PSYC 2260 Chapter 7

University of Manitoba - PSYC 2260: Chapter 7 - Finance Literacy

7.1 Samples, Populations, and the Distribution of Sample Means

• Definition of Distribution of Sample Means

  • Collection of sample means from all possible random samples of a specific size (n) from a population.

  • Ability to predict sample characteristics is based on the distribution of sample means.

  • Contains all possible sample means, necessary for computing probabilities.

  • Distinction: The values are statistics (sample means) instead of raw scores.

7.2 Sampling Distribution

• Definition: A distribution of statistics obtained from selecting all possible samples of a specific size from a population.

  • Sampling Distribution of M: Another term for the distribution of sample means.

7.3 Construction of the Distribution of Sample Means

• Steps:

  1. Select a random sample of size (n) from the population.

  2. Calculate the sample mean.

  3. Place the sample mean in a frequency distribution.

  4. Repeat with another random sample of the same size.

  5. Continue the process until a distribution of sample means is constructed.

  • Important: All samples must contain the same number of scores (n).

7.4 Characteristics of the Distribution of Sample Means

• Characteristics:

  1. Sample means should cluster around the population mean.

     • Samples will not be perfect but need to be representative.

  2. Sample means generally form a normal-shaped distribution when the number of samples is large.

     • Most samples have means close to population mean (μ).

     • Frequencies taper off as they move away from μ.

  3. Larger sample sizes yield sample means that are closer to the population mean.

     • Large samples are typically more representative and cluster near μ.

     • Small samples tend to show greater variation in means.

7.5 Example of Constructing Distribution of Sample Means

• Population Example: Scores of 2, 4, 6, 8.

  • Step 1: List all possible samples of size n=2.

  • Step 2: Calculate the mean for each of the 16 samples.

  • Example: Determine probability of obtaining a sample mean greater than 7.

7.6 Central Limit Theorem

• Statement: For any population with mean μ and standard deviation σ, the distribution of sample means for sample size n will have:

  • Mean = μ

  • Standard deviation = σ/√n

  • Approaches normal distribution as n increases.

• Defines the precise distribution characteristics for any population, regardless of the original shape.

7.7 Shape of the Distribution of Sample Means

• Conditions for a Normal Distribution:

  1. The population distribution should be normal.

  2. The sample size (n) should be sufficiently large (around 30+).

7.8 Expected Value of M

• Definition: The mean of the distribution of sample means is equal to the population mean (μ).

  • The average of all sample means equals the population mean, ensuring unbiased estimates.

7.9 Standard Error of M

• Definition: Standard deviation of the distribution of sample means (σM).

  • Indicates the average distance between a sample mean (M) and the population mean (μ).

  • Formula: σM = σ / √n.

• Characteristics:

  1. Describes distribution of sample means and indicates how much difference is expected.

  2. Smaller standard error means sample means are close together; larger means wider variation.

7.10 Law of Large Numbers

• Statement: Larger sample sizes increase the likelihood that the sample mean will more closely align with the population mean.

  • Sample size influences accuracy in representation. Larger samples reduce error between sample mean and population mean.

7.11 Population Standard Deviation and Standard Error

• When n=1, the standard error equals the population standard deviation.

• Increasing sample size improves accuracy; standard error decreases with increasing n.

7.12 Relationship Between Standard Error and Variance

• Population standard deviation (σ) and variance (σ²) are directly related.

• Example calculations demonstrate how to substitute variance into standard error formulas.

7.13 The Three Different Distributions

1. Population Distribution: Original population containing thousands/millions of scores with its specific characteristics.

2. Sample Distribution: A sample drawn from the population meant to represent it.

3. Distribution of Sample Means: Theoretical distribution of means from all possible samples of a set size.

7.14 Z-Scores and Probability for Sample Means

• Definition: Z-scores identify the location of the sample mean in the distribution of sample means.

  • Z-score characteristics:

    • Indicates whether the sample mean is above (+) or below (-) the population mean (μ).

    • Provides distance between the sample mean and μ in terms of standard errors.

• Example Problem: Calculating the probability of a sample mean greater than a specified value (e.g., Math SAT scores).

7.15 Sampling Error

• Definition: Discrepancy between the sample statistic and the population parameter.

  • Typically some sampling error is expected.

• Clarifies that half of the samples will produce means below μ, while half will exceed it.

7.16 Standard Error and Sample Representation

• Standard error quantifies the 'average' distance between sample means and the population mean.

• As sample size increases, variability in sample means decreases, resulting in a better estimate of the population.