Questions and Answers
A _____ is a number that describes some characteristic of the population.
parameter
A _____ is a number that describes some characteristic of a sample.
statistic
The _____ of a statistic is the distribution of values taken by the statistic in all possible samples of the same size from the same population.
sampling distribution
A statistic used to estimate a parameter is an _____ if the mean of its sampling distribution is equal to the true value of the parameter being estimated.
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The variability of a statistic is described by the spread of its sampling distribution. This spread is determined primarily by the size of the random sample. Larger samples give _____ spread.
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What is the mean of the sampling distribution of p̂?
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What is the formula for the standard deviation of the sampling distribution of p̂?
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What does the Central Limit Theorem (CLT) state?
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If the population distribution is Normal, then so is the sampling distribution of x̅.
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The Central Limit Theorem applies only when the population distribution is Normal.
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Study Notes
Key Statistical Terms
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Parameter: Represents a characteristic of a population; typically unknown since full population examination is unfeasible.
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Statistic: A number describing a characteristic of a sample; can be computed directly from sample data and used to estimate unknown parameters.
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Sampling Distribution: The distribution of all possible values of a statistic from samples of the same size drawn from the same population.
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Unbiased Estimator: A statistic that correctly estimates a parameter; its sampling distribution has a mean equal to the true parameter value.
Variability and Estimations
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Variability of a Statistic: Describes how spread out the sampling distribution is; primarily influenced by sample size; larger samples lead to smaller variability.
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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
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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.
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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
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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.
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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.
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Test your knowledge with these flashcards covering key terms from Chapter 7 of Statistics. Learn about important concepts such as parameters and statistics and their significance in analyzing populations and samples. Perfect for revision and quick learning.
