From a survey to a question
Statistics turns variable data into conclusions about a population. NHANES (the US National Health and Nutrition Examination Survey) measures people at mobile examination centres and asks them about their health. Start by naming the parts of a study and the type of each variable, since the type decides which summaries and methods are valid.
NHANES is an observational study: researchers record what they find and assign no treatment.
Describe each variable with a plot and numerical summaries (chapter 2).
Use probability models and sampling distributions to infer population values (chapters 3 to 6).
The study vocabulary, on real data
The parameter stays fixed. The statistic changes with every sample. That gap is what the rest of the module is about.
What type is each variable?
Pick a type for each, then check. Some are debatable; the feedback says why.
Describe the data
Categorical variables get a bar plot, proportions and the Index of Qualitative Variation. Numerical variables get a histogram, a centre (mean, median) and a spread (range, IQR, variance, standard deviation). Switch to a sample to see summaries become statistics with Latin letters.
Probability and the Binomial model
Pick a characteristic. Its share of the NHANES population is the probability p that a randomly chosen adult has it. Choose n adults independently and the count Y with the characteristic follows Binomial(n, p). Simulating real groups shows the frequentist definition at work: relative frequencies settle on the model.
Two events at once: independence and the addition rule
If A and B are independent, Pr(A ∩ B) = Pr(A) × Pr(B). If they are mutually exclusive, Pr(A ∩ B) = 0. Real health variables are rarely either.
Fit a Normal model
Adult height within one sex is close to Normal. Other variables are skewed. Compare the histogram with N(μ, σ²), check the 68, 95 and 99.7 rule against the real data, then use z-scores to find probabilities and to work backwards from a probability to a value.
Calculation
Empirical rule check
Many samples, one distribution
Draw repeated random samples of n adults and record each sample mean (or proportion). The pile of results is the sampling distribution. It centres on the population value, its spread is the standard error σ/√n, and for n ≥ 30 it is close to Normal even when the variable itself is skewed.
Confidence intervals that you can audit
One sample gives one interval. Over many samples, about 95% of 95% intervals contain μ. Here μ is known, so every interval that misses is marked. Use z when σ is known; use t with n − 1 degrees of freedom when σ is replaced by the sample s.
Latest interval, worked
Plan a study: how big must n be?
Check against the pain relief example from the lectures: σ = 2.25 hours, ME = 0.5 hours and 99% confidence give n = 135.
Case study: blood pressure in US adults
A full analysis from one sample, the way you would do it with real data where the population is unknown. At the end we reveal the population value and check the conclusion.