 # Sampling Distributions

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Sampling Distributions
Note: Homework 3 due 3/23

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.

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.

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 value The 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.

The probability of continuous variables
IQ test Mean = 100 and sd = 15 What 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)?

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.

Introduction to Statistical Inference
Chapter 11

Populations vs. Samples
The complete set of individuals Characteristics are called parameters Sample A subset of the population Characteristics are called statistics. In most cases we cannot study all the members of a population

Inferential Statistics
Statistical Inference A series of procedures in which the data obtained from samples are used to make statements about some broader set of circumstances.

Two different types of procedures
Estimating population parameters Point estimation Using a sample statistic to estimate a population parameter Interval estimation Estimation of the amount of variability in a sample statistic when many samples are repeatedly taken from a population. Hypothesis testing The comparison of sample results with a known or hypothesized population parameter

These procedures share a fundamental concept
Sampling distribution A theoretical distribution of the possible values of samples statistics if an infinite number of same-sized samples were taken from a population.

Example of the sampling distribution of a discrete variable

Continuous Distributions
Interval or ratio level data Weight, height, achievement, etc. JellyBlubbers!!!

Histogram of the Jellyblubber population

Repeated sampling of the Jellyblubber population (n = 3)

Repeated sampling of the Jellyblubber population (n = 5)

Repeated sampling of the Jellyblubber population (n = 10)

Repeated sampling of the Jellyblubber population (n = 40)

For more on this concept
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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)

Application of the sampling distribution
Sampling error The 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

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.

Confidence Intervals With our Jellyblubbers One random sample (n = 3)
Mean = 9 Therefore; 68% CI = 9 + or – 1(3.54) 95% CI = 9 + or – 1.96(3.54) 99% CI = 9 + or – 2.58(3.54)

Confidence Intervals With our Jellyblubbers One random sample (n = 30)
Mean = 8.90 Therefore; 68% CI = or – 1(1.11) 95% CI = or – 1.96(1.11) 99% CI = or – 2.58(1.11)

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.