Psych Stats

Sampling and Populations

  • Inferential Statistics: - Used to make inferences or decisions about a population based on a sample.

    • Key components include:

    • Sample: A subset of the population.

    • Population: The entire group being studied.

Overview of Sampling Methods

  • Random Sampling:

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

    • Variations:

    • Simple Random Sampling

    • Systematic Sampling

    • Stratified Random Sampling

    • Cluster or Multistage Sampling

  • Non-random Sampling:

    • Members do not have an equal chance of selection.

    • Types include:

    • Accidental Sampling: Based on convenience.

    • Quota Sampling: Sample is drawn in proportion to population characteristics.

    • Judgment or Purposive Sampling: Selection based on judgment to obtain a representative sample.

Sampling Error

  • Sampling Error: - The difference between a sample statistic and the corresponding population parameter, expected due to chance.

    • Key terms:

    • Sample Mean (X̄)

    • Population Mean (μ)

    • Sample Standard Deviation (S)

    • Population Standard Deviation (σ)

Sampling Distribution of Means

  • Properties: - Approximates a normal curve as sample size increases (n > 30).

    • The mean of the sampling distribution equals the true population mean.

    • Standard deviation of the sampling distribution is smaller than that of the population.

Standard Error of the Mean (SEM)

  • SEM: Average distance of the sample means from the population mean.

  • Formula: - SEM (σx) = σ / √N

    • As sample size increases, SEM decreases, leading to closer approximation of the population mean (μ).

Confidence Intervals

  • Used to estimate the likely range of the population mean based on sample data.

  • 68% CI, 95% CI, 99% CI: - Higher confidence levels provide a wider interval but less precision.

  • Example: - For IQ scores with μ = 60 and σ = 8, calculating for N = 64 shows the probability of obtaining a sample mean above a threshold.

Confidence Interval Calculations

  • Formula:

  • CI = X ± (z or t) * SEM

  • Adjust standard error based on whether population standard deviation is known or unknown.

Estimating Population Mean

  • If population standard deviation is known:

    • Use normal distribution (z-distribution) for CI.

  • If population standard deviation is unknown:

    • Use sample standard deviation (s) and t-distribution for CI.

The t-Distribution

  • Used when the population standard deviation is unknown.

  • Degrees of Freedom: - Calculated as df = N - 1

    • Depend on the sample size, affecting the t-values obtained from t-tables.

Practical Example

  • Finding a 95% confidence interval involves:

    1. Calculating sample mean and standard deviation.

    2. Finding the standard error.

    3. Getting t-value from the t-table based on degrees of freedom.

    4. Applying the CI formula to yield a range for which we can be confident the true mean lies within.

    • Calculation Example:

      • Given: Sample Mean (X̄) = 62, Sample Standard Deviation (s) = 10, N = 30

      • SEM = s / √N = 10 / √30 ≈ 1.83

      • Degrees of Freedom (df) = N - 1 = 30 - 1 = 29

      • For a 95% CI, the t-value (from t-table for df=29) is approximately 2.045.

      • CI = 62 ± (2.045 * 1.83) = 62 ± 3.74 \n - Thus, the 95% CI is approximately (58.26, 65.74).