AP Stats Ch 7
Summary of 7.1
A parameter is a number that describes a population. To estimate an unknown parameter, use a statistic calculated from a sample. The population distribution of a variable describes the values of the variable for all individuals in a population. The sampling distribution of a statistic describes the values of the statistic in all possible samples of the same size from the same population. Don't confuse the sampling distribution with a distribution of sample data, which gives the values of the variable for all individuals in a sample. A statistic can be an unbiased estimator or a biased estimator of a parameter. Bias means that the center (mean) of the sampling distribution is not equal to the true value of the parameter. The variability of a statistic is described by the spread of its sampling distribution. Larger samples give smaller spread. When trying to estimate a parameter, choose a statistic with low or no bias and minimum variability. Don't forget to consider the shape of the sampling distribution before doing inference.
Parameter
A parameter is 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
Statistic
A statistic is 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.
Unbiased 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.
Central limit theorem (CLT)
Draw an SRS of size n from any population with mean μ and finite standard deviation σ. The central limit theorem (CLT) says that when n is large, the sampling distribution of the sample mean X is approximately Normal.
Sampling distribution
The sampling distribution of a statistic is the distribution of values taken by the statistic in all possible samples of the same size from the same population.
Summary of 7.3
When we want information about the population mean μ for some variable, we often take an SRS and use the sample mean X to estimate the unknown parameter μ. The sampling distribution of X describes how the statistic X varies in all possible samples of the same size from the population. The mean of the sampling distribution is μ, so that X is an unbiased estimator of μ. The standard deviation of the sampling distribution of X is for an SRS of size n if the population has standard deviation σ. That is, averages are less variable than individual observations. This formula can be used if the population is at least 10 times as large as the sample (10% condition). Choose an SRS of size n from a population with mean μ and standard deviation σ. If the population distribution is Normal, then so is the sampling distribution of the sample mean X. If the population distribution is not Normal, the central limit theorem (CLT) states that when n is large, the sampling distribution of X is approximately Normal. We can use a Normal distribution to calculate approximate probabilities for events involving X whenever the Normal condition is met: If the population distribution is Normal, so is the sampling distribution of X. If n ≥ 30, the CLT tells us that the sampling distribution of X will be approximately Normal in most cases.
Summary of 7.2
When we want information about the population proportion p of successes, we often take an SRS and use the sample proportion ôop to estimate the unknown parameter p. The sampling distribution of ôop describes how the statistic varies in all possible samples from the population. The mean of the sampling distribution of ôop is equal to the population proportion p. That is, ôop is an unbiased estimator of p. The standard deviation of the sampling distribution of is for an SRS of size n. This formula can be used if the population is at least 10 times as large as the sample (the 10% condition). The standard deviation of ôop gets smaller as the sample size n gets larger. Because of the square root, a sample four times larger is needed to cut the standard deviation in half. When the sample size n is large, the sampling distribution of ôop is close to a Normal distribution with mean p and standard deviation . In practice, use this Normal approximation when both np ≥ 10 and n(1 − p) ≥ 10 (the Normal condition).