Statistics Chapter 7

Normal Condition for Sample Means

-If the population distribution is Normal, then so is the sampling distribution of x-bar. This is true no matter what the same size n is. -If the population distribution is not Normal, the central limit theorem tells us that the sampling distribution of x-bar will be approximately Normal in most cases if n≥30.

Central Limit Theorem

Draw an SRS of size n from any population with mean µ and finite standard deviation σ. The central limit theorem (CLT) states that when n is large, the sampling distribution of the sample mean x-bar 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 sample population.

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. Examples 1. The sample proportion from an SRS is always an unbiased estimator of the population proportion. 2. The sample mean is an unbiased estimator of the population mean. 3. The sample range is a biased estimator of the population range. The range of the sample tends to be much lower, on average, than the population range. 4. The same median is a biased estimator if the shape of the distribution is not symmetrical. Tidbit: Unfortunately, using an unbiased estimator doesn't guarantee that the value of your statistic will be close to the actual parameter value.

Bias, Variability, and Shape

Both bias and variability describe what happens when we take many shots at the target. Bias means that our aim is off and we consistently miss the bull's eye in the same direction. our sample values do not center on the population value. High variability means that repeated shots are widely scattered on the target. Ideally, we'd like our estimates to be accurate (unbiased) and precise (have low variability). Tidbit: Taking a larger sample doesn't fix bias. Remember that even a very large voluntary response sample or convenience sample is worthless because of bias.

Sampling Distribution of a Sample Proportion (p-hat)

Shape: In some cases, the sampling distribution of sample proportion can be approximated by a Normal curve. This seems to depend on both the sample size n and the population proportion p. Center: The mean of the distribution is µ (sub p-hat)=p. This makes sense because the sample proportion is an unbiased estimator of p. Spread: For a specific value of p, the standard deviation of p-hat gets smaller as n gets larger. The value of the standard deviation of p-hat depends on both n and p. Chose an SRS of size n from a population of size N with proportion p of successes. Let p-hat be the sample proportion of successes. Then: -The mean of the sampling distribution of p-hat is: µ(sub p-hat)=p. -The standard deviation of the sampling distribution of p-hat is shown in the diagram (derived from binomial distribution). It is true as long as the 10% condition is satisfied: n≤0.1N. Since the sample size n is under the square root sign, we'd have to take a sample four times as large to cut the standard deviation in half. -As n increases, the sampling distribution of p-hat becomes approximately normal. Before you perform Normal calculations, check that the Normal condition is satisfied: np≥10 and n(1-p)≥10.

Sampling Distribution of a Sample Mean from a Normal Distribution

Suppose that a population is Normally distribution with mean µ and standard deviation σ. Then the sampling distribution of x-bar has the Normal distribution with mean µ and standard deviation σ in the diagram, provided that the 10% condition is met.

Mean and Standard Deviation of the Sampling Distribution of x-bar (Sample Mean)

Suppose that x-bar is the mean of an SRS of size n drawn from a large population with mean µ and standard deviation σ. Then: -The mean of the sampling distribution is the same as the mean of the population (the parameter). -The standard deviation of the sampling distribution is shown in the diagram and is true as long as the 10% condition is satisfied. -The values of x-bar are less spread out for larger samples. Their standard deviation decreases at the rate of square-root-n, so you must take a sample four times as large to cut the standard deviation of x-bar in half. Tidbit: These facts are true no matter what shape the population distribution has. The shape of the sampling distribution, however, depends on the shape of the population distribution.

Population Distribution

The population distribution gives the values of the variable for all the individuals in the population. The population distribution and the distribution of sample data describe individuals. A sampling distribution describes how a statistic varies in many samples from the population.

Sampling Variability

The value of a statistic varies in repeated random sampling. 1. Take a large number of samples from the same population. 2. Calculate the statistic (like the sample mean or the sample proportion) for each sample. 3. Make a graph of the values of the statistic. 4. Examine the distribution displayed in the graph for shape, center, and spread, as well as outliers or other deviations.

Variability of a Statistic

The variability of a statistic is described by the spread of its sampling distribution. This 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.