Sampling and Statistics Concepts

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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?

Normal and expected. Signup and view all the answers

What is a sampling distribution?

The distribution of all values taken on by the statistic in all possible samples of the same size from the same population. Signup and view all the answers

To calculate the sample population, the formula is ______.

p̂ = (1/n) Σ (frequency)(pi) Signup and view all the answers

What elements do you consider when describing sampling distributions?

All of the above Signup and view all the answers

What is the overall shape of the sampling distribution?

Symmetric and approximately normal. Signup and view all the answers

What are outliers in a sampling distribution?

None. Signup and view all the answers

What should the center of a sampling distribution be close to?

Very close to the true p value. Signup and view all the answers

How is spread in a sampling distribution described?

Use standard deviation to describe its spread. Signup and view all the answers

When is a statistical estimator considered unbiased?

The mean of its sampling distribution is equal to the true value of the parameter being estimated. Signup and view all the answers

What does high bias refer to?

Far from center. Signup and view all the answers

What does low bias indicate?

Close to center. Signup and view all the answers

What is high variability in a sampling distribution?

Not clustered. Signup and view all the answers

What indicates low variability?

Clustered. Signup and view all the answers

When is the spread of the sampling distribution approximately the same for any population?

The population is at least 10 times as large. Signup and view all the answers

What produces smaller spreads in sampling distributions?

Larger samples. Signup and view all the answers

μx̅ = ______

μ (sample mean) Signup and view all the answers

μp̂ = ______

p (sample proportion) Signup and view all the answers

As n increases, what happens to the standard deviation?

Decreases. Signup and view all the answers

As n increases, what happens to x̅?

Gets closer to μ. Signup and view all the answers

What does the mean of the sampling distribution equal?

μp̂ = p. Signup and view all the answers

What is the standard deviation of the sampling distribution equal to?

σp̂. Signup and view all the answers

What is the first rule of thumb for populations?

population > 10n. Signup and view all the answers

What are the two rules of thumb regarding percent and proportions?

<ol> <li>np > 10; 2) nq > 10.</li> </ol> Signup and view all the answers

What do the rules of thumb protect against?

Skewness. Signup and view all the answers

Match the following steps in a problem to their order:

1 = Given 2 = Check for normality 3 = We can now assume normality 4 = Draw a bell curve 5 = P() 6 = Find area 7 = Go back and label z scores and area Signup and view all the answers

Z scores always have how many decimal places?

2 decimal places. Signup and view all the answers

What does the central limit theorem state?

Draw an SRS of size n from any population with μ and σ when n is large (>25) the sampling distribution of x̅ is close to N(μ, σ/√n). Signup and view all the answers

The formula for standard deviation in sample means is s = ______

σ/√n. Signup and view all the answers

What does the CLT allow regarding normal probability calculation about sample means?

The population distribution is not normal (the sample will act normally). Signup and view all the answers

The formula for Z score (sample means n=1) is ______

(x-μ)/σ. Signup and view all the answers

The formula for Z score (sample means n>1) is ______

(x̅ - μ)/(σ/√n). Signup and view all the answers

The formula for σp̂ (sample proportions) is ______

√(pq)/n. Signup and view all the answers

What is the Z score formula for sample proportions?

(p̂ - p)/σp̂. Signup and view all the answers

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.
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.