Module 9 – Hypothesis Testing

Hypothesis Testing

Basics

• Psychologists use inferential tests through a procedure called hypothesis testing.
• A hypothesis is a prediction that is tested during a research study.

Research Hypotheses

• Inferential tests require data to answer questions like, "Do college students have a higher IQ than the general population?"
• Data from a sample is used to make inferences about a population.
• A research hypothesis guides the design of a study to collect data from a sample.
• A specific prediction is needed to create a hypothesis.
• Example Research Hypothesis: Young adults entering college have a higher IQ than the general population.
  • Requires determining:
    • Mean IQ of the general population.
    • Mean IQ of first-year college students.
• Research hypotheses are testable predictions, such as:
  • Children watching a video of someone else learning to ride a bike will learn faster than those who don't.
  • People practicing mindfulness meditation have better mental imagery abilities than those who don't.
  • New mothers are more sensitive than other women to the cuteness of babies.
• Hypotheses can be expressed using symbols for means:
  • Population 1 = population of children who learn to ride with the video.
  • Population 2 = population of children who learn to ride without the video.
  • Prediction: Mean learning times should be lower for population 1 than population 2.
  • Expressed as: H1: \mu1 < \mu2

Null Hypotheses

• The null hypothesis is the prediction that the populations do not differ.
  • It suggests that the research is a complete waste of time because there is no effect of whatever is being studied.
• Expressed in symbols: $H0: \mu1 = \mu2$
• The null hypothesis is what we actually use for our inferential tests.
• When conducting an inferential test, we check to see if the null hypothesis is true.
  • We’re trying to see if there is no difference between the groups!

Exam Distributions Example

• To determine if $H0: \mu1 = \mu2$, we need to know the distribution of each population.
• SPSS was used to get descriptive statistics and plot a normal curve for exam scores from two different stats courses.
• The highest frequency of scores was around 88 on the first chart and around 78 on the second chart.
• The second chart had a shorter and wider distribution, indicating more variability.

Determining Population Membership

• If we know a student’s score, we can guess which distribution they belong to.
  • A score of 95 was more common in the black distribution (class A), so we should guess that a student with a 95 was in that class.
  • A score of 75 was more common in the blue distribution (class X), so we should guess that a student with a 75 was in that class.
  • Some scores are easier to guess than others.
• The bars never actually touch the x axis.
  • So a score of 60 probably belongs to the blue distribution, but there is still a small chance they were in the black distribution.

Inferential Tests

• Inferential tests are used to figure out if a sample looks like it came from the comparison population.
  • If yes, then $\mu1 = \mu2$; fail to reject the null hypothesis.
  • If no, then $\mu1 \neq \mu2$; reject the null hypothesis.

Sampling Distributions

• We don’t usually know the population mean and standard deviation, so we usually need to estimate our comparison distribution and refer to this estimated distribution as the sampling distribution.
• A sampling distribution is the distribution of a statistic over repeated sampling from a specific population.

Simplified Example

• Professor X is trying to improve student grades using a complex process involving radioactive cats.
  • His research hypothesis is that students in the radioactive cat condition ($\mu1$) are going to score higher on the first exam than the average stat’s student ($\mu2$).
    • \mu1 > \mu2
  • His null hypothesis is that both groups will be equal. His radioactive cats won’t improve student scores.
    • $\mu1 = \mu2$
• The mean and standard deviation for the first exam score for all students that have taken stats are known:
  • $\mu = 67.8$
  • $\sigma = 16.5$
• Professor X exposes his students to radioactive cats and then gives them the first stats exam.
• Professor X compares the results from his sample to the population to see if $\mu1 = \mu2$.
• The mean for the sample was 77.35.
  • Class X did score above the average, but half the people in the population score above the average.

Statistical Significance

• Psychologists have agreed that a good cutoff point for what we’ll consider statistically significant is 5%.
• If you look up a Z score of 1.64, you see that only 5% of scores are more extreme (percent to tail).
• This logic is very similar to what we did with our cutoffs for outliers.
  • We set a cutoff point on our comparison distribution.
  • Scores that are less extreme than the cutoff are considered to be part of our population.
    • These scores are likely to occur given that our sample came from the population.
  • Scores that are more extreme than our cutoff are considered to be part of a different population.
    • These scores were unlikely to occur if our sample came from the normal population, so it’s more likely they came from a different population.

