Author: kmcclain

Research Hypothesis

A RESEARCH HYPOTHESIS is a clear, specific, testable statement or prediction about the relationship between two or more variables (i.e., independent variable X explains variation in dependent variable Y).  The research hypothesis guides the direction of the study and outlines what the researcher expects to find.  A good hypothesis is:

• based on existing knowledge (i.e., theory drives your predictions)
• FALSIFIABLE (i.e., can be proven false if it is incorrect)

Some hypotheses are directional, positing that as values of one variable increase or decrease, values of the other variable increase or decrease:

• There is a POSITIVE RELATIONSHIP between population density (independent variable) and crime rates (dependent variable) — as population density increases, crime rates increase
• There is a NEGATIVE RELATIONSHIP between education level (independent variable) and teen pregnancy (dependent variable) — as education increases, teen pregnancy decreases

These types of hypothesis are appropriate when working with variables that have direction (i.e., ordinal-, interval- or ratio-level variables).

Some hypotheses merely posit that there is a relationship between two variables:

• Women are more likely to vote than men (respondent sex is the independent variable) to vote (dependent variable) — there is a relationship between sex and voting

This type of hypothesis is common when working with nominal variables.  Because nominal variables have no direction, the relationship between a nominal variable and another variable cannot be stated in directional terms. Instead, the hypothesis should specify the type of relationship between the variables in terms of how differences in the dependent variable are linked with differences in the independent variable.

If, based on theory and existing knowledge, there are control variables that further explain variation in the dependent variable, they should be included the research hypothesis: 

• if all variables involved have direction (i.e., ordinal-, interval-, or ratio-level variables), you would simply add the phrase “while controlling for” and then specify the control variables
• if the control variable is nominal, you should specify the expected relationship

The research hypothesis is presented as H₁.  If you have more than one research hypothesis, they would be presented as H₁, H₂, H₃, etc.

Null Hypothesis

Null means having no value.  By extension, the NULL HYPOTHESIS is a statement that there is no relationship between the variables being studied. While the research hypothesis serves as a foundation for conducting empirical research by guiding the direction of the study and outlining what the researcher or administrator expects to find, the null hypothesis forms the basis of hypothesis testing.  The null hypothesis is presented as H₀.

Alternative Hypothesis

An ALTERNATIVE HYPOTHESIS is proposed as an alternative to the null hypothesis; it indicates that there is a relationship between the variables being studied. Alternative hypotheses can be:

• directional (one-tailed), specifying a direction of the effect or difference
• non-directional (two-tailed), merely stating that there is a difference, without specifying the direction

Research hypotheses and alternative hypotheses are conceptually similar but serve different functions: a research hypothesis’s broader context is more aligned with scientific inquiry and theory testing, whereas an alternative hypothesis is specifically formulated for statistical testing against the null hypothesis. The alternative hypothesis is presented as H_a.

Confidence intervals provide a useful way to convey the uncertainty and reliability of an estimate, allowing researchers to make more informed conclusions about the population parameter.

When constructing a confidence interval for a sample mean, the critical value (t) for a given confidence level and number of degrees of freedom is multiplied by the standard error of the mean; this gives us the margin of error, which can then be added to and subtracted from the sample mean to establish the upper and lower bounds of our confidence interval. 

The T-DISTRIBUTION is a probability distribution that has a mean of 0 and is symmetrical and bell-shaped, similar to the normal distribution, but with heavier tails.  The t-distribution provides a more accurate and conservative estimates of population parameters when dealing with small samples (n<30) or when population standard deviations are unknown (which is usually the case in social science research).

The shape of the t-distribution — how tall/short the center of the distribution is and how thin/thick the tails of the distribution are (i.e., the dispersion of the distribution) — is determined by the DEGREES OF FREEDOM (df).  The degrees of freedom for a single sample is equal to the sample size, minus one; as a formula: df=n-1.  As degrees of freedom increase, the t-distribution approaches the normal distribution.

To interpret a t-distribution, you will need to reference a T-DISTRIBUTION TABLE (i.e., a T-TABLE).  Using a t-table is similar to using a z-table:

• Rows correspond to different degrees of freedom 
• Columns correspond to different confidence levels (90%, 95%, 99%) or SIGNIFICANCE LEVELS (α), which are equal to 1 minus the confidence level (α = 0.10, 0.05, 0.01)
• Table cells report the CRITICAL VALUES of the t-distribution, given the degrees of freedom and the confidence level/significance level; critical values are helpful in hypothesis testing and determining confidence intervals

The SAMPLE MEAN (X̄) is a measure of central tendency that represents the average value of a variable in sample data.  It is calculated in the same way that population mean (μ) is calculated: by summing all the observations for a variable in the sample, and dividing by the number of observations.

The SAMPLE STANDARD DEVIATION (s) is a measure of the dispersion or spread of the values in a sample around the sample mean.  It quantifies the amount of variation or dispersion of a set of values.  It is calculated in a similar manner to population standard deviation (σ), but with one notable difference: instead of dividing by the total number of observations (N), we divide by n-1. Dividing by n-1 produces a larger value (more variation) than dividing by N alone.  The reason why we would find this appealing when working with sample data is simple and straight-forward: whenever we use a sample instead of the entire population, there is the possibility of random error being introduced to our statistical analysis; calculating the standard deviation with n-1 errs on the side of caution by assuming larger variation.

The STANDARD ERROR OF THE MEAN (s.e.) is a measure of how much the sample mean (X̄) is expected to vary from the true population mean (μ) — in other words, it tells us how precise the sample mean is as an estimate of the population mean.  The standard error of the mean is calculated by dividing the sample standard deviation by the square root of the number of observations. The standard error of the mean decreases as the sample size increases.  Mathematically, the standard error of the mean is inversely related to the square root of the sample size (n). As the standard error of the mean decreases, the margin of error and confidence intervals narrow, and the sample mean becomes a more precise and reliable estimate of the population mean.  This is tied to the central limit theorem: 

Larger samples tend to provide a better representation of the population

→ The more representative a sample is, the more normally distributed the sample data is

→ As the sample mean approaches a normal distribution, we can make more accurate and robust inferences

Beware of Outliers!

Whether outliers (i.e., extreme values) are likely to skew results in a normal distribution is based in large part on sample size.  In small samples, outliers can disproportionately affect the sample mean, the sample standard deviation, and, as a result, the standard error of the mean.  This, in turn, can lead us to make generalizations about the population parameters based on inaccurate information.  Thus, it is important to identify, investigate, and decide how to handle outliers (i.e., include, exclude, adjust/transform, or consider separately) based on their potential impact and the context of the study.