Questions and Answers
What is a parameter?
A number that describes the population (μ, σ, p).
What is a statistic?
A number that describes the sample (x̅, s, p̂).
What does sampling variability refer to?
The value of a statistic varies in the repeated random sampling.
What is the expected variation in sampling variability?
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What is a sampling distribution?
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To calculate the sample population, the formula is ______.
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What elements do you consider when describing sampling distributions?
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What is the overall shape of the sampling distribution?
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What are outliers in a sampling distribution?
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What should the center of a sampling distribution be close to?
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How is spread in a sampling distribution described?
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When is a statistical estimator considered unbiased?
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What does high bias refer to?
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What does low bias indicate?
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What is high variability in a sampling distribution?
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What indicates low variability?
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When is the spread of the sampling distribution approximately the same for any population?
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What produces smaller spreads in sampling distributions?
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μx̅ = ______
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μp̂ = ______
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As n increases, what happens to the standard deviation?
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As n increases, what happens to x̅?
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What does the mean of the sampling distribution equal?
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What is the standard deviation of the sampling distribution equal to?
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What is the first rule of thumb for populations?
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What are the two rules of thumb regarding percent and proportions?
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What do the rules of thumb protect against?
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Match the following steps in a problem to their order:
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Z scores always have how many decimal places?
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What does the central limit theorem state?
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The formula for standard deviation in sample means is s = ______
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What does the CLT allow regarding normal probability calculation about sample means?
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The formula for Z score (sample means n=1) is ______
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The formula for Z score (sample means n>1) is ______
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The formula for σp̂ (sample proportions) is ______
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What is the Z score formula for sample proportions?
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Study Notes
Key Concepts in Sampling and Statistics
- Parameter: Represents a population characteristic, indicated by symbols like μ (mean), σ (standard deviation), and p (proportion).
- Statistics: Refers to a sample characteristic, denoted by symbols such as x̅ (sample mean), s (sample standard deviation), and p̂ (sample proportion).
- Sampling Variability: Indicates that the values of a statistic will vary with repeated random samples from the same population.
- Sampling Distribution: A distribution encompassing all possible values of a statistic for every possible sample of the same size taken from the population.
Understanding Sampling Distributions
- Describing Sampling Distributions: Key aspects include overall shape (symmetrical and approximately normal), the presence of outliers (ideally none), center (close to the true p value), and spread (described by standard deviation).
- Unbiased Estimate: A statistic is unbiased if the mean of its sampling distribution matches the true population parameter.
- Bias Levels: High bias indicates values are far from the center, while low bias signifies closeness to the center of the distribution.
- Variability: High variability suggests that data points are widely scattered, while low variability indicates they are clustered closely together.
Sample Size and Spread
- Effect of Sample Size: The spread of sampling distributions is similar across populations, given the population size is at least tenfold larger than the sample size (n). Larger samples lead to smaller spreads in the distribution.
- Central Limit Theorem (CLT): States that for a sufficiently large sample size (n > 25), the sampling distribution of x̅ approximates a normal distribution, even if the original population is not normal.
Z Scores and Standard Deviation
- Z scores: Indicate how many standard deviations an element is from the mean; used to compare sample means and proportions.
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Z score calculations:
- For sample means with n=1: ( Z = \frac{x - μ}{σ} )
- For sample means with n>1: ( Z = \frac{x̅ - μ}{σ/√n} )
- For sample proportions: ( Z = \frac{p̂ - p}{σ_{p̂}} ) where ( σ_{p̂} = \frac{\sqrt{pq}}{n} ).
Guidelines and Procedure
- Rules of Thumb: To ensure sufficient sample size, the population should exceed 10 times the sample size (n), and both conditions ( np > 10 ) and ( nq > 10 ) should be satisfied for proportions.
- Problem-solving Steps: Includes defining given information, checking normality, assuming normality, sketching a bell curve, calculating probabilities, and labeling z scores with areas.
Summary of Important Formulas
- Mean of Sampling Distribution: ( μ_{p̂} = p )
- Standard Deviation of Sampling Distribution for Sample Means: ( s = \frac{σ}{√n} )
- Standard Deviation for Sample Proportions: ( σ_{p̂} = \sqrt{\frac{pq}{n}} )
Central Limit Theorem Implications
- The CLT confirms that even if the original population distribution is not normal, the sampling distribution will approach normality with larger sample sizes, enabling reliable use of normal probability calculations.
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Description
This quiz covers key concepts related to sampling and statistics, including parameters, statistics, sampling variability, and distributions. Understand the characteristics of sampling distributions and the importance of unbiased estimates in statistics.