Shavelson Descriptive Statistics

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Shavelson Descriptive Statistics

• Variability
• Range
• Variance
• SD

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Shavelson Chapter 5

• S5-1. Define, be able to create and recognize
  graphic representations of a normal distribution
  (115-121).
• Normal distribution Provides a good model of
  relative frequency distribution found in
  behavioral research.

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Shavelson Chapter 5

• S5-2. Know the four properties of the normal
  distribution (120-121).
• Unimodal, thus the greater the distance a score
  lies from the mean, the less the frequency of at
  score.
• Symmetrical
• Mean, mode, and median all the same
• Aymptotic line never touches the abscissa
• Note that the mean and variance can differ, thus
  a family of normal distributions

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Shavelson Chapter 5

• S5-3. You should know what is meant by the phrase
  a family of normal distributions (121,3). I
  will also cover in class the general issues of
  distributions which are frequently used in
  statistical analyses.

Fromhttp//www.gifted.uconn.edu/siegle/research/N
ormal/instructornotes.html
5

• These distributions have the same mode, different
  median and SD
• These distributions have different mode, same
  median, different SD
• These distributions have different means, modes
  and variances
• These distributions have the same mode, mean and
  median, but different SDs

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Enter question text...

• These distributions have the same mean, but
  different SDs
• These distributions have a different means and
  medians, but the same modes and SDs
• These distributions have different means, modes,
  and
• Nothing is the same with these two!

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Shavelson Chapter 5

• S5-4. Know the areas under the curve of a normal
  distribution (roughly, e.g. 34.13, 13.59, 2.14
  and .13 on either side of the mean)

Fromhttp//www.gifted.uconn.edu/siegle/research/N
ormal/instructornotes.html
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Shavelson Chapter 5
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Shavelson Chapter 5

• S5-5a. What is a standard score (z-score)
  (123,3)? Be able to calculate the z-score, given
  a raw score, mean, and standard deviation.
• Z score X-mean
• S
• X raw score
• Mean mean of distribution
• S standard deviation
• Notice that to calculate the Z score you need the
  mean and S of a distribution of scores.

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Shavelson Chapter 5

• S5-5b. What two bits of information does the
  z-score provide us (125, 1-2)?
• Z scores provides the following information
• Size of Z scores indicates the number of standard
  deviations raw score is from the mean
• Sign ( or -) indicates if the raw score is above
  the mean () or below the mean (-)

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A z score of -1.8 means

• The mean of the distribution is 1.8
• The distribution is skewed
• The raw score lies 1.8 means above the mean
• The raw score lies 1.8 standard deviations below
  the mean
• The raw score 1.8 lies standard deviations above
  the mean

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Mean 10, X 18, S 4, what is the Z score?

• -2
• 4
• -4
• 2
• Not listed

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Shavelson Chapter 5

• S5-6. Know what a Standard(ized) distribution is.
  
• Convert all raw scores of a distribution into Z
  scores, and put into a frequency distribution.
• Mean 0
• Std. Dev. And Variance 1

-2 -1 0 1 2
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Shavelson Chapter 5

• S5-8. Know how to calculate the proportion of
  scores that lie above or below a given raw score
• Convert raw score to a Z score
• Do rough estimate on a standard normal
  distribution
• Look up in table B (swap labels on 34 if it is a
  neg Z value).
• Mean 80
• S 5
• X 69
• Let's do a few more!

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Mean 18, X 5, S 7.1. What percentile was
the person who scored the x in?

• 1.83
• 96.64
• 3.44
• 46.64

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Shavelson Chapter 8

• S8-1. Know the definition of a statistic,
  parameter, and estimator
• Statistic describes characteristic of sample
  e.g. sample mean x-bar as opposed to population
  mean mu (µ)
• Parameter describes characteristic of population
• Estimator statistic that estimates a population
  parameter

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The mean is an example of..

• Parameter
• Statistic
• Estimator
• All of the above

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Shavelson Chapter 8

• S8-2. Know the role of statistics, as well as the
  difference between inferential and descriptive
  statistics.
• Role of Stats
• Guidelines for summarizing/describing data
• Method for drawing inferences from sample to
  population
• Help set effective methodology
• Descriptive Stats
• Organize/summarize/depict/describe collections of
  data
• Inferential Stats
• Draw inferences about population from sample

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Shavelson Chapter 8

• S8-3. Know and be able to recognize and provide
  examples of the two types of questions asked
  about a population (Case 1 and Case II research).
  (217)
• Case I Research
• Was a particular sample of observations drawn
  from a particular (known) population?
• Example all students in US took GRE on same day,
  means of all scoreslook at one state in
  particularmean is higherthey are from a
  different population with a higher mean.
• Take one sample from the population (get mean)
  and compare to the overall population mean.
• Actually answer what is the probability that a
  sample was drawn from a particular (known)
  population.

