Power and Sampling Distributions

Power and Sampling Distributions

• Discussions around power often relate to sampling distributions.
• Key concepts from previous learning (e.g., covered in course 2910).

Distribution of Sample Means

• Defined as all the possible sample means from random samples of size (n) obtained from a population.
• Example: Randomly sampling 25 MUN students and calculating means (M1, M2, M3, etc.).

Key Properties of Distribution of Sample Means

1. Convergence to Population Mean: Sample means should cluster around the population mean, 𝜇.
2. Normal Distribution Shape: The distribution of sample means resembles a normal distribution.
3. Effect of Sample Size: Increasing sample size (n) results in sample means being closer to 𝜇.

Example of Sample Means Distribution

• Sample Scoring:
  • Mean Scores calculated from various sample combinations (e.g., n = 2).
• Observed behavior:
  1. Sample means pile up around 𝜇.
  2. Approximates a normal distribution.

Probability Calculations

• Calculation of probabilities related to sample means:
  • Probability of getting a mean $M > 6$: p(M > 6) = rac{2}{12} = 0.1667
  • Probability of getting a mean $M < 5$: p(M < 5) = rac{4}{12} = 0.3333

Characteristics of Sample Means

• When examining a population with different distributions (bimodal, uniform, skewed):
  • Even distributions that are not normal will tend toward normality as sample sizes increase.
  • E.g., for $n = 30$, the distribution of sample means approaches normality regardless of the population distribution shape.

Central Limit Theorem (CLT)

• The CLT states:
  • The mean of the sample means (M) equals the population mean, 𝜇.
  • The standard deviation of the sample means (standard error) is given by rac{ ext{Population SD}}{ ext{sqrt}(n)}.
  • As sample size (n) increases, the distribution of sample means approaches normal distribution regardless of the population's shape.

Standard Error of Mean (SEM)

• Definition: It describes how well an individual sample mean represents the population mean, 𝜇.
• Decreases as sample size increases:
  • ext{Standard Error} = rac{ ext{Population SD}}{ ext{sqrt}(n)}
  • Example:
  • For n=1, SEM=10; n=4, SEM=5; n=9, SEM approximately 3.33; n=64, SEM=1.25.

Example Problem Solving

• Question: Probability of sample mean exceeding certain values:
  1. Question 1 highlights a population normal distribution with given parameters (e.g., 𝜇 = 500, 𝜎 = 100).
  2. Calculate the probability that $M > 540$:
  • $z$-score calculation leads to conclusion using z-tables resulting in $p(M > 540) = 0.02275$.

Range of Expected Scores

• Question 2: Determining the expected score range with 80% probability.
  • Utilizing $z$-score boundaries to find limits around the mean yielding 474.4 to 525.6.

Probability Calculations on M

• Question 3: Analysis of n = 64 observations:
  • a) For $M < 16$, z = -2; $p(M < 16) = 0.02275$.
  • b) For $M > 23$, z = 1.5; $p(M > 23) = 0.06681$.
  • c) For $17 < M < 24$, combine probability ranges to derive $p(17 < M < 24) = 0.91044$.