Population vs Parameter

The population is fixed by the question, not by the data. Defining one means writing a membership rule precise enough to sort any individual in or out: not "voters" but "the 12,400 people registered to vote in this county on October 1". A number computed from the whole population is a parameter, written $\mu$ (mu) for a mean, $p$ for a proportion, $\sigma$ (sigma) for a standard deviation. A parameter is one fixed number, usually unknown, and it does not move when you take a different sample.

Stay with that county. The parameter is $p$, the proportion of all 12,400 registered voters who plan to vote yes. A random sample of 600 gives $\hat{p} = 0.52$ (p-hat), an estimate of $p$ and not $p$ itself. The population size enters in one place only: the sample is $600/12400 = 0.048$ of the population, under the 10 percent ceiling the standard error formula needs, so $\sqrt{(0.52)(0.48)/600} = 0.020$ can stand as the standard error.

The wrong sentence is short and common: "the population is the 600 voters who were surveyed." Those 600 are the sample. The population is the group the conclusion is about, and it exists whether or not anyone measures it. Two smaller slips travel with it. A population need not be people; it can be 4,000 laptop batteries, or every 20-minute interval in a factory's day. And a population is not everyone, because it stops exactly where the question stops, so this survey says nothing about the state.

One piece of intuition is worth killing off. A bigger population does not need a bigger sample. Precision comes from $n$; apart from that 10 percent check, the population size appears nowhere in $\sqrt{p(1-p)/n}$, so a county of 12,400 and a country of 300 million take about the same sample for the same margin of error.

The population you want and the population your method can reach are different things. The list actually drawn from is the sampling frame, and anyone in the population but off that list has no chance of selection. Random sampling is topic 1.11 in Unit 1.

A parameter is a number attached to a population. It is fixed: it does not change when you draw a different sample, and it does not change because you failed to measure it. What is missing is your knowledge of the value, not the value. The symbols are mostly Greek: $\mu$ (mu) for the population mean, $\sigma$ (sigma) for the standard deviation, $\rho$ (rho) for the correlation. The proportion is the exception, since the population value is the Roman letter $p$ and the sample value is $\hat{p}$ (p-hat).

A company has exactly five regional offices, staffed by 12, 15, 18, 20, and 25 people. If the population is those five offices, then $\mu = \frac{90}{5} = 18$ employees, and the population standard deviation divides the squared deviations by $N = 5$, giving $\sigma = \sqrt{\frac{98}{5}} = 4.43$. Both are parameters and both are known, which is unusual and entirely legitimate.

Two sentences to unlearn: "a parameter is a number you cannot know" and "if I calculated it, it must be a statistic." The label comes from which group the number describes, not from whether the value is available and not from the group's size. Treat the same five offices as a random sample of 60 offices nationwide and the same data give $\bar{x} = 18$ and $s = \sqrt{\frac{98}{4}} = 4.95$, dividing by $n - 1 = 4$. Same numbers in, different label out, and a different divisor for the spread.

Which one you hold depends on how the population was defined, and that comes from the question asked rather than from the spreadsheet in front of you. The mean of 500 rows is a parameter when those 500 are the whole group you want to describe and a statistic when they stand in for a larger one.

Inference is written about parameters: a confidence interval brackets one, and a hypothesis states a value for one, which is why $H_0: p = 0.5$ is well formed and $H_0: \hat{p} = 0.5$ is not. The sample value is already in front of you, so there is nothing left to hypothesize.