Sample fairly and estimate

Population, sample and a fair sampling frame

Before calculating, name the population, check who can enter the sampling frame, then identify how the sample is selected. Keep selection bias separate from random variation.

A population is the total set of observations of interest; a sample is a subset used to learn about it.

The lecture’s height-survey context gives three reasons to sample: some members are difficult to measure or do not respond accurately; measuring everyone takes time and values can change during collection; hiring enough interviewers costs money.

Define the population and time period before choosing a frame. A street survey restricted to people who look friendly gives some members no chance of inclusion.

For finite simple random sampling without replacement, every subset of the prescribed size has the same chance. Equal individual inclusion probabilities alone do not establish this stronger property.

The source and its teacher callout also stress “independent” selection; distinguish that later iid observation model from the dependent choices in a fixed-size draw without replacement.

Random selection reduces selection bias; it does not guarantee that every realised sample is representative, remove nonresponse bias, or make an incomplete frame complete. H2 problems in this source assume simple random sampling unless stated otherwise.

The source’s polling case studies illustrate selection and response bias. As figures reported by the lecture, its 1936 Literary Digest account says 10 million questionnaires, 2.4 million replies, a predicted 57% for Landon and a reported 62% Roosevelt result; its 1948 account contrasts major polls predicting Dewey by about 5% with a reported Truman result just below 50% and Dewey 45%.

Its extension names FiveThirtyEight, reporting 49/50 state predictions in 2008 and 50/50 in 2012, then characterising 2016 as a failure.

These accounts and the source’s bankruptcy/causation wording remain in the native transcription, not independently verified historical findings here.

The statistical lesson is to examine the frame, participation and forecasting uncertainty: one election outcome alone does not establish whether a probabilistic forecast was well calibrated. Larger sample size cannot cure systematic selection or response bias.

Compare two survey plans

A college wants to estimate the mean weekly study time of all its registered students. Plan A randomly selects 50 distinct names from the complete register and follows up every selected student. Plan B asks the first 200 visitors to the library who agree to answer. Which plan supports inference about the stated population more convincingly? Explain whether increasing Plan B to 800 responses fixes its weakness.

After your attempt: compare the reasoning

Plan A gives every size-50 subset the same chance when the random selection is implemented correctly. Its complete frame and follow-up address two sources of selection/response bias, although any realised sample can still be unusual. Plan B favours library visitors who attend at that time and choose to respond. More responses from that process can reduce variability without repairing its selection and response bias.

Statistics, parameters, estimators and estimates

A parameter, such as population mean $\mu$, variance $\sigma^2$ or coffee-drinking proportion $p$, is fixed for a specified population. A statistic is a function of a random sample: mean, median, mode, standard deviation and proportion are examples.

In the lecture’s coffee example, if $X$ counts morning coffee drinkers in a sample of size $n$, the random sample proportion is $P_s=X/n$; after collecting data its realised value is $p_s=x/n$. Distinguish the random rule from its observed value.

An estimator $T$ is a statistic used to estimate a parameter $\theta$; an estimate $t$ is its realised numerical value.

Unbiasedness means $E(T)=\theta$ over repeated sampling, not $T=\theta$ in every sample. The teacher suggests following the short mean proof first and self-reading the longer variance proof; both are retained below.