5.2 - Writing Hypotheses

The first step in conducting a hypothesis test is to write the hypothesis statements that are going to be tested. For each test you will have a null hypothesis (\(H_0\)) and an alternative hypothesis (\(H_a\)).

Null Hypothesis
The statement that there is not a difference in the population(s), denoted as \(H_0\)
Alternative Hypothesis
The statement that there is some difference in the population(s), denoted as \(H_a\) or \(H_1\)

When writing hypotheses there are three things that we need to know: (1) the parameter that we are testing (2) the direction of the test (non-directional, right-tailed or left-tailed), and (3) the value of the hypothesized parameter.

  1. At this point we can write hypotheses for a single mean (\(\mu\)), paired means(\(\mu_d\)), a single proportion (\(p\)), the difference between two independent means (\(\mu_1-\mu_2\)), the difference between two proportions (\(p_1-p_2\)), a simple linear regression slope (\(\beta\)), and a correlation (\(\rho\)). 
  2. The research question will give us the information necessary to determine if the test is two-tailed (e.g., "different from," "not equal to"), right-tailed (e.g., "greater than," "more than"), or left-tailed (e.g., "less than," "fewer than").
  3. The research question will also give us the hypothesized parameter value. This is the number that goes in the hypothesis statements (i.e., \(\mu_0\) and \(p_0\)). For the difference between two groups, regression, and correlation, this value is typically 0.

Hypotheses are always written in terms of population parameters (e.g., \(p\) and \(\mu\)).  The tables below display all of the possible hypotheses for the parameters that we have learned thus far. Note that the null hypothesis always includes the equality (i.e., =).

One Group Mean
Research Question Is the population mean different from \( \mu_{0} \)? Is the population mean greater than \(\mu_{0}\)? Is the population mean less than \(\mu_{0}\)?
Null Hypothesis, \(H_{0}\) \(\mu=\mu_{0} \) \(\mu=\mu_{0} \) \(\mu=\mu_{0} \)
Alternative Hypothesis, \(H_{a}\) \(\mu\neq \mu_{0} \) \(\mu> \mu_{0} \) \(\mu<\mu_{0} \)
Type of Hypothesis Test Two-tailed, non-directional Right-tailed, directional Left-tailed, directional
 
\( \mu_{0} \) is the hypothesized population mean
Paired Means
Research Question Is there a difference in the population? Is there a mean increase in the population? Is there a mean decrease in the population?
Null Hypothesis, \(H_{0}\) \(\mu_d=0 \) \(\mu_d =0 \) \(\mu_d=0 \)
Alternative Hypothesis, \(H_{a}\) \(\mu_d \neq 0 \) \(\mu_d> 0 \) \(\mu_d<0 \)
Type of Hypothesis Test Two-tailed, non-directional Right-tailed, directional Left-tailed, directional
 
A paired means test is comparable to conducting a one group mean test on the differences.  
One Group Proportion
Research Question Is the population proportion different from \(p_0\)? Is the population proportion greater than \(p_0\)? Is the population proportion less than \(p_0\)?
Null Hypothesis, \(H_{0}\) \(p=p_0\) \(p= p_0\) \(p= p_0\)
Alternative Hypothesis, \(H_{a}\) \(p\neq p_0\) \(p> p_0\) \(p< p_0\)
Type of Hypothesis Test Two-tailed, non-directional Right-tailed, directional Left-tailed, directional
 
\( p_{0} \) is the hypothesized population proportion
Difference between Two Independent Means
Research Question Are the population means different? Is the population mean in group 1 greater than the population mean in group 2? Is the population mean in group 1 less than the population mean in groups 2?
Null Hypothesis, \(H_{0}\) \(\mu_1=\mu_2\) \(\mu_1 = \mu_2 \) \(\mu_1 = \mu_2 \)
Alternative Hypothesis, \(H_{a}\) \(\mu_1 \ne \mu_2 \) \(\mu_1 \gt \mu_2 \) \(\mu_1 \lt \mu_2\)
Type of Hypothesis Test Two-tailed, non-directional Right-tailed, directional Left-tailed, directional
 
Note: \(\mu_1 = \mu_2\) is equivalent to \(\mu_1-\mu_2=0\)
Difference between Two Proportions
Research Question Are the population proportions different? Is the population proportion in group 1 greater than the population proportion in groups 2? Is the population proportion in group 1 less than the population proportion in group 2?
Null Hypothesis, \(H_{0}\) \(p_1 = p_2 \) \(p_1 = p_2 \) \(p_1 = p_2 \)
Alternative Hypothesis, \(H_{a}\) \(p_1 \ne p_2\) \(p_1 \gt p_2 \) \(p_1 \lt p_2\)
Type of Hypothesis Test Two-tailed, non-directional Right-tailed, directional Left-tailed, directional
 
Note: \(p_1=p_2\) is equivalent to \(p_1-p_2=0\)
Simple Linear Regression: Slope
Research Question Is the slope in the population different from 0? Is the slope in the population positive? Is the slope in the population negative?
Null Hypothesis, \(H_{0}\) \(\beta =0\) \(\beta= 0\) \(\beta = 0\)
Alternative Hypothesis, \(H_{a}\) \(\beta\neq 0\) \(\beta> 0\) \(\beta< 0\)
Type of Hypothesis Test Two-tailed, non-directional Right-tailed, directional Left-tailed, directional
Correlation (Pearson's r)
Research Question Is the correlation in the population different from 0? Is the correlation in the population positive? Is the correlation in the population negative?
Null Hypothesis, \(H_{0}\) \(\rho=0\) \(\rho= 0\) \(\rho = 0\)
Alternative Hypothesis, \(H_{a}\) \(\rho \neq 0\) \(\rho > 0\) \(\rho< 0\)
Type of Hypothesis Test Two-tailed, non-directional Right-tailed, directional Left-tailed, directional
 

Students in this course should pause here and return to complete the assignment in Canvas.