The Statistical Imagination Chapter 7. Using Probability Theory to Produce Sampling Distributions

Estimating the Parameters of a Population Point estimate – a statistic provided without indicating a range of error Point estimates are limited because a calculation made for sample data is only an estimate of a population parameter. This is apparent when different results are found with repeated sampling

Repeated Sampling Repeated sampling refers to the procedure of drawing a sample and computing its statistic, and then drawing a second sample, a third, a fourth, and so on Repeated sampling reveals the nature of sampling error An illustration of repeated sampling is presented in Figure 7-1 in the text

Symbols Sample statistics are usually noted with English letters Population parameters are usually noted with Greek letters

What Repeated Sampling Reveals 1.A given sample’s statistic will be slightly off from the true value of its population’s parameter due to sampling error 2.Sampling error is patterned, systematic and predictable 3.Sampling variability is mathematically predictable from probability curves called sampling distributions 4.The larger the sample size, the smaller the range of error

A Sampling Distribution A mathematical description of all possible sampling event outcomes and the probability of each one Sampling distributions are obtained from repeated sampling Sampling distributions are probability curves; they tell us the probability of occurrence of any sample outcome

A Sampling Distribution of Means A sampling distribution of means describes all possible sampling event outcomes and the probability of each outcome when means are repeatedly calculated on an infinite number of samples It answers the question: What would happen if we repeatedly sampled a population using a sample size of n, calculated each sample mean, and plotted it on a histogram?

Features of a Sampling Distribution of Means A sampling distribution of means is illustrated in the text in Figure 7-3. It reveals that for an interval/ratio variable, means calculated from a repeatedly sampled population calculate to similar values which cluster around the value of the population parameter A sampling distribution of means will be approximately normal with a mean equal to the actual population mean

The Standard Error The standard error is the standard deviation of a sampling distribution It measures the spread of sampling error that occurs when a population is sampled repeatedly Rather than repeatedly sample, we estimate standard errors using the sample standard deviation of a single sample

The Law of Large Numbers The law of large numbers states that the larger the sample size, the smaller the standard error of the sampling distribution The relationship between sample size and sampling error is apparent in the formula for the standard error of the mean; a large n in the denominator produces a small quotient

The Central Limit Theorem The central limit theorem states that regardless of the shape of the raw score distribution of an interval/ratio variable, the sampling distribution of means will be approximately normal in shape This is illustrated in the text in Figure 7-10

The Student’s t Sampling Distribution The sampling distribution curve used with especially small samples and/or when the standard error is estimated is called Student’s t The t-distribution is an approximately normal distribution

The t-distribution of Means For a sampling distribution of means, when the sample size is below 120, the probability curve begins to flatten into a t-distribution See Figure 7-7 in the text

Features of the t-distribution Standardized scores for the t-distribution are called t-scores and are computed just as are Z- scores The t-distribution, like the Z-distribution of the normal curve, allows us to calculate probabilities

The t-distribution Table The t-distribution table (Appendix B, Statistical Table C) is organized differently from the normal curve table See Table 7-1 in the text The t-distribution table provides t-scores for only the critical probabilities of.05,.01, and.001; that is, this table provides “critical t-values”

Degrees of Freedom Degrees of freedom ( df ) are the number of opportunities in sampling to compensate for limitations, distortions, and potential weaknesses in statistical procedures Use of the t-distribution table requires the calculation of degrees of freedom

More on Degrees of Freedom From repeated sampling we know that any single sample is only an estimate. An estimate can be distorted by limitations of the statistical procedures used to obtain it E.g., the mean is influenced by outliers The larger the sample, the greater the opportunity for an outlier to be neutralized in such a way as to not distort a sample mean A small sample is especially vulnerable to outliers

Sampling Distributions for Nominal Variables A sampling distribution of proportions is approximately normal and the t-distribution is used to obtain critical values The larger the sample size, the smaller the range of error The standard error is estimated using the probabilities of success and failure in a sample

Features of a Sampling Distri- bution for Nominal Variables The mean of a sampling distribution of proportions is equal to the probability of success in the population A sampling distribution of proportions will be approximately normal when the smaller parameter (the probability of success or failure in the population) multiplied by the sample size is greater than or equal to 5