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AP Statistics Chapter 9 - Sampling Distributions

9.1: Sampling Distributions

PARAMETER, STATISTIC
A parameter is a number that describes the population. A parameter is a fixed number, but in practice we do nopt know its value because we cannot examine the entire population.

A statistic is a number that describes a sample. The value of a statistic is known when we have taken a sample, but it can change from sample to sample. We often use a sample to estimate an unknown parameter.

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

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

VARIABILITY OF A STATISTIC
The variability of a statistic is described by the spread of its samping distribution. This spread is determined by the sampling design and the size of the sample. Larger samples give smaller spread.


9.2: Sample Proportions

SAMPLING DISTRIBUTION OF A SAMPLE PROPORTION ()

Choose an SRS of size n from a large population with population proportion p having some characteristic of interest.
Let be the proportion of the sample having that characteristic. Then:

  • The mean of the sampling distribution is exactly p.
  • The standard deviation of the sampling distribution is

RULE OF THUMB 1
Use the formula for the standard deviation of p-hat ony when the size of the population is at least 10 times as large as the sample size.

RULE OF THUMB 2
We will use the normal approximation to the sampling distribution of for values of n and p that satisfy np>=10 and n(1-p)>=10.

9.3: Sample Means

SAMPLING DISTRIBUTION OF A SAMPLE MEAN FROM A NORMAL POPULATION

Draw an SRS of size n from a population that has the normal distribution with mean and standard deviation. Then the sample mean has a normal distribution for any sample size, with

  • The mean of the sampling distribution = .
  • The standard deviation of the sampling distribution =


THE CENTRAL LIMIT THEOREM

Draw an SRS of size n from any population whatsoever with mean and standard deviation. When n is large (n=30 or greater), the sampling distribution of the sample mean is close to the normal distribution with

  • The mean = .
  • The standard deviation =


 

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