Chapter 8: Sampling Distributions
Sampling distribution
A __________________ of a statistic is a probability distribution for all possible values of the statistic computed from a sample of size n.
Distribution of the Sample Mean: Normal Population
As the size of the sample increases, the standard deviation of the distribution decreases. Suppose that a simple random sample of size n is drawn from a large population with mean μ and standard deviation σ. The sampling distribution of x-bar will have mean μ-subscript-x-bar and standard deviation σ-subscript-x-bar = σ ÷√n The standard deviation of the sampling distribution of x-bar is called the STANDARD ERROR OF THE MEAN and is denoted σ-subscript-x-bar. The standard deviation of the sampling distribution of x -bar, σx-bar, is called the standard error of the mean.
Sampling distribution of "p-hat"
For a simple random sample of size, n, with a population proportion, p: 1. The shape of the _____________________ of p-hat is approximately normal provided np(1 - p) ≥ 10. 2. The mean of __________________________ of p-hat is μp-hat = p. 3. The standard deviation of _________________ of p-hat is σp = p(1-p) ÷ n.
Central Limit Theorem
Regardless of the shape of the underlying population, the sampling distribution of "x bar" becomes approximately normal as the sample size, n, increases.
Shape, Center, & Spread of the Sampling Distribution of x-bar
Shape, Center & Spread of Population: POPULATION IS NORMAL with mean μ and standard deviation σ. Shape: Regardless of the sample size n, the shape of the distribution of the sample mean is normal. Center: μx-bar - μ. Spread: σx - σ ÷√n Shape, Center & Spread of Population: POPULATION IS NOT NORMAL/NONNORMAL with mean μ and standard deviation σ. Shape: As the sample size increases, the distribution of the sample mean becomes approximately normal. Center: μx-bar - μ. Spread: σx - σ ÷√n
Normal Sampling Distribution
Shape: the shape of the distribution of the sample mean is normal. Center: the mean of the distribution of the sample equal the mean of the population. Spread: the standard deviation of the sample is less than the standard deviation of the population.
Sample proportion, "p-hat"
Suppose that a random sample of size, n, is obtained from a population in which each individual either does or does not have a certain characteristic. This is given by "p-hat" = x ÷ n (where x is the number of individuals in the sample with the specified characteristic. The __________________________ is a statistic that estimates the POPULATION PROPORTION.
Mean and Standard Deviation of the Sampling Distribution of x-bar
Suppose that a simple random sample of size n is drawn from a large with population with the mean μ and a standard deviation of σ. The sampling distribution of x-bar with have μ-subscript-x = μ and the standard deviation σ-subscript-x = σ ÷√n
Central Limit Theorem (CLT)
The Central Limit Theorem (CLT) for a "relatively large" sample size ( n ≥ 30), the variable x-bar is normally distributed regardless of the distribution of the variable. This result gets more accurate as n increases.
Sampling Distribution of the Mean
The __________________________ is the distribution of of the variable x-bar (all possible sample means) for a random variable X and a fixed sample n. 1. If the distribution of X is normal, the distribution of x-bar will be normal too. 2.The Central Limit Theorem (CLT) for a "relatively large" sample size ( n ≥ 30), the variable x-bar is normally distributed regardless of the distribution of the variable. This result gets more accurate as n increases. 3. For samples of any size, the mean of the variable x-bar is the same as the mean of the variable X, that is μx = μ 4. For the samples of any size, the standard deviation of the variable x-bar is the standard deviation the variable X divided the the square root of the size of the repeated samples n, that is σx-bar = σ ÷√n 5. Since σx-bar represents the error/variation from estimating μ with x-bar, it's often called the "standard error of the mean"
Sampling distribution of the sample mean, "x bar"
The ________________________________ is the probability distribution of all possible values of the random variable x-bar computed from a sample of size n and from a population with the mean μ and standard deviation σ .
Standard Error of the Mean
The sampling distribution of the sample mean x-bar is the probability distribution of all possible values of the random variable x-bar computed from a sample of size n from a population with mean μ and standard deviation σ. Suppose that a simple random sample of size n is drawn from a large population with mean μ and standard deviation σ. The sampling distribution of x-bar will have mean μ-subscript-x = μ and standard deviation σ-subscript-x = σ √n. The standard deviation of the sampling distribution of x-bar, σ-subscript-x, is called the __________________________.
Distribution, To Describe
To _____________________ we need to know: 1. Shape 2. Mean 3. Standard Deviation