Midterm 2 (Chpt 6-8) Psychology 2220: Data and Statistical Analysis

Z score vs Z Stat

Z score = single score
Z Stat = sample mean

Characteristics of The Normal Curve

-unimodal
-bell-shaped
-symmetric
-defined mathematically

Normal Distribution

A distribution of values having a specific shape that is symmetric, unimodal, and bell-shaped.

The Standard Normal Distribution

a normal distribution with a mean of 0, and standard deviation of 1
A specific version of the normal distribution that is defined to have a mean of 0 and standard deviation of 1

Standard Normal Distribution Assumptions

median & mode are 0
variance is 1
1/2 the values will be negative
will be symmetric

Z Score

(X-µ)
z=––––––
σ
standardized raw scores

Conversion: Raw Scores to Z Scores

subtract the mean from each data point
then divide the differences by the deviation

Z Scores and Percentiles

z = 0: the mean of a distribution
z= ±1: one standard deviation above/below the mean
approximately 68% of values fall between these z
scores
z=±1.96: approximately 95% of values fall between these z scores

The Central Limit Theorem

How a distribution of sample means is a more normal distribution than a distribution of scores, (when the population distribution of scores is not normally distributed)

Distribution of Means

A distribution composed of means that are calculated from all possible samples of a given size, taken from the same population

Properties of a Distribution of Means

1) the distribution of means has the same mean as the sample population distribution of scores
2) the distribution of means has a less variability than the distribution of scores. this also tends to reduce the range of observed values in the distribution of means
3) a sufficient sample size, the distribution of means becomes more normally distributed (assuming that the parent population is not already perfectly normally distributed)

Standard Error

σ
σm=––––––
√N
the standard deviation of a distribution of a distribution of means for a specific sample size N

Computing a Z Statistic

(M-µm) σ
z=––––––– σm=––––––
σm √N
finds a single mean in a distribution of means

The Z Table

-Percentage of normally distributed scores between the mean (50% mark) and a given z score
-Percentage of normally distributed scores beyond a given z score, in the tail of the distribution

Assumptions

– Assumptions represent characteristics that we want the population that we are sampling from to have
– Meeting the assumptions helps us to make accurate inferences
– Always check the assumptions before running your test (assuming that you have a parametric test.)

Parametric Test

Inferential statistical test based on assumptions about a population

Nonparametric Test

Inferential statistical test NOT based on assumptions about the population

Assumptions of the Z Test

– Dependent variable is measured as a scale variable
– Participants are randomly sampled from the population
– The distribution of scores in the population of interest is
normally distributed
– The independent variable is nominal

Alpha/p Level

our "acceptable level of risk for a Type I error in a study and its analyses, typically set to 0.05 or 5%

Critical Value

A test statistic value beyond which we reject the null hypothesis (also called a cutoff)

Critical Region

The area in the tail(s) of the comparison distribution in which the null hypothesis can be rejected

One-Tailed Test

– directional test
– critical region in one tail of the distribution
– α = .05, we put all 0.05 in the one tail (which could be
the upper OR lower tail, based on the hypothesis)

Two-Tailed Test

– Nondirectional test
– critical region divided between the two tails of the
distribution
– more conservative. Reduces power for a particular tail
(relative to a one tail test)

Calculating z score hypothesis testing

(X-µ)
z=––––––
σ
Get a single observation for X from a normal distribution with a known σ and µ
Compute using the z score formula
Look at the critical z value(s) for the desired α
determine wether the score is beyond the critical values or not

The Steps of Hypothesis Tetsing

1) Identify the populations, distribution, and assumptions,
and then choose the appropriate hypothesis test
2) state the null and research hypothesis, in both words
and symbolic notation
3) Determine the characteristics of the comparison
distribution
4) Determine the critical values, or cutoffs, that indicate
the points beyond which we will reject the null
hypothesis
5) Calculate the test statistic
6) Decide whether to reject or fail to reject the null
hypothesis

