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

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Z score vs Z Stat

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Z score vs Z Stat

Z score = single score Z Stat = sample mean

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Characteristics of The Normal Curve

-unimodal -bell-shaped -symmetric -defined mathematically

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Normal Distribution

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

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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

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Standard Normal Distribution Assumptions

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

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Z Score

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

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Conversion: Raw Scores to Z Scores

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

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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

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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)

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Distribution of Means

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

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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)

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Standard Error

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

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Computing a Z Statistic

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

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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

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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.)

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Parametric Test

Inferential statistical test based on assumptions about a population

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Nonparametric Test

Inferential statistical test NOT based on assumptions about the population

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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

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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%

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Critical Value

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

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Critical Region

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

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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)

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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)

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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

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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

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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 ofthe population mean (µ)

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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

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Interval Estimate

A range of plausible values for population parameter

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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

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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)

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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

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Steps for Creating CI for z Distributions

1. draw a picture that will include the confidence interval, centered on your sample mean

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

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Effect Size

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

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Z Score

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

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Cohen's d

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

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Conventions of Effect sizes for d

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

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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)

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Statistical Power

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

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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

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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

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Meta-Analysis

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

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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

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Converting z scores

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

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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

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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

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