Recap of Normal DistributionSampling Distribution

EstimationSteps of Hypothesis Testing

Hypothesis Testing

Normal curve: brief review“Standardized scores” brief review

Z-scoresTransformed scores (e.g., T-score, IQ, SAT)

Sampling DistributionsEstimation: Confidence IntervalsHypothesis Testing

Agenda

Normal CurveAreasZ-scoresTransformed scores (e.g., T-scores)

Normal Distribution

Common shape in nature for many traitsSymmetrical distribution with one modeContains 100% of all cases in populationDetermined by two parameters:

Mean (center). Symbols: (pop.) and M (sample)

Standard Deviation (spread)Symbols: (pop.) and s (sample)Variance = 2 (pop) and s2 (sample)

Z-score for individual score:Transformed scores with new M, new s:New X = new M + (z)(new s) (Example: T scores)

Probability and Normal Distribution

s

X-MXz or

Z-score for individual score:Transformed scores with new M, new s:New X = new M + (z)(new s) (Examples: T scores, IQ, SAT and GRE, etc.)

Probability and Normal Distribution

s

MXXz

or

Sampling Distribution of the MeanShape of the distributionStandard Error

No longer dealing with individual scoresInstead, considering the behavior of a statistic

(computed on samples) relative to parameter in the population.

Sampling Distribution = distribution of that statistic if a very large (infinite) number of samples are drawn.

Can be distribution of mean, variance, median.

Sampling Distribution of the Mean is common.

Change of Perspective

Samples from Normal & Skewed Distributions

Samples drawn from Uniform and Irregular Distribution

Sampling Distribution of any Parameter

Bias and Variability Simulation

from the Estimation Chapter

When many samples are drawn, the distribution of their means is normal.

This is true regardless of the shape of the population’s distribution.

The Mean of the sampling distribution is equal to the Population Mean

Central Limit Theorem

NNN MM

222 or so

Variance of sampling distribution is equal to population variance divided by N (sample size)

Standard deviation of the sampling distribution is called the “Standard Error”(of the parameter). It is square root of the variance of the sampling distribution.

Variance and Standard Error

NNN MM

222 or so

EstimationPoint EstimatesInterval Estimates

Use our knowledge of sampling distribution to make a guess about the population when we only have information from a sample.

First examples are artificially simple: they give us the population standard deviation while we don’t know the population mean.

Estimation overview

The mean of the distribution of sample means is equal to the population mean .

The mean of the sampling distribution is called the expected value of the statistic

The sample mean is an unbiased estimator of the population mean .

Point Estimate

The mean of a single sample is an unbiased estimator of the population mean

The Standard Error of the Mean (when we know the population standard deviation ) gives us information about the expected deviation score of our sample mean M from the population parameter

We use this information, and the areas of the normal curve, to compute an interval estimate.

Interval Estimates

M = 30, = 9, N =25, 90% confidence intervalWe find z.90 from the Inverse Normal

Calculator (here) for the Unit Normal Curve (z = 1.6449 for a 90% confidence interval)

According to the Central Limit theorem, the middle 90% of sample means will be within1.6449 M of the population mean

Therefore, the probability that the population mean will be within 1.6449 M is also 90%

Sampling distribution example

An interval around the sample mean that is 1.6449 * M will contain 90% of the time.

Confidence Interval Simulator lets us see the effect of different sample sizes on intervals for 95% and 99% confidence (here)

Sampling distribution example (2)

Calculate the Standard Error of the Mean:M = /N = 9/ 25 = 9/5 = 1.80

Calculate Confidence Interval by hand: Lower limit = M – (1.6449*1.80) = 35-2.96082Upper limit = M + (1.6449*1.80) = 35+2.9608290% Confidence Interval

= [32.03918, 37.96082 ]

Sampling distribution example (3)

Remember: M is the Standard Deviation of the Sampling Distribution, so we will use M in the Inverse Normal Calculator.

Inverse Normal Calculator

Hypothesis TestingNull hypothesis and Alternate HypothesesReject a Null hypothesis when it is very unlikely

Logic of hypothesis test

1. State a null hypothesis about a population2. Predict the characteristics of the sample

based on the hypothesis3. Obtain a random sample from the population4. Compare the obtained sample data with the

prediction made from the hypothesis If consistent, hypothesis is reasonable If discrepant, hypothesis is rejected

Jury trial: an analogy• Trial begins with null hypothesis

(innocent until proven guilty).• Police and prosecutor gather evidence (data)

about probable innocence.• If there is sufficient evidence, jury rejects

innocence claim and concludes guilt.• If there is not enough evidence, jury fails to

convict (but does not conclude defendant is innocent).

Four steps of Hypothesis Testing

State the hypothesesNull hypothesis or H0

Alternate hypothesis or HA or H1

Set the criteria for a decisionPick an alpha level (.05 is common)

Collect data and compute sample statisticsNeed mean and standard deviation

Make a decision

Step 1: State hypothesesNull hypothesis (H0) states that, in the

general population, there is no change, no difference, or no relationship

Alternative hypothesis (H1) states that there is a change, a difference, or a relationship in the general population

Step 2: Set the criteria for decision

Distribution of sample means is dividedThose likely if H0 is trueThose very unlikely if H0 is true

Alpha level, or level of significance, is a probability value used to define “very unlikely”

Critical region is composed of the extreme sample values that are very unlikely

Boundaries of critical region are determined by alpha level.

Division of distribution of sample means

Critical regions for α = .05

Step 3: Collect data and Compute sample statistics

Data collected after hypotheses statedData collected after criteria for decision setThis sequence assures objectivity

Compute a sample statistic (z-score) to show the exact position of the sample.

Step 4: Make a decisionIf sample data are in

the critical region, the null hypothesis is rejected

If the sample data are not in the critical region, the researcher fails to reject the null hypothesis

Uncertainty and Errors in Hypothesis Testing

Hypothesis testing is an inferential processUses limited information to reach a general

conclusion, often leading to actionSample data used to draw conclusion

about a population that cannot be observed.

Errors are possible and probable

Type I and Type II Errors

Actual Situation

No EffectH0 True

Effect ExitsH0 False

Experimenter’s Decision

Reject H0 Type I error Decision correct

Retain H0 Decision correct Type II error

Location of critical region boundaries

Factors influencing hypothesis testSize of difference between sample mean

and original population meanAppears in numerator of the z-score

Variability of the scores Influences size of the standard error

Sample SizeInfluences size of the standard error

Directional (One-Tailed) Hypothesis Tests

In a two-tailed test, the critical region is divided on both tails of the distribution.

Researchers usually have a specific prediction about the direction of a treatment effect before they begin.

In a directional hypothesis or one-tailed test, the hypotheses specify an increase or decrease in the population mean

Critical region for One-Tailed Test

Video of an example (here)

Homework & QuizzesIt all makes sense when a teacher walks through it.The ideas become yours when you work with them.

Recap of Normal DistributionSampling Distribution

EstimationSteps of Hypothesis Testing

Hypothesis Testing