Statistics Chapter 7 Flashcards

Statistics Concepts

• A statistic is a number describing a characteristic of a sample.
• A parameter is a number that describes a characteristic of a population.
• X̄ represents the sample mean, estimating the population mean.
• P̂ denotes the sample proportion, estimating the population proportion.
• Sx symbolizes the sample standard deviation, estimating the population standard deviation.

Sampling Distributions

• When sample proportions (P̂) are calculated, if the sample size ( n ) is small and P̂ is close to 0, the distribution is right skewed.
• Conversely, if P̂ is close to 1 in a small sample, the distribution becomes left skewed.
• For large ( n ) or when P̂ is around 0.5, the sampling distribution of P̂ is considered approximately Normal.
• The 10% condition must be met for the standard deviation of the sampling distribution to be calculated as ( \sqrt{p(1-p)/n} ).
• The Large Counts Condition establishes that the sampling distribution of P̂ is approximately Normal if certain thresholds are met.

Sample Mean Distribution

• The standard deviation of the sample mean (X̄) will be ( \sigma/\sqrt{n} ) if the 10% condition is satisfied.
• This standard deviation (σx̄) measures the typical distance between the sample mean and the population mean.
• If the population distribution of P̂ is approximately normal, the sample distribution will also be approximately Normal.
• For large ( n ), the sample mean X̄ is approximately normal.

Central Limit Theorem

• The Central Limit Theorem asserts that for sufficiently large samples, the sampling distribution of the sample mean will be approximately Normally distributed, irrespective of the population distribution shape.
• The unbiased statistic has a mean of its sampling distribution equal to the true value of the parameter being estimated.

Light Bulb Burnout Times

• For a distribution of burnout times that is strongly right skewed, with larger sample sizes, the mean burnout time's distribution tends towards Normal.

Chipmunk Weight Example

• For a wildlife biologist's sample of 9 chipmunks weighing on average 84 gm (mean) with an SD of 18 gm, the sampling distribution mean is 84 gm, but the SD is unknown, indicating it is approximately Normal.

Estimation and Sampling

• In a car sample where 19% of cars are white from a population where 25% are white, the adequate characterization is knowing 19% is a statistic and 25% is a parameter.
• An effective statistic for estimating a parameter should have low bias and low variability.
• Given a population mean IQ of 112 with a standard deviation of 20, the correct approximation for a sample mean from 20 adults results in an SD of approximately 1.414.

Acid Rain Study

• In examining the impact of acid rain on trees, the sample proportion indicating damage is considered approximately Normal.

Sampling Conditions

• For any sample, the requirement that np ≥ 10 and n(1-p) ≥ 10 together form conditions for approximating sampling distributions.
• The sampling distribution of the sample mean from a sample size of 25 will have the same mean (130) with a reduced standard deviation (4), signifying the link to its population characteristics.

Test your knowledge of key concepts from Statistics Chapter 7 with these flashcards. Each card contains a term along with its definition, helping you grasp important statistical vocabulary. Perfect for students looking to reinforce their understanding of statistics.