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Chapter I^
☐ Statistics : branch of mathematics that focuses
on the (^) organization, (^) analysis + (^) interpretation of a group
of numbers
1.) (^) Descriptive statistics → (^) summarizes a (^) group of (^) numbers from (^) a research (^) study.
2.) Inferential statistics →^ draw conclusions /make inferences that go beyond the scores from a research
study.
° Variables -
condition or^ characteristic that^ can have different values .
- (^) can (^) vary (what is being measured) ° (^) Values → (^) possible numbers (^) or
categories that^ can^ be^ assigned to^ a^ variable^.
☒ Values for variables fall in a
meaningful (^) range of^ numbers^ or^ categories.
° Score → a particular individual's value on a variable
ex. (^20) years old (^) , 150 pounds Levels of^ measurement 1.) (^) Numeric variable (quantitative) → (^) Equal - interval variables : (^) the numbers stand for approximately equal^ amounts^ of^ what^ is^ being^ measured.
→ Rank order or ordinal variables : values are ranks
2.) (^) Categorical variables (non^ - quantitative) (^) How to make a
frequency table
ex. eye color, gender, brand^ preference
1.) make a list down the page of each possible
value from lowest to (^) highest.
2.) Go one by one through scores , making a mark for each
next to (^) its value (^) on the list (^). 3.) make (^) a table (^) showing how (^) many times each valve on the list is used (^).
4.) figure the percentage of scores for each value .
→ (^) take the (^) frequency for (^) that (^) value and divide it (^) by the total number of^ scores and (^) multiply it^ by 100.
Frequency Table^ Histogram ° Grouped (^) frequency tables^ →^ frequency table^ using intervals^ based^ on^ valves
- (^) intervals must be (^) equivalent across the (^) range of (^) scores ° Histogram are^ in^ numerical^ order
* the bars do not have to be in a certain order.
° Frequency Polygons^ are^ useful^ for^ comparing sets^ of^ data
Unimodal =^ one high area Bimodal = two high areas
Chapter 2
- Measures of (^) Central tendency " sum of^ " ✗ (^) Scores of 1.) Mean → (^) sum of^ all scores divided (^) by the number of (^) scores ex. variable ← ✗ 2.) (^) median → the (^) score that (^) divides the distribution in half M = number
3.) F^ of^ scores
Mode →^ common (^) single value^ (for^ nominal^ values)^ mean
average of (^) squared deviations
Chapter
3- 2 scores
° comparing one score to the mean of^ a distribution does indicate whether a (^) score is above or below (^) average. ° comparing a^ score^ to^ the^ mean^ &^ standard^ deviation^ of^ a^ distribution^ indicates^ how^ much^ above^ or^ below average a^ score^ is^ in^ relation^ to^ the^ spread^ of^ scores^ in^ the^ distribution^.
° 2 score → describe
a (^) particular score^ in^ terms^ of^ where^ it^ fits^ into^ the^ overall^ group of^ scores.
- (^) transforms the ordinary
score to^ better describe the score's location in a distribution .
° 2 score → number of standard deviations the actual score is above or below the mean
.
- raw score : ordinary score^ opposed^ to^2 score^ (which^ is^ scaled) Formula to (^) change Raw^ score to (^2) score 2=2 score
* 2 = ✗ = raw score
M= mean
SD =^ standard^ deviation
Formula to (^) change 2 score to a Raw (^) score 2=2 score
- ✗ =^ G)(SD)^ -1M ✗ =^ raw^ score M= mean
SD = standard^ deviation
ex . Jerome's 2 score =^ 1.09 , what is his raw score?
m= (^) 3.40 If (^) a 2 score... SD =^ 1. ( ✗ =^ (2) D)+^ M^ 1.)^ has^ a^ value^ of^0 , it^ is^ equal to ✓ ✗ =^ G.^09 )(1-47)+3.40 =^ 1.60+3.40^ =^5 the^ group mean^. 2.) (^) is (^) positive; it is above the (^) group mean. 3.) is (^) negative ;
° In
a normal curve, scores fall near center , fewer at extremes .^ 4.)^ is^ equal to^ +1^ ,^ it^ is 1 SD above the mean .
5.) is equal to +2 , it is 2 SDS above the mean.
6.) (^) is (^) equal to -1 (^) , it^ is 1 SD (^) below the mean (^).
7.) is equal to -2,^ it^ is^2 Sds below the mean.
