review & hypothesis testing
DESCRIPTION
Central Limit Theorem, Confidence Intervals, and Basics of Hypothesis TestsTRANSCRIPT
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