Unit 7: Inference for Quantitative Data: Means

This unit uses sample means to estimate and test claims about population means. Because population standard deviations are usually unknown in real studies, AP Statistics emphasizes the t-distribution.

z Procedures Versus t Procedures

Use a z procedure for a population mean only when the population standard deviation \(\sigma\) is known:

\[z = \frac{\bar{x}-\mu_0}{\sigma/\sqrt{n}}.\]

In most real problems, \(\sigma\) is unknown, so use the sample standard deviation \(s\) and a t-distribution:

\[t = \frac{\bar{x}-\mu_0}{s/\sqrt{n}}.\]

The t-distribution is symmetric and bell-shaped like the normal distribution, but it has heavier tails. As degrees of freedom increase, the t-distribution approaches the standard normal distribution.

t distribution placeholder

One-Sample t-Interval For A Mean

Use a one-sample t-interval to estimate a population mean \(\mu\):

\[\bar{x} \pm t^*\frac{s}{\sqrt{n}}.\]

Degrees of freedom:

\[df = n-1.\]

Conditions:

  1. Random sample or random assignment.
  2. Independence: if sampling without replacement, \(n \le 0.10N\).
  3. Normal/large sample condition: population is approximately normal, or sample size is large enough for the Central Limit Theorem. Check graphs for strong skew and outliers when \(n\) is small.

Interpretation: “We are __% confident that the true population mean __ is between __ and __.”

One-Sample t-Test For A Mean

Use a one-sample t-test for a claim about one population mean:

\[H_0:\mu=\mu_0.\]

The test statistic is

\[t = \frac{\bar{x}-\mu_0}{s/\sqrt{n}},\]

with

\[df=n-1.\]

The alternative may be \(\mu>\mu_0\), \(\mu<\mu_0\), or \(\mu\ne\mu_0\). The p-value is found from the t-distribution with the correct degrees of freedom.

Two-Sample t-Interval For Difference In Means

Use a two-sample t-interval to estimate \(\mu_1-\mu_2\) for two independent groups:

\[(\bar{x}_1-\bar{x}_2) \pm t^* \sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}.\]

Conditions:

  1. Random samples or random assignment.
  2. The two samples/groups are independent.
  3. Independence within each group: \(n_1 \le 0.10N_1\) and \(n_2 \le 0.10N_2\) if sampling without replacement.
  4. Normal/large sample condition for each group.

Degrees of freedom can be found with technology. If doing by hand, use the conservative choice:

\[df = \min(n_1-1,\ n_2-1).\]

Two-Sample t-Test For Difference In Means

For independent samples, test

\[H_0:\mu_1-\mu_2=0\]

with

\[t = \frac{(\bar{x}_1-\bar{x}_2)-0} {\sqrt{s_1^2/n_1+s_2^2/n_2}}.\]

Use technology for degrees of freedom unless told otherwise. AP problems often care more about setup, conditions, and conclusion than hand-calculating df.

Do not pool variances unless a problem specifically says to use a pooled two-sample t procedure. Modern AP Statistics generally uses unpooled two-sample t procedures.

Matched Pairs t Procedures

A matched pairs design compares paired observations: before/after measurements on the same subject, twins, matched individuals, or two treatments applied to each unit in random order.

For matched pairs, convert the data to differences:

\[d_i = \text{value}_{1,i} - \text{value}_{2,i}.\]

Then run a one-sample t procedure on the differences.

Interval:

\[\bar{d} \pm t^*\frac{s_d}{\sqrt{n}}.\]

Test statistic:

\[t = \frac{\bar{d}-\mu_{d,0}}{s_d/\sqrt{n}}.\]

Degrees of freedom:

\[df=n-1.\]

Matched pairs design placeholder

Choosing The Correct Mean Procedure

Situation Procedure
One quantitative sample, \(\sigma\) unknown One-sample t
One quantitative sample, \(\sigma\) known One-sample z
Two independent quantitative samples Two-sample t
Paired quantitative measurements Matched pairs t

If the data are categorical counts or proportions, use Unit 6 or Unit 8 instead.

Confidence Intervals And Tests Together

A two-sided hypothesis test at significance level \(\alpha\) corresponds to a \((1-\alpha)100\%\) confidence interval. If the null value is outside the interval, reject \(H_0\). If the null value is inside the interval, fail to reject \(H_0\).

For one-sided tests, this direct interval comparison requires more care, but the logic is still connected: values far from the interval’s plausible range are less compatible with the data.

Calculator Notes

Common calculator tools:

  • TInterval: one-sample t confidence interval.
  • T-Test: one-sample t test.
  • 2-SampTInt: two-sample t confidence interval.
  • 2-SampTTest: two-sample t test.
  • For matched pairs, enter the list of differences and use TInterval or T-Test.

Calculator output should be translated into statistical language: parameter, conditions, statistic, interval or p-value, and conclusion in context.

Working Checklist

  1. Identify the parameter: \(\mu\), \(\mu_1-\mu_2\), or \(\mu_d\).
  2. Decide whether the samples are independent or paired.
  3. Check random, independence, and normal/large sample conditions.
  4. Use t unless population \(\sigma\) is known.
  5. Compute the interval or test statistic/p-value.
  6. Conclude in context, including units.

Key Equations

Idea Equation
One-sample t interval \(\bar{x}\pm t^*s/\sqrt{n}\)
One-sample t test \(t=(\bar{x}-\mu_0)/(s/\sqrt{n})\)
Two-sample t interval \((\bar{x}_1-\bar{x}_2)\pm t^*\sqrt{s_1^2/n_1+s_2^2/n_2}\)
Two-sample t test \(t=\frac{(\bar{x}_1-\bar{x}_2)-0}{\sqrt{s_1^2/n_1+s_2^2/n_2}}\)
Matched pairs interval \(\bar{d}\pm t^*s_d/\sqrt{n}\)
Matched pairs test \(t=(\bar{d}-\mu_{d,0})/(s_d/\sqrt{n})\)
Made by Joshua Bie