Lecture 26 - Population Parameter Sample & Estimate

LECTURE 26 Population, parameter, sample, estimate Assuming a given population: - The subjects eligible to vote in the US We are interested in a specific characteristic of the population, that we call parameter - The percentage of voters for the Democratic party We cannot observe the entire population, but we can randomly select some subjects: the sample - We observe the characteristics of interest of those subjects - We may ask their vote preferences Using this information, we obtain an estimate of the parameter of interest - The percentage of voters for the given party in the sample Example: - How many people are going to vote for the Democratic party during the next elections? - How widespread is the dance among 15-17 years old? % of voters for the Democratic party among the 240 million eligible voters % of subjects that went to disco at least once in the last year among the residents of 15-17 years % of voters for the Democratic party among the 1000 randomly selected voters % of subjects that went to disco at least once in the last year among a random sample According to ISTAT, the percentage is 60% (Multi-purpose survey on families, aspects of daily life) Population-sample-parameter-estimate In general Normal distribution case Binomial distribution case Parameter Estimate Any sample function is called statistic, i.e. sample proportion and sample mean. When a statistic is used to estimate a parameter, it is called estimator. If the parameter is θ and the estimator is θ̂, then it is often denoted as θ̂. The estimator value computed on the sample is the estimate. The probability distribution of the statistic is called sampling distribution. Example: The week of study The POPULATION we have a population of 86 units, represented by 86 slips of paper. A number is written on each slip. The PARAMETER is the mean of the 86 numbers. The SAMPLE: Extract 5 slips of paper from the population, this will be our sample. Write the numbers extracted and compute the sample mean, this is the ESTIMATE X₁ X₂ X₃ X₄ X₅ 6 20 5 25 40 mean=19.22 a bit different mean mean=20 Now we have to repeat the sampling step and computation of the estimator 1000 times Then check the results and compare it with the population The 86 means derived from our 1000 samples are distributed as follows: They concentrate around the true value of the parameter, in the sense that values close to the population mean, 16.2, are more likely to be observed than values far away. Imagine we have an infinite number of samples, we can move on to the density. In real life, however: - we do not know the population mean - we observe only one sample Sampling distribution We know the distribution but we don't know where it is. We observe a sample and its mean. We can compare the sample with the possible sample mean distributions. What happens if the sample size n in the population N increases? - The larger the sample, the more concentrated the estimates around the mean: X̄ₙ ≈ N(μ, σ²/n) Point estimate and confidence intervals In the point estimate, we compute a likely value for the parameter from the sample. What is the probability that we obtain exactly the parameter value? P(X̄ = μ) = ? X̄ is a continuous random variable, then P(X̄ = μ) = 0 So with probability equal to 1 we are wrong. We know that the estimate is likely close to the parameter value, and we are able to evaluate how far away it is, using sampling distribution. We can use this information to get an interval estimate. An interval is a way of expressing the estimate and the uncertainty together. CI for the mean of a normal (σ² known) Consider the estimator X̄ for the mean of a normal distribution We start from: D: ~N(0,1) For which: P(-zα/2 < Z < zα/2) = 1 - α P(-zα/2 < (X̄ - μ) / (σ / √n) < zα/2) = 1 - α ⇒ P(X̄ - zα/2 * σ / √n < μ < X̄ + zα/2 * σ / √n) = 1 - α The interval [X̄ - zα/2 * σ / √n; X̄ + zα/2 * σ / √n] that contains, with probability 1 - α, the parameter value μ for each α and σ² is called confidence interval for μ of level 1 - α. Instead of a single number (X̄), we estimate μ with an interval of values; this accounts for the uncertainty around μ that (partially) remains also after observing the sample (the interval has random bounds that depend on the sample). Example: μ̂ = X̄ ~ N(μ, σ²/n), μ̂ = μ₀λ