Topic 9

PSYC 220 - Psychological Statistics: Topic 9: CLT Practice and Standard Error

Overview of Distributions

  • Distribution of X (Population Distribution)

    • Represents the original distribution of individual scores related to a variable of interest.

    • The mean () and standard deviation () are key summaries of this distribution.

  • Distribution of Sample Means (Sampling Distribution)

    • Comprises sample means derived from repeated random samples of size n drawn from the population.

    • Mean of this distribution is denoted as _X.

    • Standard deviation of this distribution is known as the standard error (S.E.).

Central Limit Theorem (CLT)

  • The CLT serves as a critical link between the distribution of the population (X) and the distribution of sample means (X̄).

  • Formulation of CLT includes:

    • Z=Xˉ−μσnZ = \frac{X̄ - \mu}{\frac{\sigma}{\sqrt{n}}}

    • Where:

    • Z = Z-score

    • X̄ = sample mean

    • μ = population mean

    • σ = population standard deviation

    • n = sample size

Key Examples of Distributions

1. Exam Scores Distribution

  • A professor tracked scores; mean score (μ) = 82, standard deviation (σ) = 16.

  • Notably, the exam scores show a positive skew in the distribution.

2. Youtuber Viewers' Time

  • A Youtuber randomly selected n = 16 subscribers, average viewing time was 20 minutes.

  • The average viewing time from multiple samples n makes up a distribution graph.

3. Weight of Shiba Inus (n=1)

  • Weight follows a normal distribution with μ = 30 and σ = 2.

  • Probabilities:

    • P(X≥32)P(X \geq 32) for selecting a big dog, which requires calculating the Z-score.

4. Weight of Shiba Inus (n=4)

  • Again with weights, when sampling 4 dogs, calculate the probability of mean weight () being 32 lbs or more.

  • Utilize the standard deviation (σ) for the calculation.

5. Normal Distribution Scenarios

  • Given μ = 20 and σ = 4, determination of probability for various selections is needed:

    • P(X > 22) and P(X̄ > 22) for single and sample mean respectively, elaborating Z-score formulations.

Sampling Distribution Shape

  • The sampling distribution of X̄ becomes normal when:

    • The population distribution of X is normal, or

    • Sample size n ≥ 30, enabling normality regardless of original population shape.

Decision Tree for Distribution Shapes

  • Check if Distribution of X is Normal:

    • If No, assess if n ≥ 30. If both are No, sampling distribution is not normal.

    • If Yes, the sampling distribution is normal.

Review of Sampling Distribution Parameters

  • If population parameters (μ and σ) are known, this informs the distribution of sample means (X̄) and standard error (S.E.).

Standard Error (S.E.)

  • Definition: Standard error is the standard deviation of the sampling distribution.

    • Mathematical expression: S.E.=σnS.E. = \frac{\sigma}{\sqrt{n}}

  • Interpretation: S.E. measures the average distance between a sample mean and the population mean.

    • Smaller S.E. indicates that sample means are closer to the population mean, enhancing estimation accuracy.

Components and Implications of S.E.

  • Acts as an indicator of sampling accuracy and variability in relation to the population mean.

  • Average sampling error is presented across multiple samples.

In-Class Practices and Applications

Practice I

  • Given:

    • Population: μ = 50, σ = 16

    • Sample size n = 16

    1. Determine mean for sample means: μXˉ=μ=50\mu_{X̄} = \mu = 50

    2. Determine standard deviation for sample means: σXˉ=σn=1616=4\sigma_{X̄} = \frac{\sigma}{\sqrt{n}} = \frac{16}{\sqrt{16}} = 4

Practice II

  • Given a normal distribution μ = 70, σ = 8, find probabilities for:

    • Selecting a score greater than 72

    • Sample mean accuracy relative to 72.

Practice III

  • Compare two researchers with different sample sizes (100 vs. 25) and mean comparisons of 80 or higher to determine likelihood based on Central Limit Theorem.

Practice IV

  • From population μ = 35, σ = 15, analyze sample of n = 25 scores for:

    • Standard Error, differences, and sampling error assessments.

Magnitude and Relationship Insights of Standard Error

  • Law of Large Numbers states larger sample sizes assure closeness of sample mean to population mean.

  • Smaller population variance yielded better precision for sample means.

  • As sample size increases, decreases in standard deviation of sampling distribution occur:

    • An illustrative graph shows S.E. decline with rising n.

Conclusion and Quiz Preparation

  • Example quiz question on S.E. and average differences between sample means and population means to reinforce understanding of the learned material.