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SYRACUSE / Psychology / PSY 252 / What is a hypothesis test?

What is a hypothesis test?

What is a hypothesis test?

Description

School: Syracuse University
Department: Psychology
Course: Statistical Methods II
Professor: A. park
Term: Fall 2015
Tags:
Cost: 50
Name: PSY 252: Exam 2 Study Guide
Description: All material covered on Exam 2. Includes examples, definitions, terms, and diagrams.
Uploaded: 11/02/2015
10 Pages 75 Views 4 Unlocks
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PSY 252 Exam #2 Study Guide


What is a hypothesis test?



10/27 (Monday) 10:35 to 11:30 (regular class time) at HBC Kit (regular classroom)

This exam will cover all material that you learned AFTER exam#1 and this is NOT cumulative: Chapter 5, 6, and 7, and Chapter 8, and Chapter 14 correlation part (up to page 464).

It will be in a same format as Exam#1, consisting of 25 multi-choice questions (with a few questions same as or similar to end-of-lecture questions)

Most of questions will be based on lecture notes. You may want to study lecture notes first, followed by Aplia HW questions and a textbook, and then review lab materials. Reading lecture notes won’t be enough for getting a good score in exams.​You need to PRACTICE questions​in textbook, Aplia HW (personalized reviews for each chapter) & lab materials. I already have described what kinds of questions that I would ask in exam at many occasions during lectures.


A sample of n=25 has a se of 2 points which is the sd for the population from which the sample is obtained?



All lecture notes will be available on Aplia from 48 hours before the exam time (this Saturday morning); all Aplia HWs and lab materials are available for your review on Aplia at any time (not on Blackboard).

You will be given a z-table (the unit normal table) but NO formula sheet.

Bring (1) a No. 2 pencil, (2) an eraser, and (3) a scientific calculator that cannot store information.

You are only allowed to use calculators that cannot store information.​No cell phone calculators are allowed during an exam. Use of cell phones or the incorrect type of calculator during an exam can be considered cheating. Incorrect calculators include all graphing calculators​, as they can actually perform the statistical analysis you’re doing on the test.​Scientific calculators (usually less that $15) and really cheap calculators that have a square root button (usually less than $5) are good.


What do you call a single statistic that is calculated from the sample data and that is used to test the hypothesis?



We also discuss several other topics like What is the difference between interspecific competition and intraspecific competition?

Only those who have submitted a documented excuse will be allowed to take a makeup exam. If you happen to miss this in-class exam without a documented absence, you will receive a zero point for this exam. To replace the zero point, you may choose to take the optional final exam in the end of this semester.

Page 1​of 10

PSY 252 Exam Study Guide

Definition of standardized scores

Definition and unit of z-scores 

z-score- standardized score Don't forget about the age old question of What are the pros and cons of electroencephalography (eeg)?

- # of SD’s from the mean

- units= standard deviations

Population Scores: Sample Scores:

z = (X ­ μ) / σ z= (X­M)/ s 

Properties of a z distribution

● gives the location of X in its distribution

○ (-) below the mean

○ (+) above the mean

-can compare scores from different distributions (w/ diff. means and SDs)

Properties of normal distribution

● the shape of the distribution remains the same after standardized scores are re-labeled Relationship between probability (proportion) and normal distribution Don't forget about the age old question of What type of dyslexia that can read non­words and regularly spelled words better than irregularly spelled words?

Probability (proportion): proportion of population scores that fall in each section of the normal distribution We also discuss several other topics like What is mendel's hypotheses all about?

p = # of outcomes classified as x

______________________________________

total # of possible outcomes

Normal Distribution: the most commonly observed frequency

● bell-shaped

● symmetrical

● uni-modal

● mean=median=mode

● extends to +/- infinite numbers

Probability & Normal Distribution 

1. the proportion of a certain range of z-scores = their area under the normal curve 2. the sum of all proportions/probabilities is 1

3. a proportion of .5 lies to the left of the mean, and a proportion of .5 lies to the right 4. the proportion of the LEFT side is EXACTLY SAME as the proportion to the RIGHT

Probability (proportion) of z-score distribution assuming normal distribution 

● different portions of the normal distribution are indicated by z-score units ● a z-score corresponds to a specific proportion under the normal curve

( use the unit normal table)

Sampling error 

Page 2​of 10Don't forget about the age old question of In what year did the fda approve the first oral contraceptive for use by women?

● the discrepancy ( amount of error) that exists by chance between statistics and parameters ● a sample typically will NOT provide a perfectly accurate representation of its population Don't forget about the age old question of Why technological process is a source of sustained growth in gdp?

