| Front |
Back |
| Sample |
Subgroup of the population |
| Sampling |
Process of selecting sample from population |
| Random sampling |
Independent selection |
| Descriptive vs. Inferential Statistics |
– Descriptive: primary purpose is to describe some aspect of the data
Inferential: primary purpose is to infer (to estimate or to make a decision, test a hypothesis) |
| All inferential statistics have the following in common: |
– use of some descriptive statistic
– use of probability
– potential for estimation
– sampling variability
– sampling distributions
– use of a theoretical distribution
– two hypotheses, two decisions, two types of error |
| Research defined |
Structured Problem Solving |
| Scientific methods: steps (cyclic) |
– 1. encounter and identify problem
– 2. formulate hypotheses, define variables
– 3. think through consequences of hypotheses
– 4. design & run study, collect data, compute statistics, test hypotheses
– 5. draw conclusions |
| Variable |
entity that is free to take on different values |
| ndependent variable (IV) |
its values are manipulated by the researcher, comes first in time |
| Dependent variable (DV) |
measured by researcher, follows the IV in time |
| Population |
Target group for inference |
| Extraneous variable (EV) |
controlled by researcher
• randomization of subjects to groups
• keep all subjects constant on EV
• include EV in the design of the experiment |
| Predictor variable (PV) |
comes first in time but there is no manipulation, analogous to IV. |
| Criterion variable (CV): |
follows PV in time, analogous to DV. |
| Causal relationship: |
IV causes the DV |
| Predictive relationship: |
PV predicts the CV |
| 2 Types of research |
1. experimental 2. observational |
| True experiment |
• manipulation of IV
• randomization of subjects to groups
• causal relationship between IV and DV |
| Observational research |
• no manipulation
• minimal control of EV
• predictive relationship between PV and CV
|
| Stem and Leaf Display |
• The first digit(s) of a score form the stem, the last digit(s) form the leaf.
• We want 10-20 total number of stems.
• Number of stems per digit depends on total number of stems: can do 1, 2, or 5 stems per digit.
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| Description With Statistics
Aspects or characteristics of data that we can describe are: |
– Middle
– Spread
– Skewness
– Kurtosis |
| Other words that describe Middle |
central tendency, location, center |
| Statistics that Measure middle are: |
mean, median, mode
• “Middle” is the aspect of data
we want to describe.
• We describe/measure the middle of data in a population with the parameter m (‘mu’); we usually don’t know m, so we estimate it with X-bar. |
| Other words that describe Spread |
variability, dispersion, skatter |
| Statistics that Measure spread are: |
range, variance, standard deviation, midrange
• “Spread” is the aspect of data we want to describe.
• Any statistic that describes/measures spread should have these characteristics: it should
– Equal zero when the spread is zero.
– Inc |
| Skewness |
=departure from symmetry
– Positive skewness = tail (extreme scores) in positive direction
– Negative skewness = tail (extreme scores) in negative direction
(The Few name the Skew) |
| Kurtosis |
peakedness relative to normal curve |
| Sample Mean |
-The sample mean is the sum of the scores divided by the number of scores, and is symbolized by X-bar, X = SX/N
-For example, for X1=4, X2=1, X3=7, N=3, SX=12 and X = SX/N = 12/3 = 4
• Characteristics:
– X-bar is the balance point |
| Sample Median |
• The median is the middle of the ordered scores, and is symbolized as X50.
• Median position (as distinct from the median itself) is (N+1)/2 and is used to find the median.
• Example: X1=4, X2=1, X3=7, then N=3.
• Characteristic |
| Sample Mode |
• The mode is the most frequent score.
• Examples:
– 1 1 4 7, the mode is 1.
– 1 1 4 7 7, there are two modes, 1 and 7.
– 1 4 7, there is no mode.
• Characteristics:
– Has problems: more than one, or none; maybe not in the mid |
| Spred cont. |
• We describe/measure the spread of data in a sample with the statistics:
– Range = high score-low score.
– Midrange, MR.
– Sample variance, s*².
– Sample standard deviation, s*.
– Unbiased variance estimate, s².
– s.
• We des |
| Midrange (MR) |
• Formula is MR=UH-LH
– UH=upper hinge
– LH=lower hinge
– Hinges cut off 25% of the data in each tail
• Hinge position is ([median position]+1)/2.
– [median position] is the whole number part of the median position (remember, median p |
| Hinge position |
([median position]+1)/2
– [median position] is the whole number part of the median position (remember, median pos.=(N+1)/2)
• Use hinge position to count in from the tails to find the hinges.
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|
Sample Standard Deviation, s*Sample Variance, s*² |
• Definitional formula: s*²=S(X-X)²/N, the average squared deviation from X-bar.
Sample Standard Deviation= s*
Unbiased Variance Estimate, s² |
| Box-plots |
• A pictorial description that uses a box to show the middle of the data and lines called whiskers to show the tails of a distribution.
|
| 3 Parts to Box Plot |
1.) Box
2.) Wiskers
3.) Outliers |
| Box |
– Upper end is at the UH, lower end is at the LH - Line across the middle is X50 |
| Whiskers |
– Whiskers are lines drawn from the ends of the box (the hinges) to adjacent values, UAV & LAV.
– Adjacent values are the first real data values inside the inner fences.
