Eksempel på brug af normalfordelingen De besøgende på en hjemmeside bruger i gns. 300 sekunder på forsiden, før de klikker videre til en underside. Besøgstiden er normalfordelt med en standardafvigelse på 50 sekunder. Hvad er sandsynligheden for at tilfældig besøgende højest bruger 240 sekunder på forsiden? X = 240, μ = 300, σ = 50, Hvad er P(X<240)? Svaret er 0,115069732. 1

Eksempel på z-score X angiver tiden før der klikkes videre fra forsiden til en underside. X er normalfordelt med μ = 300 og σ = 50. X’s z-score angiver hvor mange (antal) standardafvigelser, X ligger fra μ Hvad er sandsynligheden for at en tilfældig besøgende bruger mindre end 240 sekunder på forsiden? z = (X – μ) / σ = (240 – 300) / 50 = -60 / 50 = -1,2 P(z<-1,2) = 0,1151. 2

Øvelse i z-score X angiver tiden før der klikkes videre fra forsiden til en underside. X er normalfordelt med μ = 250 og σ = 50. Vi vil finde sandsynligheden for at en tilfældig besøgende bruger mindre end 200 sekunder på forsiden. Hvad er z-scoren? Hvad er P(z<z-score)? 3

Learning Objectives Statistic vs. Parameter Sampling Distributions Mean and Standard Deviation of the Sampling Distribution Standard Error Central Limit Theorem

Learning Objective 1: Statistic and Parameter A statistic is a numerical summary of sample data such as a sample proportion or sample mean A parameter is a numerical summary of a population such as a population proportion or population mean. In practice, we seldom know the values of parameters. Parameters are estimated using sample data. We use statistics to estimate parameters. Hvad er følgende? μ s σ Population Sample

Learning Objective 2: Sampling Distributions Example: Prior to counting the votes, the proportion in favor of Mr. Barack Obama was an unknown parameter. An exit poll of 3.160 voters reported that the sample proportion in favor of a recall was 0.54. If a different random sample of about 3000 voters were selected, a different sample proportion would occur. The sampling distribution of the sample proportion shows all possible values and the probabilities for those values.

Probability vs. sampling distribution The probability distribution of a random variable specifies its possible values and their probabilities. The sampling distribution of a statistic is the probability distribution that specifies probabilities for the possible values the statistic can take.

Learning Objective 2: Sampling Distributions The sampling distribution of a statistic is the probability distribution that specifies probabilities for the possible values the statistic can take. Sampling distributions describe the variability that occurs from study to study using statistics to estimate population parameters

Learning Objective 3: Mean and SD of the Sampling Distribution of a Proportion For a random sample of size n from a population with proportion p of outcomes in a particular category, the sampling distribution of the proportion of the sample in that category has

Learning Objective 4: The Standard Error To distinguish the standard deviation of a sampling distribution from the standard deviation of an ordinary probability distribution, we refer to it as a standard error. Makkerøvelse. Hvad er forskellen på probablity distribution og sampling distribution standard deviation og standard error

Learning Objective 3: The Sampling Distribution of the Sample Mean The sample mean, x, is a random variable. The sample mean varies from sample to sample. By contrast, the population mean, µ, is a single fixed number.

Learning Objective 3: The Sampling Distribution of the Sample Mean For a random sample of size n from a population having mean µ and standard deviation σ, the sampling distribution of the sample mean has: Center described by the mean µ (the same as the mean of the population). Spread described by the standard error, which equals the population standard deviation divided by the square root of the sample size: standard error of

Learning Objective 5: CLT: Impact of increasing n

Learning Objective 5: Central Limit Theorem (CLT) For random sampling with a large sample size n, the sampling distribution of the sample mean is approximately a normal distribution. This result applies no matter what the shape of the probability distribution from which the samples are taken.

Fordelinger Empiriske fordelinger Population distribution, N. Populationens parametre og fordeling er ukendte. Vi udtager en stikprøve fra populationen for at få viden om populationen, typisk parametrene μ og σ. Sample distribution, n. Stikprøven er en delmængde af N. Den består af data / observationer, u1, u2,..,un. Stikprøven kan beskrives grafisk og numerisk, f.eks. ved hjælp af gns. ū og std.afv. s. Jo større stikprøven er, des mere ligner den populationen Teoretiske fordelinger (fx binomial- eller normalfordelingen) Sandsynlighedsfordelinger viser sandsynligheden for at en variabel har ét bestemt udfald (sandsynligheden er udfaldets ”andel” i det lange løb). ”Standard deviation” er et spredningsmål. En ”samling distribution” er sandsynlighedsfordelingen for et statistisk mål, (typisk gennemsnit og standardafvigelse). Den bruges til at finde de sandsynlige værdier af det statistiske mål i populationen. ”Standard deviation” kaldes ”Standard error” 15