Sampling distribution 

● the distribution of a statistic obtained from all possible samples from the population ○ distribution of sample means

○ distribution of sample variances

Central Limit Theorem 

● a mathematical proposition that provides a description about the sample mean distribution of ALL possible random samples

The distribution of sample means: definition, shape, mean and SD 

distribution of sample means :

● collection of sample means for ALL POSSIBLE random samples of a particular size (n) obtained from the population of interest

● one type of sampling distribution

● the distribution of sample means obtained from all possible random samples of a specific size from the population

1) Shape

the distribution of sample means will be NORMALLY Distributed if EITHER of the below are met ● the population is normally distributed

● a sample size n>30 even if the population distribution isn’t normally distributed

2) Mean

the mean of the distribution of sample means is:

● same as population mean (M)

● called expected value of sample mean (m)

● an UNBIASED estimate of μ 

** on AVERAGE, the sample mean produces a value that EXACTLY MATCHES the population mean

3) Standard Deviation (SD)

the SD of a sampling distribution is known as the standard error (σM )

● the standard( average/expected) amount of difference between M and µ

● due to random chance

formulas:

z= (M - μ ) / σM mean = μ 

σM = σ/ √n SD = σM 

Standard error 

● the measure of “standard” distance from the population mean (SD) and a sample mean by chance ● SD of the sample mean distribution

Formula:

σM = σ/ √n

Law of large numbers 

● bigger sample = less sampling error

Page 3​of 10

● the more individuals you obtain from the population, the more representative the sample is Definition of hypothesis testing 

Hypothesis Test- a statistical method that uses sample data to evaluate a hypothesis about a population

1. state hypothesis about a population

2. set the decision criteria by locating the critical region ( setting alpha level) 3. collect data and compute a statistic

4. compare the obtained sample data w/ the prediction that was made to make a decision about a population

Null and alternative hypotheses 

Null Hypothesis (H0):

● NO change

● NO difference

● NO relationship

● the IV has NO effect on the DV

Alternative Hypothesis (H1):

-does state a relationship, change, or effect of the IV On the DV

- also called scientific hypothesis

Directional versus non-directional alternative hypotheses 

Directional

-one-tailed

we can make a directional hypothesis when:

● past research, theory should decide BEFORE testign the hypothesis

● without good theory/ research suggesting a directional hypothesis, use a non-directional hypothesis

H0 (null): µ w/ banana ≤or ≥100

H1(alt.): states difference WITH a SPECIFIC DIRECTION µ w/ banana > or < 100

Non-Directional

-two-tailed

H1: states difference but no direction: µ w/ banana ҂100

-H0: µ w/ banana = 100

One-tailed versus two-tailed testing 

One -tailed

- directional

- critical region is BIGGER, better Testing

Two-Tailed

-non-directional

Alpha( a) level 

● level of significance

Page 4​of 10

● a probability value used to separate very unlikely samples from likely samples, when H0 is true ○ researchers separate the most UNLIKELY 5% of the sample means

Critical region 

● the region of the sample mean distribution where the sample means that are extremely UNlikely to occur ( a very low probability) if H0 is true

○ If M does NOT fall in the critical region:

■ the M is similar to u

■ we conclude that H0 is true we RETAIN H0

If M DOES FALL in the critical region

● the M is VERY different from u

● thus we conclude the H0 is NOT TRUE we REJECT H0

Meaning of numerator and denominator of z-test statistic 

z-test statistics- a single statistic that is calculated from the sample data and that is used to test the hypothesis

z= sample mean -hypothesized population mean

_________________________________________________________ z= (M - μ ) / σM standard error between the sample and population means

Meaning of p values of z-test statistic 

● the P value or significance level, of your obtained test statistic tells the strength of the evidence AGAINST H0

○ the SMALLER the p-value, the STRONGER the evidence AGAINST H0

Effect size 

● a measurement of the absolute magnitude of a treatment effect, independent of the samples being used

○ statistically SIGNIFICANT treatment effect does NOT necessarily mean a substantial treatment effect

■ Significant treatment: SD is very small and/or sample size is very BIG

■ measures: how big an effect is

not a yes / no decision

Cohen’s d: definition, Cohen’s d formula for z-test, and interpretation 

Cohen’s d= mean difference M - u

____________________ = __________ Cohen’s d: effect size can be standardized by mean standard deviation σ difference in terms of SD

small effect: d= .2

medium effect: d= .5

large effect: d= .8

Type I and Type II errors​​(​when we retain or reject the null hypothesis)

H0​is TRUE ( NO Effect)

H0​is FALSE ( Effect exists)

Page 5​of 10

Type 1​Error

Decision Correct

Decision Correct

Type 2 ​Error

Type 1:

● reject a TRUE null hypothesis

○ actually was due to sampling error, but erroneously concluded that there is a tx effect ○ False positive (+) = False significant finding (claims it’s true but it isn’t)

● Probability of Type 1 error= alpha level

○ sample mean is in critical region by chance

Type 2:

● retain (accept) the null hypothesis

○ actually was a real effect, but we erroneously concluded that there was NO effect ○ False negative (-) = false non-significant finding

● Probability of Type 2 error = beta level

○ impossible to determine a single exact probability

○ can be calculated based on STATISTICAL power

Statistical power: definition and its relationships with Type II error, effect size, sample size, alpha level, one- versus two-tailed testing, and SE (or SD) 

Statistical Power: 

● the probability to correctly reject a false H0 

○ to find an effect when there is a REAL effect

● When there is a real effect( FALSE H0​)

○ if Type 2 error = Beta then the probability to correctly reject the False H0 ( power) = 1 - beta ● Power = 1- Type 2 Error (1 - Beta)

● Beta = 1- power

Relation… 

● if Z is BIGGER - MORE likely to reject null →GREATER power

● if Z is bigger in the NUMERATOR- MORE likely to reject the null→ GREATER power ● if Z is bigger in the DENOMINATOR- LESS likely to reject null→ LESS power

How to determine Power?