✗ =^ (2-0)^ /^ o)^ -
°
the shape of^ a normal curve is standard t well defined ; there is a known percentage of scores above or below
any particular^ point^.
° Normal (^) curve with approximate (^) percentages of^ scores^ between^ the^ meant^ 1,2^ and^ more^ than^2 SDS^ above^ +
below the^ mean.
you can^ figure^ the^ exact^ percentage^ of^ scores^ between^ any two^ points^ on^ the^ normal^ curve. I % in tail % mean^ to^ Z F % (^) mean to^2 % (^) in (^) tail steps to^ figuring the^ percentage of^ scores^ above^ or^ below a (^) particular Raw or 2 score 1.) If (^) you are (^) beginning with (^) a raw (^) score (^) , first (^) change it to (^) a 2 score (^). 2.) Draw^ a (^) picture of^ a (^) normal curve, decide where the (^2) score falls (^) on it, and shade in the (^) area for which (^) you are (^) finding the (^) percentage. 3.) make (^) a rough estimate^ of the (^) shaded area's (^) percentage based on (^) the 50%-34%-14-1. percentages. 4.) (^) Find exact
percentage using^ normal^ curve^ table,^ adding 50%^ if^ necessary.
5.) Check^ that^ your exact percentage is^ within^ the^ range of^ your rough estimate from^ step 3.
Steps for^ figuring 2 or^ Raw^ scores^ from^ percentages 1.) (^) Draw (^) a (^) picture of (^) the normal (^) curve (^) , and shade in (^) the (^) approximate area for your percentage (^) using 50%-34%-14-1. percentages. 2.) (^) Make (^) a rough estimate^ of^ the^2 score^ where^ the^ shaded^ area^ stops^. 3.) Find the exact 2 score (^) using the normal (^) curve table (subtracting 50% from (^) your percentage if^ necessary before looking up^ the^
2 score ) .
4.) (^) check that your exact^2 score^ is^ within^ the^ range of^ your rough^ estimate^ from^ step^2. 5.) (^) If (^) you want to find (^) a raw score (^) , change it^ from^ the^2 score^.
Chapter 4 :^ compare 1 individual^ score^ against the^ population
chapter 5 :^ compare (^1) group of^ individuals^ against the^ population
Chapter 6 :^ making sense^ of^ significance (decision^ errors^ , effect^ size^ , statistical^ power)
Chapter 7 :^ compare sample group against hypothetical mean^ , compare 2 conditions^ within^ a^ sample (within^ - subjects)
Chapter 8 :^ compare 2 groups from^ the^ sample
Chapter 9 :^ compare 3 or^ more groups from^ the^ sample (between^ - subjects)
Chapter 5 :^ Hypothesis (^) testing with^ means^ of^ samples Hypothesis testing Process
1.) Restate the question as a research hypothesis and a null hypothesis
2.) (^) Determine the characteristics of the (^) comparison distribution
3.) Determine the cutoff sample score on the comparison distribution of the null hypothesis should be rejected
4.) (^) Determine (^) your sample's (^) score on the (^) comparison distribution 5.) (^) Decide (^) whether to (^) reject the null (^) hypothesis ° Hypothesis Testing with^ a^ distribution^ of^ means^ →^ procedure^ in^ which^ there^ is^ a^ single sample^ and^ the^ mean^ and^ population
variance is known .^ CZ^ tests)
° Hypothesis (^) testing involving means^ of^ groups of^ scores
- a score must be (^) compared with a distribution of (^) scores
- a mean must be compared with^ a^ distribution^ of^ means
° Distribution of means
- same mean with less (^) variance
° Have to
report the^ mean^ , the^ spread^ and^ shape.
- the mean : is the (^) same as the mean of (^) the population of^ individuals^.^ ÑM=^ μ)
- the (^) spread : standard (^) deviation of (^) a distribution of (^) means (ex (^). Standard Error of (^) the mean)
→ the variance is the variance of the population divided
by the^ number^ of^ individuals^ in^ each^ sample^.
shape :^ is^ it^ normal^ or^ not^?^ Gm^ =^ JEM)
→ (^) the distribution of the (^) population of (^) individuals (^) corresponds to a normal distribution or each (^) sample
includes 30 or more individuals
Population's^ Distribution^ Particular^ Sample's^ Distribution^ Distribution^ of^ means mean →^ μ=%- Mean^ → m=G# Mean →^ μm=μ variance →^ o2=d§ Variance^ →^ ofn =o÷
standard → • =^ Toa SD
- =¢(xμ-M)Z]- standard (^) → deviation m=fÑ deviation SD =^ Fiz
° Statistical Power → probability that the (^) study will (^) produce a statistically significant result if the research
hypothesis is^ true
why its^ important?
