Topcis

1. What is a Sampling Distrubution?
   • A parameter is a number that describes a population, a statistic is a number that describes some characteristic of a sample
     • Statistics come from samples, parameters come from populations
   • The population distribution of a variable describes the values for all individuals in the population. The sampling distribution describes the values of the statistic in all possible samples of size n from the population.
     • The distribution of sample data shows a individual sample of the population. All of the distributions of sample data combined make up the sampling distribution.
   • A statistic used to estimate a parameter is an unbiased estimator if the mean of its sampling distribution is equal to the value of the parameter being estimated. If the mean is not equal, it is a biased estimator
   • The variability of a statistic describes the spread of the sampling distribution.
     • Larger samples give smaller spread
   • When we are estimating a parameter, the statistic we choose should have low or no bias and minimum variability.
2. Sample Proportions
   • The sampling distribution of $\hat{p}$ describes how the sample proportion varies in all possible samples from the population
   • The mean of the sampling distribution of $\hat{p}$ is equal to the population proportion p. This means $\hat{p}$ is an unbiased estimator of p
   • The standard deviation of the sampling distribution of $\hat{p}$ is $\sqrt{p(1-p)/n}$
     • This formula can only be used if the population is at least x10 times as large as the sample (the 10% condition)
     • The standard deviation of $\hat{p}$ gets smaller as the sample size n gets larger
   • We can only assume it is a normal approximation if $np \geq 10$ and $n(1-p) \geq 10$ (Large Counts condition)
3. Sample Means
   • The sampling distribution of $\bar{x}$ describes how the statistic x varies in all possible samples of size n from the population
   • The mean of the sampling distribution is $\mu$, so $\bar{x}$ is an unbiased estimator of $\mu$
   • The standard deviation of the sampling distribution of $\bar{x}$ is $\frac{\sigma}{\sqrt{n}}$
     • This formula can only be used if the population is at least x10 times as large as the sample (the 10% condition)
   • If the population distribution is normal, so is the sampling distribution of the sample mean $\bar{x}$. If its not normal, the central limit theorem (CLT) states that when n is large, the sampling distribution is approximately Normal.
     • $n \geq 30$

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