AP Statistics Chapter 7: Sampling Distributions

How is the size of a sample related to the spread of the sampling distribution?

Variability decreases for larger samples. Larger samples give smaller spreads.

The Central Limit Theorem (The CLT)

When n is large, the sampling distribution of the sample mean x̄ is close to the Normal Distribution.

Unbiased Estimator

When p̂=p

Variability

Repeated samples do not give very similar results.

Standard Deviation of the sampling distribution of p̂

***as long as the 10% condition is satisfied: n≤(1/10)N

Important Summary Type Things

-We often use a statistic to estimate an unknown parameter. -as sample size increases, spread decreases -the normal condition is helpful -the 10% condition: n≤(1/10)N -the CLT states that when n is large, the sampling distribution with mean x̄ is approximately Normal

Statistic

A number that describes some characteristic of a sample. The value of a statistic can be computed directly from the sample data. We often use a statistic to estimate an unknown parameter.

Parameter

A number that describes some characteristic of the population. In statistical practice, the value of a parameter is usually not known because we cannot examine the entire population.

Biased Estimator

A statistic used to estimate a parameter is an unbiased estimator if the mean of its sampling distribution is equal to the true value of the parameter being estimated.

p̂ and n relationship

For a specific value of p, the standard deviation σp̂ gets smaller as n gets larger (as sample size increases, spread decreases).

Population Distribution

Gives the values of the variable for all the individuals in the population.

The variability of a Statistic

Is described by the spread of its sampling distribution. The spread is determined primarily by the size of the random sample. Larger samples give smaller spread. The spread of the sampling distribution does not depend on the size of the population, as long as the population is at least 10 times larger than the sample

Bias

The aim is off and we consistently miss in the same direction.

Sampling Distribution (of a statistic)

The distribution of values taken by the statistic in all possible samples of the same size from the same population.

Sample Proportion

The statistic that we use to gain information about the unknown population parameter p.

Sampling Variability

The value of a statistic varies in repeated random sampling.

µ,x̄,p,p̂

µ-population mean x̄-sample mean p-population proportion p̂-sample proportion ***p̂ is used to estimate the unknown parameter p

What are the mean and standard deviation of the sampling distribution of the sample mean x̄?

mean: µx̄=µ standard deviation:σx̄=σ/√n ***as long as the 10% condition is satisfied: n≤(1/10)N

Sampling Distribution of p̂

mew of p hat equals p

The Normal Condition

np≥10 n(1-p)≥10

The 10% condition

n≤(1/10)N