Writing parameters in AP Statistics

Inference is about population parameters, while the data give us sample statistics. A clear solution should distinguish those two before any interval or test is carried out.

Parameter versus statistic

A parameter is a fixed numerical characteristic of a population, and a statistic is calculated from a sample. For a proportion, \(p\) denotes the population proportion, while \(\hat p\) denotes a sample proportion. For a mean, \(\mu\) denotes the population mean, while \(\bar x\) denotes a sample mean.

Hypotheses are statements about parameters, so \(H_0:p=0.60\) is meaningful. A null hypothesis written as \(H_0:\hat p=0.60\) is not the same claim, because \(\hat p\) is already observed from the sample.

A useful definition pattern

A parameter definition should name three things:

• the symbol
• the variable or characteristic being measured
• the population

For example:

\[\begin{aligned} p &= \text{the true proportion of all students at the school who bike to school}\\ \text{or}\quad \mu &= \text{the true mean wait time, in minutes, for all visits to the coffee shop} \end{aligned}\]

The word “true” is useful shorthand for emphasizing that the quantity belongs to the population rather than the sample.

One proportion

Suppose a school claims that 60% of students bike to school and you suspect the true proportion is lower. Define \(p = \text{the true proportion of all students at the school who bike to school}\). Then \(H_0:p=0.60\) and \(H_a:p<0.60\).

One mean

Suppose a coffee shop advertises a mean wait time of 4 minutes. Define \(\mu = \text{the true mean wait time, in minutes, for all visits to the shop}\). A two-sided test would use \(H_0:\mu=4\) and \(H_a:\mu\neq4\).

Two proportions

Suppose we compare first-year and second-year students. Define \(p_1 = \text{the true proportion of first-year students who use the tutoring center}\) and \(p_2 = \text{the true proportion of second-year students who use the tutoring center}\). Then a no-difference null is \(H_0:p_1-p_2=0\), and the order of the groups should remain consistent throughout the problem.

Two means

For two independent populations, \(\mu_1-\mu_2\) describes the difference in population means. For example, \(\mu_1 = \text{the true mean weekly study time for athletes}\) and \(\mu_2 = \text{the true mean weekly study time for non-athletes}\). A no-difference null is \(H_0:\mu_1-\mu_2=0\).

Paired data

Paired data use one population of differences. If each student has a before and after score, define a difference such as \(d=\text{after}-\text{before}\). Then \(\mu_d = \text{the true mean after-minus-before difference for the population}\). A null hypothesis may be \(H_0:\mu_d=0\). This is a one-sample mean problem on the differences, not a two-independent-sample mean problem.

Confidence intervals use the same parameters

A confidence interval estimates a parameter, so the parameter should still be defined in context. An interval for \(\mu\) is not an interval for \(\bar x\). The sample mean is already known, and the uncertainty concerns the population mean.