Statistics Chapter 7 Flashcards

Key Statistical Terms

• Parameter: Represents a characteristic of a population; typically unknown since full population examination is unfeasible.

• Statistic: A number describing a characteristic of a sample; can be computed directly from sample data and used to estimate unknown parameters.

• Sampling Distribution: The distribution of all possible values of a statistic from samples of the same size drawn from the same population.

• Unbiased Estimator: A statistic that correctly estimates a parameter; its sampling distribution has a mean equal to the true parameter value.

Variability and Estimations

• Variability of a Statistic: Describes how spread out the sampling distribution is; primarily influenced by sample size; larger samples lead to smaller variability.

• Sampling Distribution of a Sample Proportion: For a simple random sample (SRS) size n from a population with proportion p:

  • Mean: μp̂ = p
  • Standard Deviation: σp̂ = √[p(1−p)/n]
  • Follows the 10% condition: n must be ≤ 1/10N for large populations.

Sample Means

• Mean and Standard Deviation of Sampling Distribution of x̅: For an SRS of size n drawn from a population with mean μ and standard deviation σ:

  • Mean: μx̅ = μ
  • Standard Deviation: σx̅ = σ/√n, applying the 10% condition.

• Sampling Distribution of a Sample Mean from a Normal Population: If a population is Normally distributed, the sampling distribution of the sample mean x̅ is also Normally distributed, regardless of sample size, given the 10% condition.

Central Limit Theorem (CLT) and Normal Conditions

• Central Limit Theorem (CLT): States that for a sufficiently large sample size n drawn from any population, the sampling distribution of the sample mean x̅ approaches a Normal distribution.

• Normal Conditions for Sample Means:

  • If the population is Normally distributed, the sampling distribution of x̅ is also Normal for any sample size n.
  • If the population is not Normal, the distribution of x̅ will still be approximately Normal for sample sizes n ≥ 30 due to CLT.