1.) (^) Figuring power when
planning a^ study^ helps^ you decide^ how^ many^ participants^ you need
2.) Understanding power^ is^ extremely^ important^ for making sense^ of^ results^ that^ are^ not^ significant^ or^ results^ that^ are statistically but^ not practically significant. Steps for (^) figuring power 1.) (^) Gather the needed (^) information : (^) ten and (^) 0M of (^) Population 2 Comparison distribution) and (^) the (^) predicted mean (M (^) predicted)
of Population 1.
2.) (^) Figure the raw - score cutoff (^) point of the (^) comparison distribution to (^) reject the null (^) hypothesis. 3.) (^) figure the (^2) score example.^ Whether^ being told^ a^ person has^ positive^ personality qualities^ increases^ ratings of^ the^ physical
attractiveness of^ that person
Population I^ →^ N=^64 ,^ M^ predicted =^208
Students who are told that the (^) person has^ positive (^) personality qualities Population 2 →^ μ= (^200) , ⑧^ =^48
Students in^ general
step 1 Pop.^1 μm^ =^ 4= N=64 02m^ =NÉ= (^) 644¥ = -26%4= M (^) predicted = (^208) 0m =^ =^ 356=
step 2
For the 5% (^) significance level (^) , positive one - (^) tailed (^) hypothesis, the Z (^) score cutoff (^) is +1. μm =^200 On =^6 Raw (^) score : ✗ = (^) (2)Com)-1μm = (^) (1.64×6)+200 Raw (^) score cutoff (^) point on the distribution of (^) means = (^) 209.
= (^) 209. 5% 188 194 200 206 f 212 218
p
step3""hy step 4
M predicted =^208
On =^6 2 score =^ • 31 took^ at^ table) cutoff raw (^) score t (pop^ 2)^ =^ 209.
Cutoff z (^) score =^ = (209-2,4--208)
t.gg#-- •^31 :É 63%
Confidence Intervals
° NHST is the main focus in the course. But there is another kind of statisical
question related^ to^ the^ distribution^ of^ means^ that
is (^) also (^) important in Psychology : (^) estimating the (^) population mean based (^) on the scores in (^) a (^) sample. ° Confidence interval (G) → (^) the
range of^ scores^ that^ is^ likely to^ include^ the^ true^ population^ mean^.
° Confidence limit →
Upper or^ lower^ value^ of^ a^ confidence^ level.
Steps for^ figuring confidence^ limits^ : 1.) (^) Figure the standard (^) error of the (^) mean (om) oñ=% (^) om=E 2.) (^) Figure raw scores G) for 1.96 (^) standard errors (95% confidence interval) (^) or 2.58 standard (^) errors (99% (^) confidence
interval) above and below the sample mean
- Upper confidence limit : (^) ✗ = M + (^) (2)(om)
- Lower (^) confidence limit : (^) ✗ = (^) M - (2)(om)
99% l
l l
l l
l I
A. A
1 1 I
2 -^ I 0 1 2
v V^ v^13
- (^) 1.96 +1.96 (^) V +2.
- (^) 2.
ex . Find the 95% confidence interval for those students who are told that the person has positive personality qualities
- (^) Population I →^ A- 64 M= 220
* Population 2-7 4=
1.) (^) Figure standard error : ozμ= = 46¥ =^ 236¥^ = (^36) on = (^) #= (^) 536=
Upper confidence^ limit^ :^ ✗^ =^ M^ +^ (2)(om)^ =^ 220+11.76=231.
Lower confidence limit^ :^ ✗ =^ M -^ (2)(om)^ =^220 -^ 11.76=208.
* Based on the
sample of^64 students^ , you can^ be^ 95%^ confident^ that^ an^ interval^ from^ 208.24^ to^ 231.76^ includes^ the^ true
population.
° If the confidence interval does not include the mean of the null
hypothesis distribution^ , then^ the^ result^ is^ statistically (^) significant t (^) t ✗ =^ 208.24^ ✗^ =^ 231. ° Should (^) we use confidence intervals or null^ hypothesis (^) significance (^) testing?
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