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Shavelson Chapter 8

• S8-5. Know the general approach for conducting
  case I and case II hypothesis testing. That is,
  you should be able to list and briefly describe
  the steps your author lists at the end of each
  section (case I, 220-221, 4 steps case II
  223-224, 5 steps). Be able to describe the
  various alternative hypotheses (step two of each)
• Case II research
• Are the observations from two different samples
  drawn from the same population?
• (do observations on two groups of subjects differ
  from one another)
• Actually answer given that a difference exists
  between two samples (e.g. the means) what is the
  probability that this difference is caused by
  chance alone? If not from chance alone, they must
  be from different populations e.g. our treatment
  changed them!

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Shavelson Chapter 8

• S8-5
• Case 1 research steps
• 1. Set your hypotheses
• Ho µ specific value
• H1 µ ? some specific value usually pop mean
  (two tailed)
• H1 µ gt some specific value (one tailed)
• H1 µ lt some specific value (one tailed)
• 2. Randomly select participants for your study
• 3. decide to reject the null or not based on the
  comparison of the sample mean to the population
  mean
• Reject null means that the difference between the
  population mean and the sample mean is not likely
  to have occurred by chance (it was probably due
  to whatever you were studying!)
• Failure to reject the null means there is a
  fairly good chance that the difference between
  the sample mean and the population mean could
  have occurred simply by chance (not due to
  whatever you were studying)

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Shavelson Chapter 8

• Case II
• 1. Set your hypotheses
• Ho µe µc
• µe Experimental group
• µc Control Group
• H1 µe ? µc (two tailed)
• H1 µe gt µc (one tailed)
• H1 µe lt µc value (one tailed)
• 2. Randomly Select then Randomly Assign
  participants to experimental and control groups.
• 3. Perform the experiment apply the IV and
  measure the DV
• 4. Decide to reject the Null Hypothesis or not
• Reject the null means that the difference between
  the experimental and control group is not likely
  to have occurred by chance (thus was probably
  your IV!)
• Failure to reject the null means that it is
  likely that the difference between the control
  group and the experimental group was due to
  chance and not your IV.

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Shavelson Chapter 8

• S8-6. Know the two types of statistical errors
  Type 1 and type 2. Be able to prove and recognize
  examples of each.
• Types of errors in statistical inference
• Type I Reject the null when it is true (say
  there is a treatment effect when there is not)
• Type II Not reject null when it should have been
  (say there is no treatment effect when there was)

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Shavelson Chapter 8

• S8-6. Know the two types of statistical errors
  Type 1 and type 2. Be able to prove and recognize
  examples of each.

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Shavelson Chapter 9

• Probability
• Event any specified outcome
• Outcome space all possible outcomes
• P(e) the probability of some event
• P(e) events/ outcomes in outcome space
• Ex. Dice
• Outcome space 1,2,3,4,5,6 ( six items)
• E 2 (1 item)
• Probability of getting a 2 1/6.17

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Probability what is the probability of getting a
two by chance alone?
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Shavelson Chapter 10

• 10-1. Two fundamental ideas of conducting case I
  research
• The null hypothesis is assumed to be true.
• (that is, the difference between the sample and
  population mean is assumed to be due to chance
  alone)
• A sampling distribution is used to determine the
  probability of obtaining a particular sample
  mean.
• In this case the sampling distribution is
  composed of group means

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Shavelson Chapter 10

• 10-2. What is the central limit theorem?
• The Central Limit Theorem is a statement about
  the characteristics of the sampling distribution
  of means of random samples from a given
  population. That is, it describes the
  characteristics of the distribution of values we
  would obtain if we were able to draw an infinite
  number of random samples of a given size from a
  given population and we calculated the mean of
  each sample.
• The Central Limit Theorem consists of three
  statements
• 1 The mean of the sampling distribution of
  means is equal to the mean of the population from
  which the samples were drawn.
• 2 The variance of the sampling distribution of
  means is equal to the variance of the population
  from which the samples were drawn divided by sqrt
  of the size of the samples.
• 3 If the original population is distributed
  normally (i.e. it is bell shaped), the sampling
  distribution of means will also be normal. If the
  original population is not normally distributed,
  the sampling distribution of means will
  increasingly approximate a normal distribution as
  sample size increases. (i.e. when increasingly
  large samples are drawn)

