Chapter 6 Flashcards
As sample size __, shape of distribution become more like __ curve.
increases,
normal
Larger samples show more clear n__ c__ and d__.
Normal Curves,
Distributions
Examples of normally distributed variables:
- h__
- w__
- i_
- r__ t__
N__( c__) variables CAN’T be normally distributed.
height weight IQ reaction time nominal (categorical)
Normal Curve:
Specific b__-shaped curve that is u__, s__, and defined m__.
-Describes the distributions of m__ variables.
-As the size of a s__ approaches the size of the p__, the distribution resembles a normal curve (as long as the population is n__ d__).
bell-shaped, unimodal, symmetric, mathematically.
many
sample, population
normally distributed
Standardization: allows for fair c__ when variables are on different s__.
- comparing z-scores: number of s__ d__ a score is from the m__.
- z distribution: n__ distribution of s__ scores.
comparisons, scales
standard deviations, mean
normal, standardized
A z score is the number of s__ d__ a particular score is from the m__.
A z score is part of its own distribution, the _ distribution, just as a raw score, such as a person’s height, is part of its own distribution, a distribution of heights.
standard deviations, mean
z
Most DV’s are assumed to be __ distributed.
normally
If a variable is normally distributed, we can know stuff about l__ of o__.
likelihood, occurence
Most stats are based on a__ of n__.
assumption, normality
In the Z distribution, the mean is _ and the standard deviation is _.
0, 1
Linear Transformations:
-when you add a constant to each score (e.g: x+4)
what happens to the mean?
what happens to the SD?
-when you subtract a constant from each score (e.g: x-6)
what happens to the mean?
what happens to the SD?
-when you divide each score by a constant (e.g. 2)
what happens to the mean?
what happens to the SD?
increases by 4
stays the same
decreases by 6
stays the same
decreases by 1/2
gets smaller by 1/2
What is μ
Population Mean (i.e. 0 for z distribution)
What is σ
Standard deviation (i.e. 1 for z distribution)
To convert RAW scores into their Z scores steps:
- subtract the mean of the __ from the r__ score.
- divide by the s__ d__ of the population.
Practice writing out the equation
population, raw
standard deviation
z= x-μ/σ
ex converting raw score into z score:
mean: 6.07
SD: 1.62
raw score: 5
Find the z score.
z= 5-6.07/1.62 = -0.66
To convert Z scores into their RAW scores steps:
- multiply the _ score by the _ _ of the population.
- add the __ of the population
Practice writing out the equation
z, SD
mean
x=z∗σ+μ
ex converting z score into raw score:
mean: 6.07
SD: 1.62
z score: 1.81
x=1.81∗1.62+6.07 = 9.00
If we can standardize raw scores on 2 different scales, by converting both scores to z scores, we can then __ the scores directly.
compare
Ex of standardizing two raw scores on different scales:
-Lab A: measured reaction time after 1 drink of alcohol, the population mean was 2.1 seconds, with an SD of 0.3. Larry’s reaction time was 1.4 seconds.
-Lab B: measured reaction times after 2 drinks of alcohol. The population mean was 3.1 seconds, with an SD of 0.6.
Bill’s reaction time was 1.5 seconds.
When using a standardized scale, who has the larger (slower) reaction time?
Larry:
1.4-2.1/0.3 = -2.33
Bill:
1.5-3.1/0.6= -2.67
Larry has the larger Z score and slower time.
-Larry’s score is less negative so LARGER TIME. (larger is longer with time)
Transforming Z scores into PERCENTILES:
- z scores tell you __ a values fits into a __ distribution.
- based on a normal distribution, there are rules about where scores with a _ value will fall, and how it will relate to a p__ rank.
- use a__ under normal curve to calculate percentiles for any score.
where, normal
z, percentile
area
Break down of percentiles associated with each standard deviation:
DRAW OUT
between 0 and 1/-1—> 34% each side
between 0 and 2/-2—> 14% each side
between 0 and 3/-3—> 2% each side
Practice percentile problems:
1) -What percentage falls within 1 standard deviation of the mean?
2) -What percentage falls within 2 standard deviations of the mean?
3) -What percentage is less than the mean?
4) -If you have a raw score equal to z= -1, what is your percentile score?
5) -What is your percentile if z= +1
6) -What is your percentile if z=1.5
7) -What is your percentile if z=0.25
8) -What is your percentile if z= -2.5
1) 34% + 34% = 68%
2) 14+14+34+34=96 OR 100-4=96
3) 50%
4) 16th percentile because you add the percent from -1 and below.
5) 84th percentile
6) 91st percentile
7) 58.5th percentile
8) 1st percentile
Practice percentile problems continued:
1) If the pop. has a mean of 125 and std deviation of 10, if you are at the 16th percentile, what would your score be?
2) If your score is 135, what is your percentile?
1) x = -1(10) + 125 = 115
so z = -1
(or just look at chart and where 16% is)
2) z = (135-125)/10 = 1, 84th percentile
Check your learning problems:
Check Your Learning
•If the population mean score on a quiz is 10 and the standard deviation is 2:
1) •If a student’s score is 8, what is z?
2) •If a student’s z score is 1.4, what is the raw score?
1) z= -1
2) x=1.4 * 2 +10
=12.8
The Central Limit Theorem:
-Distribution of s__ means is n__ distributed even when the p__ from which it was drawn is not n__.
- A distribution of m__ is less variable then a distribution of i__ scores.
- Variability c__.
Sample, normally
population, normal
means, individual
changes
The mean of the d__ tends to be the mean of the p__.
mean=μ
distribution, population
Standard deviation of the d__ tends to be __ than the standard deviation of the p__ of scores.
distribution, less
population
What is :
M
N
M=sample mean
N=number of elements in the sample
Standard error formula:
SD * Population Mean= SD/square root of number of elements in sample
The standard error is the standard deviation of the p__ divided by the square root of the sample s__, __. The formula is.
Write out in notation.
population
size, N
σ(M)=σ/√ N
Standard Error: s__ d__ of the distribution of m__.
-S__ than SD; takes on smaller value as _ increases.
standard deviation, means.
smaller, N (number of elements in sample)
Write out the Z statistic
z=(M-μ*M)/σM
aka z=(sample mean-population mean)/
standard deviation * sample mean
The mathematical magic of large samples:
-Distribution of m__ follows c__ l__ theorem.
The shape of distribution approximates n__ curve IF:
- p__ of scores has n__ shape OR
- size of each s__ of distribution is n=__+
means, central limit
normal
population, normal
sample, 30+
Example calculating the z statistic using population data:
A pizza delivery chain claims that it delivers its pizzas to any location within a 15 mile radius within an average of 30 minutes, with a standard deviation of 12 minutes. Suppose a researcher employed by a competitor orders 16 pizzas to be delivered to different locations with a 15 mile radius, and computes a sample mean delivery time of 38 minutes. What is this mean delivery time expressed as a z statistic? Interpret the z statistic.
What if sample size was 36?
Standard error: σ/√ N
so 12/√16=3 —> plug into z statistic
z statistic: z=(M-μ*M)/σM
so
z=38-30/3
standard error=12/4= 3
z statistic=38-30/3= 2.67
Thus, the mean delivery time is 2.67 standard errors from the mean.
There would be a smaller SD, and the Z statistic would be bigger.
