PSYC2001 - Inferential Statistics Notes

Course Outline

• Read the course outline on Moodle.

• Co-convenors and lecturers: Kelly Garner, Peter Lovibond.

• PDFs for lectures, tutorials, and labs are posted weekly.

• Lectures are recorded, but attending and taking notes is recommended.

• Keep up with material; weeks 1-4 are critical.

• Computing labs (TU2) start Week 1 – do Jamovi module beforehand.

• Stats tutorials (TU1) start Week 2.

• Don’t arrange travel during the exam or supplementary period.

Inferential Statistics

• Population: A large group of observations (people, objects) about which the researcher wants to draw conclusions.

  • Populations can be real or hypothetical.

  • Example: all patients receiving a particular treatment.

  • Populations are usually large and can’t be directly measured.

• Sample: A subset (small group) of the population.

  • We use samples to draw conclusions about the population.

Random Sampling

• The sample must be representative of the population.

• Achieved with random sampling:

  • Selection of any one observation from the population is independent of the selection of any other observation.

  • Every element in the population has an equal chance of being selected.

• Contrasted with biased sampling (e.g., convenience sampling, snowball recruitment).

Descriptive Statistics

• Used to describe a sample (e.g., its mean and variability).

• Mean: $M = \frac{\sum X}{n}$

• Variance: $S^2 = \frac{\sum (X - M)^2}{n}$

• Standard deviation: $S = \sqrt{\frac{\sum (X - M)^2}{n}}$

Inferential Statistics

• Based on descriptive statistics but allow inferences (conclusions) about the population.

• Use sample statistics to make inferences about population parameters.

Statistics vs. Parameters

• Statistic (sample):

  • Mean: $M$

  • Variance: $S^2$

  • Standard deviation: $S$

• Parameter (population):

  • Mean: $\mu$

  • Variance: $\sigma^2$

  • Standard deviation: $\sigma$

Estimation / Hypothesis Testing

• Two inferential procedures:

  • Estimation: Estimating a population parameter (e.g., $\mu$) through a confidence interval.

  • Hypothesis testing: Deciding whether to accept or reject a statement about a population parameter.

• Conclusions always relate to the population.

• Inferential statistical procedures determine whether a sample outcome has occurred by chance.

Example

• Is a treatment for anxiety effective?

  • Experiment: Randomly allocate participants to treatment or control groups.

  • Measure anxiety score (DV).

  • Suppose M1 = 20 and M2 = 22 (M1 – M2 = -2).

  • Is the treatment effective, or did the difference occur by chance?

  • Need to determine the likelihood of obtaining a difference as large as 2 points if the treatment is not effective, which requires looking at sampling variability.

Sampling Variability

• The value of a statistic varies from sample to sample due to chance.

• Example: Population mean $\mu$ = 30.

  • Randomly select a sample of n=10 and calculate the sample mean M.

  • M is likely near 30 but not exactly 30 (e.g., 28 or 33).

  • The same principle applies to other statistics estimating a population parameter.

  • Quantify the margin of error in our estimate.

Sampling Distribution

• Hypothetical distribution of a sample statistic formed by repeatedly drawing samples of n observations from a population.

• Focus on the sampling distribution of the mean.

Sampling Distribution Example

• Imagine drawing many samples of n=10 and calculating M for each.

• Form a frequency distribution from these M values.

• The sampling distribution is the frequency distribution you would get with an infinite number of samples (hypothetical).

Properties of Sampling Distribution of Mean

• Mean is $\mu$ (M is an unbiased estimator of $\mu$).

• Variance is $\frac{\sigma^2}{n}$

• Standard deviation is $\frac{\sigma}{\sqrt{n}}$ (standard error of the mean).

• Normal distribution.

Effect of n on $\sigma_M$

• As n increases, the standard error of the mean decreases, leading to a narrower sampling distribution.

Effect of $\sigma$ on $\sigma_M$

• As $\sigma$ increases, the standard error of the mean increases, leading to a wider sampling distribution.

Shape of the Sampling Distribution

• If the population is normally distributed, the sampling distribution of the mean is also normally distributed.

• Central Limit Theorem: The sampling distribution of the mean tends towards a normal distribution as n increases, regardless of the population distribution.

• If n is reasonably large, assume the sampling distribution is normal.