Introduction to Statistical Inference

Chapter 11

Announcement: Read chapter 12 to page 299

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.