– Inner fences, upper and lower
• Upper, UIF=UH+1.5MR
• Lower, LIF= L |
| Outliers |
Outliers: outside whiskers, marked with |
| Midrange (MR) |
UH- LH |
| z Scores |
• The aspect of the data we want to describe/measure is relative position. • z scores are statistics that describe the relative position of something in its distribution. |
| Z score formula |
z is something minus its mean divided by its standard deviation. |
| z score characteristics |
– The mean of a distribution of z scores is zero.
– The variance of a distribution of z scores is one.
– The shape of a distribution of z scores is reflective, the shape is the same as the shape of the distribution of the Xs. |
| Characteristics of Normal Distributions |
– Symmetric, continuous, unimodal.
– Bell-shaped.
– Scores range from -¥ to +¥ .
– Mean, median, and mode are all the same value.
– Each distribution has two parameters, m and s². |
| Use of Z score |
• We use this distribution to get probabilities associated with a z score (probability, proportion, and area under the curve are synonymous).
- look up z in table to find probabilities. |
| Correlation |
– Defined as the degree of linear relationship between X and Y. – Is measured/described by the statistic r. |
| Regression |
– Is concerned with the prediction of Y from X Forms a prediction equation to predict Y from X
Uses the formula for a straight line, Y’=bX+a.
– Y’ is the predicted Y score on the criterion variable.
– b is the slope, b=DY/ D X=rise/run.
– |
| r= |
r=SzXzY/N, the average product of z scores for X and Y
– Works with two variables, X and Y
– -1
|
| r2= |
proportion of variability in Y that is explained by X. |
| Correlation: Undefined |
If there is no spread in X or Y, then r is undefined. Note that any z is undefined if the standard deviation is zero, and r=SzXzY/N. |
| Population correlation coefficient, |
r (rho) |
| regression cont. |
• Linear only.
• Generalize only for X values in
your sample.
• Actual observed Y is different from Y’ by an amount called error, e, that is, Y=Y’+e.
• Error in regression is e=Y-Y’.
• Many different potential regression |
| Line of Best Fit |
The statistics b and a are computed so as to minimize the sum of squared errors, – Se2=S(Y-Y’)2 is a minimum. – This is called the Least Squares Criterion. |
| Partition total spread |
– Total = Explained + Not Explained
– This is true for proportion of spread and amount of spread.
• Proportion: 1 = r2 + (1-r2)
• Amount: s2y = s2y r2 + s2y(1-r2) |
| Probability |
Defined as relative frequency of occurence. |
| Sample space |
all possible outcomes of an experiment |
| Elementary event |
a single member of the sample space |
| Event |
any collection of elementary events |
| p(elementary event |
1/(total number) |
| p(event) |
(number in the event)/(total number) |
| Conditional probability |
• p(A|B)=(number in [A and B])/(number in B)
• The probability of A in the redefined (reduced) sample space of B. |
| Big 3 Probability Rules |
1. independence 2. mulitplication, mutually exclusive 3.) addition |
| Independence (1) |
events A and B are independent if
• p(A|B)=p(A)
• The A probability is not changed by
reducing the sample space to B. |
| Multiplication (And) Rule (2) |
• p(A and B)=p(A)p(B|A)=p(A|B)p(B) |
| Mutually exclusive: |
• Events A and B do not have any elementary events in common.
• Events A and B cannot occur simultaneously.
• p(A and B)=0 |
| Addition (Or) Rule (3) |
p(A or B)=p(A)+p(B)-p(A and B) |
| The sampling distribution of X-bar |
– Has the purpose of any sampling distribution: to obtain probabilities…
– Has the definition of any sampling distribution: the distribution of a statistic.
– Has specific characteristics:
• Mean: mX = m
• Variance: s2X =s2/N
• Shape i |
| Hypothesis testing |
is the process of testing tentative guesses about relationships between variables in populations. These relationships between variables are evidenced in a statement , a hypothesis, about a population parameter. |
| Test statistic |
a statistic used only for the purpose of testing hypotheses; e.g. zX |
| Assumptions |
conditions placed on a test statistic necessary for its valid use in hypothesis testing;– for zX, the assumptions are that the population is normal in shape and that the observations are independent. |
| Null hypothesis |
the hypothesis that we test; Ho. |
| Alternative hypothesis |
where we put what we believe; H |
| Significance level |
he standard for what we mean by a “small” probability in hypothesis testing; a.
The significance level is the small probability used in hypothesis testing to determine an unusual event that leads you to reject Ho.
– The significance level is sym |
| Direcetional v. Non-Directional Hypothesis |
>,<, or =
• Directional hypotheses specify a particular direction for values of the parameter.
– IQ of deaf children example: Ho: m>100, H1: m<100.
• Non-directional hypotheses do not specify a particular direction for values of the paramet |
| One- and two-tailed tests |
– A one-tailed test is a statistical test that uses only one tail of the sampling distribution of the test statistic.
– A two-tailed test is a statistical test that uses two tails of the sampling distribution of the test statistic. |
| Critical values |
values of the test statistic that cut off a or a/2 in the tail(s) of the theoretical reference distribution. |
| Rejection values |
the values of the test statistic that lead to rejection of Ho |
| p-Value Decision Rules |
• Reject Ho if
– ½ the SAS p-value
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