STATS mid 2 class

Inferential Statistics

Used to make inferences or draw conclusions about a population based on a sample of data.

Null Hypothesis (H0)

The hypothesis that states there is no effect or difference between groups.

Research Hypothesis (H1)

The hypothesis that proposes a significant effect or difference exists.

Type 1 Error

Rejecting the null hypothesis when it is true; the probability of this error is denoted as α.

Type 2 Error

Failing to reject the null hypothesis when it is false; the probability of this error is denoted as β.

P-value

The probability of obtaining a test statistic as extreme as the one observed, under the assumption that the null hypothesis is true.

Confidence Interval (CI)

A range of values that is likely to contain the true population parameter, typically calculated at a 95% confidence level.

As N increases CI gets narrower (because the SEM (standard error of the mean) will be smaller)

as N decreases, CI gets wider (because the mean is easier to estimate)

Sampling Distribution of the Mean

The probability distribution of all possible sample means from a given population.

Standard Error of the Mean (SEM)

An estimate of how much the sample mean is likely to differ from the true population mean.

Key Points:

• Measures the variability of sample means around the population mean.

• Smaller = more precise estimate of the population mean.

• Calculated as - Standard Deviation divided by the square root of sample size).

Correlation

A statistical measure that expresses the extent to which two variables are linearly related.

Covariance

A measure of how much two random variables vary together.

Pearson's Correlation Coefficient (r)

A measure of the strength and direction of association between two continuous variables. - measures a linear relationship -

involves assumptions :

-the relationship between x and y is linear

-both x and y are continuous

-x and y are both normally distributed

Effect Size

A quantitative measure of the magnitude of a phenomenon or the strength of a relationship.

• Measures magnitude of an effect (how meaningful, not just significant).

• Cohen’s d: Expresses effect in SD units.

  • Example: 1.3 psychopathy points increase = 2 SD change.

• When to use Cohen’s d: Use when it makes sense to express in SD.

One-sample t-test

A statistical test used to determine if the mean of a single sample differs significantly from a known population mean.

Two-sample t-test

A statistical test used to compare the means of two independent groups.

Paired-samples t-test

A statistical test used to compare means from the same group at different times/conditions.

also known as a repeated measure t-test

we calculate the difference score (D) for each participant, representing the change between the 2 conditions being compared

Degrees of Freedom (df)

The number of independent values or quantities which can be assigned to a statistical distribution.

always n-1 because its always predicting one parameter

Regression Analysis

A statistical process for estimating the relationships among variables.

Residuals

The differences between observed values and the values predicted by a regression model.

finding distance of each point on Y from line

Y-Y(hat)

Standardized Regression Coefficient (β)

A measure that expresses the strength of the relationship between the dependent variable and independent variable(s) in standardized units.

tells us how many standard deviations the dependent variable will change for a one standard deviation change in the independent variable.

One tailed test

a statistical test that rejects extreme scores in one tail of a distribution (aka - directional test)

testing if something is bigger or smaller

two-tailed test

a statistical test that rejects extreme scores in both tails of a distirbution (aka-nondirectional test)

tetsing if something is different

statistically significant

a statistical test with a p-value less than the alpha level value (typically the alpha level value = 0.05 unless stated otherwise)

symbol: α

hypothesis example

Your Hypotheses:

• Null Hypothesis (H₀): There is no relationship between how much you eat and how much you weigh. (Eating more does not affect weight.)

• Alternative Hypothesis (H₁): Eating more leads to weight gain.

Which One Do You Want to Reject?

To support your hypothesis (more eating → more weight), you want to reject the null hypothesis (H₀). This means your data should show a statistically significant relationship between food intake and weight gain.

How Do You Reject H₀?

• You collect data on eating habits and weight.

• Perform a statistical test (like regression or correlation analysis).

• If the p-value from the test is less than α (e.g., 0.05), you reject H₀ and conclude that eating more is associated with weight gain.

validity

how well a test, measurement, or study measures what it is supposed to measure. If a test is valid, its results are accurate and meaningful.

pooling variances

• Combines variances from two or more groups into a single estimate, assuming equal variances.

  • Used in tests like the two-sample t-test.

