upper end of the interval; the sample mean sits in the middle, at 4.0 hours.
  • B. Correct. The center is (3.4 + 4.6) / 2 = 4.0 hours, and the margin of error is half the width, (4.6 - 3.4) / 2 = 0.6 hours.
  • C. Incorrect. The center 4.0 is right, but 1.2 hours is the full width; the margin of error is half of it.
  • D. Incorrect. 3.4 hours is the lower end, not the center, and 1.2 hours is the full width, not half of it.

Explain Explain it

Task verbs: interpret, calculate

Interpret the interval with three things: the confidence level, the parameter in context, and the interval. For example: we are 95% confident that the interval from 35.9 to 41.3 minutes captures the true mean commute time of all employees at the company.

Interpret the confidence level as a statement about the method: if we took many random samples of the same size and built an interval from each, about 95% of those intervals would capture the true mean. Any one interval either captures it or does not.

Width: the margin of error shrinks when n grows (because of n\sqrt{n}), grows when the confidence level goes up, and grows when the data are more spread out.

Worked example Medium

A fictional company, Larkfield Logistics, has 600 employees. A manager selects 25 of them at random and records each one's one-way commute. The sample mean is 38.6 minutes and the sample standard deviation is 6.5 minutes. A dotplot of the 25 times is roughly symmetric with no outliers.

Which is a 95% confidence interval for the mean commute time of all 600 employees?

  1. 25.18 to 52.02 minutes
  2. 35.92 to 41.28 minutesAnswer
  3. 36.05 to 41.15 minutes
  4. 38.06 to 39.14 minutes

How to solve it

  1. Conditions: random sample; 25 is less than 10% of 600; n is under 30, but the dotplot shows no strong skew or outliers.
  2. Standard error: sn=6.55=1.3\frac{s}{\sqrt{n}} = \frac{6.5}{5} = 1.3 minutes.
  3. df = 24, so t∗=2.064t^* = 2.064, and the margin of error is 2.064×1.3≈2.682.064 \times 1.3 \approx 2.68.
  4. Interval: 38.6±2.6838.6 \pm 2.68, from about 35.92 to 41.28 minutes.

Why each choice is right or wrong

  • A. Incorrect. 25.18 to 52.02 uses t∗⋅st^* \cdot s with no n\sqrt{n}, so it uses the spread of single commutes instead of the standard error of the mean.
  • B. Correct. The standard error is 6.525=1.3\frac{6.5}{\sqrt{25}} = 1.3, t∗=2.064t^* = 2.064 for df = 24, and 38.6±2.064(1.3)=38.6±2.6838.6 \pm 2.064(1.3) = 38.6 \pm 2.68.
  • C. Incorrect. 36.05 to 41.15 uses z∗=1.96z^* = 1.96; with s in place of the population value, the critical value must come from t with df = 24.
  • D. Incorrect. 38.06 to 39.14 divides s by n = 25 instead of by 25=5\sqrt{25} = 5, which makes the interval far too narrow.

Interpret Your turn

A random sample of 40 adult members of a fictional gym gives a 95% confidence interval of 3.1 to 3.9 for the mean number of workouts per week. Interpret the interval in context, and say what 95% confidence means here.

Write it, then compare. Nothing you type here is saved or graded.

Compare with a model answer

A full answer does this

  • Says we are 95% confident the interval from 3.1 to 3.9 captures the true mean number of weekly workouts for all adult members
  • Explains 95% confidence as the long-run capture rate of the method over many random samples

Model answer

We are 95% confident that the interval from 3.1 to 3.9 workouts per week captures the true mean number of weekly workouts for all adult members of the gym. The 95% describes the method: if we took many random samples of 40 adult members and built an interval from each, about 95% of those intervals would capture the true mean.

Apply Apply it

Practice this topic. AP Practice questions on exactly what this lesson covers, with an explanation for every answer.