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Shavelson Chapter 10

• 10-3. Know the characteristics of a sampling
  distribution of means.
• Characteristics of Sampling distribution of means
• 1. normally distributed (even if pop. is skewed -
  if N 30 or more)
• 2. sampling mean population mean
• 3. standard dev (standard error of the mean)
  Pop S.D.
• N

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Shavelson Chapter 10

• 10-4. Know what happens to the SEM as sample size
  increases.
• SEM decreases as N increases

SEM Pop S.D. N
s x s N
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Shavelson Chapter 10

• 10-5. Know how one could create a sampling
  distribution of means
• Sampling Distribution of means
• A distribution composed of sample means
• How to conduct
• 1. Pull a sample from population of N size
• 2. Find the mean of the sample
• 3. Repeat this many times (all samples of size N)
• 4. Create a frequency distribution of the means
• (actual convert if to relative frequencies
  proportions!)

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Shavelson Chapter 10

• 10-5. What is the functions of a sampling
  distribution of means?
• Used as a probability distribution to determine
  the likelihood of obtaining a particular sample
  mean, given that the null hypothesis is true.
• null hypothesis is true same thing as by
  chance alone

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Shavelson Chapter 10

• S10-6. As your author does, be able to calculate
  the probability of obtaining a particular sample
  mean, given the appropriate data (e.g. the mean
  of the sampling distribution and the standard
  error). If I ask for this on the test I will
  either supply table B or will have the Zx fall on
  a whole value (e.g. 1 or 2, or 3). You should
  thus review the probabilities under the normal
  curve as you will be expected to be able to apply
  this information) (260-262)
• µ 100 (mean of the population and the sampling
  distribution)
• s x 25
• X (mean of the sample we used in our study)
• What is the probability of obtaining a sample
  mean of 175 by chance alone (i.e. when the null
  is true Ho µ x)
• Zx mean of the sample pop mean X - µ
  175-100
• SEM
  s x 25
• Use table b if needed!

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Shavelson Chapter 10

• S10-7. What meant by the terms "unlikely" and
  "likely"? You should be able to answer this in
  terms of accepting or rejecting the null
  hypothesis, or in terms of what is meant by
  "significance level" (263-264)
• Level of significance what we consider to be
  unlikely
• Generally set at 5 or 1 chance of obtaining a
  sample mean by chance alone
• Alpha .05 or alpha .01
• Thus decisions to reject the null are based on
  your alpha level
• Reject null if your sample mean is equal too, or
  less than your alpha level.

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You get all the scores of the folks in CA who
took the GRE and find that their average score is
675 (for verbal). The overall (entire population)
mean is 500 and the SEM is 100. Is the California
mean statistically significant (the diff from the
pop mean). Alpha .05

• Yes
• No
• Huh?

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Shavelson Chapter 10

• S10-7
• Decisions to reject the null are based on your
  alpha level
• Reject the null hypothesis if the probability of
  obtaining a sample mean is less than or equal to
  .05 (.01) otherwise, dont reject the null
  hypothesis

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Shavelson Chapter 10

• 10-8 Calculating Zx (critical)
• (The Zx score at which we say it is unlikely to
  obtain this value by chance alone)
• at the alpha .05 level of significance Zx
  (critical) 1.65 (from table B)
• at the a .01 level of significance (critical)
  2.33 (from table B )
• Example
• µ 42
• sx 8
• X 30
• Reject the Ho or not at the .05 level of
  significance?
• translate alpha level into z-score

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Shavelson Chapter 10

• 10-8 Calculating Zx (critical)
• Two ways to reject the null Find the probability
  of obtaining the Z score (obtained), or find the
  Z scored that lies at the alpha level (critical).
  Then
• Either compare the probability of getting the
  Zobtained (e.g. .03) to the alpha level (e.g.
  .05). In this case you would say reject the null
  - we show statistical significance
• Or, compare the Zobtained to the Zcritical in
  this case, 1.88 (obtained) and 1.65(critical). In
  this case since the Zobtained is greater than
  Zcritical we reject the null - we show
  statistical significance