  • Provides a more accurate estimate of variability.

confound

factor that makes it look like there's a relationship between two things (like cause and effect), when in reality, it's another variable that’s influencing both.

• A third variable that affects both the IV and DV, distorting the true relationship between them.

• Can bias results and make false conclusions.

• Example: Diet in a study of exercise and weight loss.

• Control confounds through design or statistical methods.

z score

tells you how many standard deviations a specific value is away from the mean of a dataset. It’s used to understand where a particular data point stands relative to the rest of the data.

When to Use It:

• To standardize values for comparison across different datasets.

• To check how unusual or typical a value is in relation to the overall data.

• In normal distribution to find percentiles or probabilities.

chi-square

• A statistical test used to assess if there is a significant association between categorical variables. It compares the observed frequencies in categories with the expected frequencies if there were no association.

When to Use It:

• To test for the independence of two categorical variables (e.g., Are gender and voting preference related?).

• To assess how well an observed distribution fits an expected distribution (e.g., testing if a dice is fair).

ANOVA

• A statistical test used to compare the means of three or more groups to see if there is a significant difference between them.

When to Use It:

• When you have more than two groups and want to compare their means.

• To test if the variability between groups is greater than the variability within groups (i.e., if group means differ significantly).

between subjects t-test

• A statistical test used to compare the means of two independent groups to determine if there is a significant difference between them.

When to Use It:

• When you have two separate groups and you want to test if their mean scores differ on some variable.

Welchs t-test

• What it does: Compares the means of two independent groups when their variances are unequal(heteroscedasticity).

• When to use: When normality is met, but the assumption of equal variance is violated.

• Difference from Student’s t-test: More reliable when sample sizes and variances are different.

Mann-Whitney U test

• What it does: A non-parametric (doesnt assume normal distribution) test for comparing two independent groups.

• When to use: When data is not normally distributed.

Spearmans correlation coefficient

measures the strength and direction of the relationship between two variables based on their ranks. It evaluates how well the relationship between the variables can be described using a monotonic function (i.e., as one variable increases, the other variable either always increases or always decreases).

Kendalls tau

another non-parametric measure that assesses the strength and direction of association between two variables. It is based on the concept of concordant and discordant pairs of data points:

• A concordant pair is one where the ranks for both variables are in the same order.

• A discordant pair is one where the ranks are in the opposite order.

Regression

finding out how x and y are related - best linear description of data - the regression changes if the dependent variable changes

regression line

represents our theory about the data

the line is our best prediction of the relationship

Y(-hat)

predicted value - the value of Y that the regression equation predicts for a given X.

• It is not the actual observed value of Y but rather the value that the model estimates or "predicts" for that specific input.

R2

measure of how closely the independent variable (X) and the dependent variable (Y) are related in a regression model. - variance

one sample t-tests

used to determine whether the mean of a single sample is significantly different from a known or hypothesized population mean. It compares the sample mean to the population mean to see if any difference observed is likely due to chance or if it reflects a true difference.

comparing a single sample mean to a known μ (pop mean)

μ

population mean

power

probability that a statistical test will correctly reject a false null hypothesis (i.e., detect a true effect when one exists). In other words, power is the ability of a test to detect an effect if there is one.

1−β - correct rejection

correct acceptance

1-a (when you correctly accept it)

independent sample t-test

known as a Two-Sample t-test) is a statistical test used to compare the means of two independent groups to determine if there is a significant difference between them. The two groups are considered independent because the measurements in one group have no relationship to the measurements in the other group.

trivial effect

smaller than 0.2 !!!!

A small, insignificant effect that has little practical impact, even if statistically significant.

• Measurable but of minimal real-world relevance.

• Often found in research but doesn't influence decisions or outcomes meaningfully.

2-sample t-test

related to independent sample T-test

statistical test used to compare the means of two independent groups to determine whether there is a statistically significant difference between them. The groups are considered independent because the data points in one group are not related to or influenced by the data points in the other group.

error bars

• Visual representation of variability in data.

• Show range of uncertainty around a data point (e.g., mean).

• Indicate the precision of the estimate (e.g., standard error, confidence interval).

• Longer = greater uncertainty; shorter = more precise estimate.

• Commonly used in graphs like bar charts or scatter plots.