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 – the distribution is symmetric about this

value – The standard deviation – 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

• Population– 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

Binomial sampling distribution of an unbiased coin tossed 10 times

0

0.05

0.1

0.150.2

0.25

0.3

0 1 2 3 4 5 6 7 8 9 10

Number of heads in 10 tosses

p(x

)

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

• Visit– http://www.ruf.rice.edu/~lane/stat_sim/sampling_dist/index.html

Central Limit Theorem

• Proposition 1:– 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)

x

Nx

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

x Nx

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.

Nes

x

..

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)

54.33

132.6..

xes

Confidence Intervals

• With our Jellyblubbers– One random sample (n = 30)

• Mean = 8.90– Therefore;

• 68% CI = 8.90 + or – 1(1.11)• 95% CI = 8.90 + or – 1.96(1.11)• 99% CI = 8.90 + or – 2.58(1.11)

11.130

132.6..

xes

Hypothesis Testing (see handout)

1. State the research question.

2. State the statistical hypothesis.

3. Set decision rule.

4. Calculate the test statistic.

5. Decide if result is significant.

6. Interpret result as it relates to your research question.