Chapter 7 Review
Sampling Distribution Sampling distribution: distribution of summary statistics you get from taking repeated random samples of same fixed size, n
Reasonably Likely vs Rare Events What are these?
Reasonably Likely vs Rare Events Reasonably likely events: values that lie in the middle 95% of sampling distribution If normal distribution: mean ± 1.96(standard deviation)
Reasonably Likely vs Rare Events Reasonably likely events: values that lie in the middle 95% of sampling distribution If normal distribution: mean ± 1.96(standard deviation) Rare events: values that lie in outer 5% of sampling distribution If normal distribution: outside the interval mean ± 1.96(standard deviation)
A parameter is __________________.
A parameter is a summary number describing a population.
A statistic is ____________________.
A statistic is a summary number calculated from a sample taken from a population.
A sampling distribution is ______________.
A sampling distribution is the distribution of a sample statistic in repeated random samples of the same fixed size, n.
Properties of Sampling Distribution of the Sample Mean If a random sample of size n is selected from a population with mean and standard deviation , then:
Properties of Sampling Distribution of the Sample Mean Center The mean, x, of the sampling distribution of x equals the mean of the population, : x =
Properties of Sampling Distribution of the Sample Mean Center The mean, x, of the sampling distribution of x equals the mean of the population, : x = In other words, the means of random samples are centered at the population mean.
Properties of Sampling Distribution of the Sample Mean Spread The standard deviation, x, of the sampling distribution, sometimes called the standard error of the mean, equals the standard deviation of the population, , divided by the square root of the sample size n. x =
Properties of Sampling Distribution of the Sample Mean Spread The standard deviation, x, of the sampling distribution, sometimes called the standard error of the mean, equals the standard deviation of the population, , divided by the square root of the sample size n. X = When sample size increases, spread ________.
Properties of Sampling Distribution of the Sample Mean Spread The standard deviation, x, of the sampling distribution, sometimes called the standard error of the mean, equals the standard deviation of the population, , divided by the square root of the sample size n. X = When sample size increases, spread decreases
Properties of Sampling Distribution of the Sample Mean Shape The shape of the sampling distribution will be approximately normal if the population is approximately normal.
Properties of Sampling Distribution of the Sample Mean Shape The shape of the sampling distribution will be approximately normal if the population is approximately normal. For other populations, the sampling distribution becomes more normal as n increases (Central Limit Theorem).
Properties of the Sampling Distribution of the Sum of a Sample Three properties based on the following premise: If a random sample of size n is selected from a distribution with mean and standard deviation , then:
Properties of the Sampling Distribution of the Sum of a Sample Three properties based on the following premise: If a random sample of size n is selected from a distribution with mean and standard deviation , then: (1) The mean of the sampling distribution of the sum is sum = n
Properties of the Sampling Distribution of the Sum of a Sample Three properties based on the following premise: If a random sample of size n is selected from a distribution with mean and standard deviation , then: (2) The standard error of the sampling distribution of the sum is sum = Note: SE does not decrease as n increases
Properties of the Sampling Distribution of the Sum of a Sample Three properties based on the following premise: If a random sample of size n is selected from a distribution with mean and standard deviation , then: (3) The shape of the sampling distribution will be approximately normal if the population is approximately normally distributed. For other populations the sampling distribution will become more normal as n increases.
Sampling distributions of the mean Which distribution represents the population, n = 4, and n = 10? How do you know?
Page 437, P7 population n = 4 n = 10
Properties of the Sampling Distribution of the Number of Successes If a random sample of size n is selected from a population with proportion of successes p, then the sampling distribution of the number of successes X:
Properties of the Sampling Distribution of the Number of Successes If a random sample of size n is selected from a population with proportion of successes p, then the sampling distribution of the number of successes X: has mean x = np
Properties of the Sampling Distribution of the Number of Successes If a random sample of size n is selected from a population with proportion of successes p, then the sampling distribution of the number of successes X: has mean x = np has standard error
Properties of the Sampling Distribution of the Number of Successes If a random sample of size n is selected from a population with proportion of successes p, then the sampling distribution of the number of successes X: has mean x = np has standard error will be approximately normal as long as n is large enough
Properties of the Sampling Distribution of the Sample Proportion If a random sample of size n is selected from a population with proportion of successes p, then the sampling distribution of p has these properties:
Properties of the Sampling Distribution of the Sample Proportion If a random sample of size n is selected from a population with proportion of successes p, then the sampling distribution of p has these properties: Mean of the sampling distribution is equal to the mean of the population, or
Properties of the Sampling Distribution of the Sample Proportion If a random sample of size n is selected from a population with proportion of successes p: Standard error of the sampling distribution is equal to the standard deviation of the population divided by the square root of the sample size:
Properties of the Sampling Distribution of the Sample Proportion If a random sample of size n is selected from a population with proportion of successes p, then the sampling distribution of p has these properties:
Properties of the Sampling Distribution of the Sample Proportion If a random sample of size n is selected from a population with proportion of successes p, then the sampling distribution of p has these properties: As the sample size gets larger, the shape of the sampling distribution becomes more normal and will be approximately normal if n is large enough (both np and n(1 – p) are at least 10).
Questions? You may use both sides of one 4x6 note card for the test—not two note cards stapled or taped together
Properties of the Sampling Distribution of the Sum Suppose two values are taken randomly from two populations with means μ1 and μ2 respectively and variances σ12 and σ22.
Properties of the Sampling Distribution of the Sum Suppose two values are taken randomly from two populations with means μ1 and μ2 respectively and variances σ12 and σ22. Then the sampling distribution of the sum of the two values has mean: μsum = μ1 + μ2
Properties of the Sampling Distribution of the Sum Suppose two values are taken randomly from two populations with means μ1 and μ2 respectively and variances σ12 and σ22. If the two values were selected independently, the variance of the sum is: σ2sum = σ12 + σ22
Properties of the Sampling Distribution of the Difference Suppose two values are taken randomly from two populations with means μ1 and μ2 respectively and variances σ12 and σ22.
Properties of the Sampling Distribution of the Difference Suppose two values are taken randomly from two populations with means μ1 and μ2 respectively and variances σ12 and σ22. Then the sampling distribution of the difference of the two values has mean: μdifference = μ1 - μ2
Properties of the Sampling Distribution of the Difference Suppose two values are taken randomly from two populations with means μ1 and μ2 respectively and variances σ12 and σ22. If the two values were selected independently, the variance of the difference is: σ2difference = σ12 + σ22
The shapes of the sampling distributions of the sum and difference depend on the shapes of the two original populations.
The shapes of the sampling distributions of the sum and difference depend on the shapes of the two original populations. If both populations are normally distributed, so are the sampling distributions of the sum and the difference.