Agenda 1.Informationer 1.Excel i fb.m. projekt 2 2.Reserver tid til projekt 2 3.Øvelse: a / b = c 2.Opsamling fra sidst 3.Estimation (konfidensintervaller) 4.Hypotesetest 5.Overordnede spørgsmål til projekt 2 6.Arbejde med projekt 2 i grupper 7.Feedback på projekt 1

Opsamling på forrige gang A statistic vs. a parameter (s vs. σ) What do you use statistics for? What is a sampling distribution? The sampling distribution of a statistic is the probability distribution that specifies probabilities for the possible values the statistic can take. Standard deviation vs. standard error Population Sample

Learning Objectives 1.Point Estimate and Interval Estimate 2.Confidence Intervals 3.Margin of Error 4.Examples 5.Logic of Confidence Intervals

Point Estimate and Interval Estimate Ex: På et spm. svarer 74% i enighedssiden A point estimate is a single number that is our “best guess” for the parameter An interval estimate is an interval of numbers within which the parameter value is believed to fall.

Point vs. Interval Estimate And Confidence Interval A point estimate doesn’t tell us how close the estimate is likely to be to the parameter An interval estimate is more useful –It incorporates a margin of error which helps us to gauge the accuracy of the point estimate A confidence interval is an interval estimate containing the most believable values for a parameter Oprationally the confidence interval for the mean is –mean ± margin of error

Standard error 95% of a normal distribution falls within 1.96 standard deviations of the mean ≈ 2. 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, se. With probability 0.95, the sample mean falls within about 1.96 standard errors of the population mean. Margin of error = 1.96 x se (for a 95% confidence interval)

Standard Errors in Practice In practice, standard errors, se, are estimated –Standard errors, se, have exact values depending on parameter values, e.g., se = for a sample mean se = for a sample proportion –In practice, these parameter values are unknown. Inference methods use standard errors that substitute sample values for the parameters in the exact formulas above These estimated standard errors are the numbers we use in practice.

Ledelsen i et firma drøfter, om de ansatte må være på FB i løbet af arbejdsdagen og vil kende tidsforbruget. En undersøgelse blandt 36 ansatte viser, at de i gns. bruger FB i 15 minutter pr. dag. Standardafvigelsen i stikprøven er 6 minutter. Et 95% konfidensinterval er –gns. ± 1,96 x (s / √ n ) –15 ± 1,96 x (6 / √ 36) = 15 ± 1,96 x (6/6) –15 ± 1,96 x 1 = 15 ± 1,96 = [13,04 – 16,96] Eksempel på konfidensinterval for μ

En undersøgelse blandt 49 brugere af en ny app viser, at de i gns. bruger den i 10 minutter pr. dag. Standardafvigelsen i stikprøven er 3,5 minutter. Beregn et 95% konfidensinterval for tidsforbruget Opgave i konfidensinterval for μ

Example: CI for a Proportion In a survey respondents were asked whether they agreed with the statement “The texts on the site is written in clear and simple language”. 19% totally agreed. Calculate a 95% confidence interval for the population proportion who totally agreed with the statement. –The standard error, se, is sqrt ( p * (1-p) / n) = 0.01 –Margin of error = 1.96*se=1.96*0.01=0.02 –95% CI = 0.19±0.02 or (0.17 to 0.21) We predict that the population proportion who agreed is somewhere between 0.17 and 0.21.

CI - problem for a proportion In a survey respondents were asked whether they agreed with the statement “It is easy to navigate the website”. 25% totally agreed. Calculate a 95% confidence interval for the population proportion who totally agreed with the statement and interpret

95% of a normal distribution falls within 1.96 standard deviations of the mean. –With probability 0.95, the sample mean falls within about 1.96 standard errors of the population mean. –The distance of 1.96 standard errors is the margin of error (in calculating a 95% confidence interval for the population proportion). Oprationally the confidence interval for the mean is mean ± margin of error Logic of Confidence Intervals