Factor 1:

● Group effect/size of treatment 

○ if Z is Bigger in the NUMERATOR, BETTER power

● SE: Variability (SD) of scores 

○ if Z is Smaller in the Denominator, Better power

● Sample Size 

○ if Z denominator is BIGGER, power is BETTER

Factor 2:

● Alpha Level: ( .05, .01, .001) 

● the BIGGER an alpha level is, the critical region becomes BIGGER

● More likely to reject the null, Greater power

● One-vs. Two- Tailed Testing 

○ look at the right-side critical region

■ 1- tailed critical region is bigger​→ better power

Page 6​of 10

Correlation Coefficient: Relationship & Strength 

Correlation Coefficient: 

● a single number that describes a relationship​between 2 variables that naturally exist by it’s direction & strength

● ranges from -1 → +1

○ a relationship does NOT mean changes in one variable will cause changes in another (causation)

○ a statistical relationship can exist​even though one variable does not​cause or influence the other

Characteristics of a Relationship: 

● Direction-sign ​of correlation coefficients

○ Positive association (+)

○ Negative association (-)

● Strength​(magnitude)- size ​of correlation coefficients

○ 0: NO ​relation

○ 1: perfectly consistent​relationship

○ the larger ​the values are, the more​closely​related they are

Rule of Thumb:

Size of Coefficient

General Interpretation

0.8 → 1.0

very strong​relationship

0.6 → 0.8

strong​relationship

0.4 → 0.6

moderate​relationship

0.2 → 0.4

weak​relationship

0.0 → 0.2

very weak ​relationship

CORRELATION is NOT CAUSATION

Pearson correlation notation and usage 

Pearson’s r 

● describes the degree of linear relationship between two interval/ratio​variables ● symbol = r

r​= degree to which x & y vary together ​/ degree to which x & y vary separately or

r​= covariability ​of x & y / variability ​of x & y separately

Definition of coefficient of determination: 

-Explains proportion of variance

● Coefficient of Determination= r​2 

Page 7​of 10

○ a proportion of variability in 1 variable that is determined by the other variable ○ same as an effect size measure in z, t, F tests

Questions:

1. A sample of n=25 has a SE of 2 points

Which is the SD for the population from which the sample is obtained?

● B 10

2. Which of the following accurately describes the critical region?

○ A Outcomes with a very low probability if H0 is true

3. A scatter plot shows a set of data points that are widely scattered around a line that slopes down to the right. Which of the following values would be closest to the correlation ( graph with negative correlation and r=4)

○ C 0.40

Page 8​of 10

Calculations ​(and practice questions from textbook)

To transform X values to z-scores

Practice question: Learning Check #1 on p. 130

To transform z-scores to X values

Practice question: Learning Check #2 on p. 130

To transform X values to a distribution with a predetermined µ and ​σ Step1: Transform X values to z-scores

Step 2: Transform z-scores to new X scores with the pre-determined µ and σ Practice question: Learning Check #1 on p. 138

To find a probability (proportion) of z-scores

: using the unit normal table

Practice question: lecture notes on Oct 7th and 9th and Lab material on Oct 10th/11th 

To find z-scores with a given probability (proportion)

: using the unit normal table

Practice question: Learning Check #2 on p. 162

To transform X values to a probability (proportion)

Step 1: transform X values to z-scores

Step 2: transform z-scores to a probability (proportion) using the unit normal table Practice question: Learning Check #1 on p. 168

To transform a probability (proportional) to X values

Step 1: transform a probability (proportional) to z-scores using the unit normal table Step 2: transform z-scores to X values

Practice question: Learning Check #2 on p. 168

To transform sample means to a probability (proportion)

Step 1: calculate σM 

Step 2: transform the sample means to z-scores

Step 3: identify a probability (proportion) of the z-scores using the unit normal table Practice question: Learning Check #2 on p. 189

To transform a probability (proportion) to sample means

Step 1: calculate σM 

Step 2: identify z-scores with a given probability (proportion) using the unit normal table Step 3: transform the z-scores to sample means

Practice question: Learning Check #3 on p. 197

z-test statistics to compare a sample mean with a population mean

Practice question: Learning Check #1, #2, #3 on p.213

Page 9​of 10

Formulas to memorize

To transform X VALUES​to z-scores

Population values:

Sample values:

To transform z-scores to X VALUES

Population values:

Sample values:

Standard error

To transform SAMPLE MEANS​to z-scores where (this is also a formula for z-test statistic)

To transform z-scores to SAMPLE MEANS

M = z*σM + µ where

Cohen’s d for z test

Transformation from Person r to coefficient of determination Coefficient of determination = r2 

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