Point Estimates

A single-number summary statistic from a sample that is used as an estimate of the population parameter
–One sample mean (M) can serve as a point estimate of
the population mean (µ)

σm (sigma m)

the standard error of the sampling distribution of the mean (for a particular sample size)
used in the calculation of an interval estimate

Interval Estimate

A range of plausible values for population parameter

Confidence Interval (CI)

An interval estimate (based on a sample statistic) that includes the population mean a certain percentage of the time if we sample from the same population (with the same ample) *repeatedly*
They give a plausible range of values for a population parameter
Provide the same information as a hypothesis test, but they also provide additional information

95% Confidence Interval

A range of values centered on a sample mean, and constructed so that 95% of independently sampled means would generate corresponding ranges containing the true mean of the population (i.e., the population from which they were randomly sampled)

How to Find the CI with a specific confidence level

Lower Endpoint: Mlower = Msample - Zcritical (α/2) x σm
Upper Endpoint: Mupper = Msample + Zcritical (α/2) x σm

Steps for Creating CI for z Distributions

1) draw a picture that will include the confidence interval,
centered on your sample mean
2) Indicate the bounds of your confidence interval on your
drawing
3) Determine the z statistic that fall at each boundary. We
will always use critical values that correspond to a two-
tailed test when creating CIs
4) Turn the z statistics back into raw means

Effect Size

The size of a difference (or strength of an effect) that is unaffected by sample size

Z Score

a statement about how far some raw score is from its population's mean, expressed in standard deviations

Cohen's d

(M - µ)
d=––––––
σ
describes how far a sample mean is from another mean (in standard deviation units)

Conventions of Effect sizes for d

small 0.2 85%
medium 0.5 67%
large 0.8 53%

Reasons for a Non-Significant Result

1) There isn't an effect (e.g., there is no difference
between means)
2) There is an effect and you didn't find evidence of it.
(there could be several reasons why this occurred)

Statistical Power

a measure of the likelihood that we will reject the null hypothesis, given that the null hypothesis is false

Ways to Increase Power

1) Increase alpha (also increases risk of a type I error)
2) Use a one-tailed test (instead of a two-tailed test)
–only increases power if you correctly predict the
direction of the difference
3) Increase the mean difference between populations
(with a more extreme manipulation of the independent
variable)
4) Increase Sample Size (N)
5) Decrease Variability (e.g., the standard deviation)
– This might be accomplished by using more reliable
measures or sampling from a more homogenous
population

Steps to Calculate Power (for a one-tailed test)

1) determine the information needed to calculate
statistical power – the hypothesized ( or observed)
mean for the sample, sample size, population mean,
population standard deviation and the standard error
(based on your sample size)
2) Determine a critical value in terms of the z distribution
and the raw mean
3) Calculate statistical power – the percentage of the
distribution of means for your hypothesized mean that
fall beyond the critical value

Meta-Analysis

A study or method that involves the calculation of a mean effect size from the individual effect sizes of many studies

Steps of a Meta-Analysis

1) choose a topic of interest and specify criteria for
including relevant studies in analysis (before beginning
track down potentially relevant studies)
2) Locate all existing studies (published and unpublished)
that meet your criteria
3) Compute a relevant effect size (e.g., cohen's d) for each
study
4) Calculate statistics about the set of effect sizes

Converting z scores

Z x SD -> scores -> sample mean
Z x SE -> population -> population mean

Hypothesis testing (One Tail)

5% of scores fall beyond the z in the tail of a distribution
Null – Ho: µ1 = µ2 OR H1: µ1 ≠ µ2
Alt. – Ho: µ1 ≥ µ2 OR H1: µ1 < µ2

Hypothesis testing (Two Tail)

2.5% of scores fall beyond the z in either tail of the distribution (5% total)
Null – Ho: µ1 = µ2 OR H1: µ1 ≠ µ2
Alt. – Ho: µ1 ≤ µ2 OR H1: µ1 > µ2