Presentation on theme: "Sampling Distributions"— Presentation transcript:
1 Sampling Distributions Note: Homework 3 due 3/23
2 Probability distributions If we measure a random variable many times, we can build up a distribution of the values it can take.Imagine an underlying distribution of values which we would get if it was possible to take more and more measurements under the same conditions.This gives the probability distribution for the variable.
3 Continuous probability distributions Because continuous random variables can take all values in a range, it is not possible to assign probabilities to individual values.Instead we have a continuous curve, called a probability density function, which allows us to calculate the probability a value within any interval.This probability is calculated as the area under the curve between the values of interest. The total area under the curve must equal 1.
4 Normal (Gaussian) distributions Normal (also known as Gaussian) distributions are by far the most commonly used family of continuous distributions.They are ‘bell-shaped’ –and are indexed by two parameters:The mean m – the distribution is symmetric about this valueThe standard deviation s – this determines the spread of the distribution. Roughly 2/3 of the distribution lies within 1 standard deviation of the mean, and 95% within 2 standard deviations.
5 The probability of continuous variables IQ testMean = 100 and sd = 15What is the probability of randomly selecting an individual with a test score of 130 or greater?P(X ≤ 95)?P(X ≥ 112)?P(X ≤ 95 or X ≥ 112)?
6 The probability of continuous variables (cont.) What is the probability of randomly selecting three people with a test score greater than 112?Remember the multiplication rule for independent events.
7 Introduction to Statistical Inference Chapter 11
8 Populations vs. Samples The complete set of individualsCharacteristics are called parametersSampleA subset of the populationCharacteristics are called statistics.In most cases we cannot study all the members of a population
10 Inferential Statistics Statistical InferenceA series of procedures in which the data obtained from samples are used to make statements about some broader set of circumstances.
11 Two different types of procedures Estimating population parametersPoint estimationUsing a sample statistic to estimate a population parameterInterval estimationEstimation of the amount of variability in a sample statistic when many samples are repeatedly taken from a population.Hypothesis testingThe comparison of sample results with a known or hypothesized population parameter
12 These procedures share a fundamental concept Sampling distributionA theoretical distribution of the possible values of samples statistics if an infinite number of same-sized samples were taken from a population.
13 Example of the sampling distribution of a discrete variable
14 Continuous Distributions Interval or ratio level dataWeight, height, achievement, etc.JellyBlubbers!!!
21 Central Limit Theorem Proposition 1: Proposition 2: Proposition 3: The mean of the sampling distribution will equal the mean of the population.Proposition 2:The sampling distribution of means will be approximately normal regardless of the shape of the population.Proposition 3:The standard deviation (standard error) equals the standard deviation of the population divided by the square root of the sample size. (see 11.5 in text)
22 Application of the sampling distribution Sampling errorThe difference between the sample mean and the population mean.Assumed to be due to random error.From the jellyblubber experience we know that a sampling distribution of means will be randomly distributed with
23 Standard Error of the Mean and Confidence Intervals We can estimate how much variability there is among potential sample means by calculating the standard error of the mean.
24 Confidence Intervals With our Jellyblubbers One random sample (n = 3) Mean = 9Therefore;68% CI = 9 + or – 1(3.54)95% CI = 9 + or – 1.96(3.54)99% CI = 9 + or – 2.58(3.54)
25 Confidence Intervals With our Jellyblubbers One random sample (n = 30) Mean = 8.90Therefore;68% CI = or – 1(1.11)95% CI = or – 1.96(1.11)99% CI = or – 2.58(1.11)
26 Hypothesis Testing (see handout) State the research question.State the statistical hypothesis.Set decision rule.Calculate the test statistic.Decide if result is significant.Interpret result as it relates to your research question.