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Shavelson Chapter 10

• 10-9. Know the difference between directional and
  non directional tests, and when to use each!
• 1. A one tail may be supported by previous
  research or theory
• 2. When in doubt, choose two tailed!
• Tails are specified by alternative hypotheses.
• Ho xbarmu
• H1 xbar ?? µ (2 tailed both)
• Or
• H1 xbar lt µ (1 tailed left)
• Or
• H1 xbar gt µ (1 tailed right)
• Easier to show statistical significance with
  1-tailed test.
• Directional vs. non-directional tests
• Directional uses only one tail of the sampling
  distribution
• Non-directional uses both tails
• Thus If alpha .05 and one tail all .05 (1.65)
  is in one tail (or -1.65)

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Shavelson, Chapter 10

• If conducting case II research, how could you
  determine the probability of getting a particular
  difference between 2 means
• (which is what we are looking at for case II).

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Shavelson Chapter 10

• Sampling distribution of differences between
  means gives the probability of obtaining a
  particular difference between means. (Case II)
• Theoretically you could.
• Make sampling distribution of differences between
  means,
• then find a z-score, compare to alpha level,
  accept or reject the null hypothesis.
• Case II
• xbar1-xbar2 2
• xbar1-xbar2 3
• xbarz-xbar2 -1
• graph frequency of each difference
• make freq distribution of differences between
  sample means
• Can also calculate SD, determine likelihood of
  obtaining difference between means by chance
  alone

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Shavelson Chapter 10

• Characteristics of the sampling distribution of
  differences between means
• 1. normally distributed
• 2. Mean0
• 3. Standard Deviation (called the standard error
  of the difference between means)
• Is equal to
• sx1-x2 sx12 sx22
• Note variance sigma squared

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Shavelson Chapter 10

• Calculate a Z score for diff between means
• Z x1-x2 Xe Xc
• 
  sx1-x2
• Example
• Xe 24
• Xc 30
• sx1-x2 2.8
• H1 Xe ? Xc
• Z crit?
• Z obs?

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Shavelson Chapter 11

• S11-1. Know the definition and recognize/generate
  examples of the two types of errors (Type I and
  Type II)(also see table 11-1)This is similar to
  what we did last unit. How does one adjust the
  probability of making a type I error? (313).

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Shavelson Chapter 11

• S11-2. Know the definition of "power" and how it
  is calculated. (314)
• Power 1-Beta
• The probability of correctly rejecting a false
  null hypothesis. OR Power is the probability of
  you detecting a true treatment effect.
• (What researchers are really interested in!
  Detecting a true difference if it exists.)
• Power .27 (27)very low. Want higher power,
  want higher number.

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Shavelson Chapter 12

• S12-1. What is the purpose of a t test in general
  (334,3). Also how is a t test used for case I
  research? (that is, what question does it
  answer?(334,3). As in previous chapters the
  function of the t test is to determine the
  probability of observing a particular sample
  mean, given that the null hypothesis is true. You
  should know this point. You should also know how
  the standard deviation is estimated for the
  population when using the t distribution (334)
• T-test is used to
• A. Determine the probability that a sample was
  drawn from a hypothesized population (given a
  true Ho)
• B. Used when the population standard deviation is
  not known
• C. Calculated standard deviation (SEM) is
• How would one go about doing this?
• Standard Dev. Of Sample Sx
  s
• Sq. Root of sample size
  N

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Shavelson Chapter 12

• S12-2. You should be able to describe the t
  distribution and what it is used for (determining
  the probability of obtaining a particular sample
  mean)(335-336). Know the important differences
  between the t distribution and the normal
  distribution. (335,5,-335,7) (there are three
  points made).
• T(observed)
• X µ
• sx
• the number of standard deviations that a
  particular t lies from the mean)
• The t distribution is created from numerous same
  sized samples from the population just like a
  sampling distribution!
• The t(observed) can be compared to the t
  distribution to determine the probability of
  obtaining that particular sample mean (given the
  Ho is true)

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Shavelson Chapter 12

• T-distribution vs. Normal Distribution
• 1. T has a different distribution for every
  sample size (N)
• 2. More values lie in the tails of t thus
  critical values for t are higher than Z
• 3. As sample size increases t becomes closer
  closer to normal distribution.

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Shavelson Chapter 12