Recap

• Hypothesis testing allows us to figure out whether it is unlikely that our sample came from a population with a certain distribution.
• If it is unlikely, then we have evidence that our sample came from another (imaginary) population with a different mean and standard deviation.
• So now we know a little more about the (imaginary) population our sample came from than we did before we studied our sample.

5 Basic Steps to Hypothesis Testing

• These basic steps will be followed for every inferential test, so remember them.
• The exact details of carrying out the 5 steps differ depending on which inferential test you are using.
  1. State the research hypothesis and null hypothesis.
  2. Determine the characteristics of the comparison distribution.
  3. Determine the cutoff sample score on the comparison distribution at which the null hypothesis should be rejected.
  4. Determine your sample’s score on the comparison distribution.
  5. Decide whether to reject the null hypothesis.

Important Considerations

• We've glossed over some things like sampling distributions and why we use 5%. We’ll talk more about these things later.
• The most important things you need to learn from this module are the 5 basic steps and an understanding of how hypothesis testing works.

Example Using the 5 Steps

• We’re curious if stressed children show more behavior problems than normal children.
  • We know the mean and standard deviation for normal children on a behavioral problems test, so we give the test to a group of stressed children.

Step 1: State the Research Hypothesis and the Null Hypothesis

• We have two populations:
  • $\mu1$: Stressed children
  • $\mu2$: Normal children
• Research hypothesis:
  • Stressed children have more behavioral problems than normal children.
  • Expressed in terms of populations:
    • Mean of population 1 > mean of population 2
  • Expressed in symbols:
    • \mu1 > \mu2
• Null hypothesis:
  • Stressed children have the same number of behavioral problems as normal children.
  • Expressed in terms of populations:
    • Mean of population 1 = mean of population 2
  • Expressed in symbols:
    • $\mu1 = \mu2$

Step 2: Determine the Characteristics of the Comparison Distribution

• The comparison distribution is what we’re calling $\mu2$ and represents the normal children.
• Let’s pretend that the mean score for the average child is 50 with a standard deviation of 3.
  • This comparison distribution will depend on which inferential test we use.

Step 3: Determine the Cutoff Sample Score on the Comparison Distribution at Which the Null Hypothesis Should Be Rejected

• Let’s use the score that separates the least common 5% of all scores from the rest of the scores.
  • If we’re using Z scores that would be 1.64 according to your Z table.
  • If the sample score is more extreme/unusual than the cutoff score, then you reject the idea that the null hypothesis is true.

Step 4: Determine Your Sample’s Score on the Comparison Distribution

• Since we’re using Z scores, we would need to find out our sample’s Z score in the comparison distribution.
• To find out, change your sample’s mean score to a Z score using the mean and SD of the comparison distribution.
• Example:
  • Suppose the mean score for our stressed children ($\mu1$) was 56.
  • $(56-50)/3 = 2$

Step 5: Decide Whether to Reject the Null Hypothesis

• Is your sample’s Z score more extreme/unusual than the cutoff score?
  • Compare sample Z to cutoff Z.
  • If sample Z more extreme/unusual than cutoff Z, reject null hypothesis.
• Example:
  • Sample Z = 2.0
  • Cutoff Z = 1.64
  • Sample Z is more extreme/unusual
  • Reject null hypothesis

Interpreting Results

• Our results never prove or disprove the research hypothesis.
• If your sample score is so extreme/unusual that it occurs less than 5% of the time in the comparison distribution, you can do three things:
  1. Reject null hypothesis
  2. Say your sample result supports (but does not prove) the research hypothesis
  3. Call your result statistically significant
• If your sample score is not so extreme/unusual, then you:
  • Fail to reject null hypothesis
  • Cannot say your result supports the research hypothesis
  • Call your result statistically non-significant

Alternative Hypothesis Example

• What if our Research hypothesis had been: stressed children have less behavioral problems than normal children.
  • \mu1 < \mu2
• For Step 3 we would:
  • Determine the cutoff sample score on the comparison distribution at which the null hypothesis should be rejected
    • If we’re using still using 5% then it would be -1.64 according to your Z table.
    • If the sample score is more extreme/unusual than the cutoff score, then you reject the idea that the null hypothesis is true