Module 6 - Sampling Distributions

Descriptive statistics

Describes the quantifiable characteristics of a sample for each measurement variable. Each measurement has its own set of descriptive statistics which are labeled using the latin alphabet (p, m, s).

Population parameters

Are any quantifiable characteristic of a statistical population, describe attributes of the stat. population. Ie., if studying first year students from U of T on mean hours of sleep, the mean number of hours of sleep of all students in the statistical population is a population parameter

Key elements of population parameters

Population parameters are any quantifiable characteristic of a statistical population.

Each measurement variable has its own set of population parameters.

Population parameters are labeled using the greek alphabet, such as μ for the mean or σ for the standard deviation.

Population parameters values are fixed.

Estimation

The process by which the descriptive statistics of a sample become the population parameters of the statistical population. Sample descriptive statistics provide an estimate of the population parameter. First step of inferential statistics

Sampling distribution

The probability distribution of a descriptive statistic that would emerge if a statistical population was sampled repeatedly a large number of times. Critical, allow us to make statements about things that have not yet been seen. The machinery that gives us statistical inference

Characteristics of a sampling distribution

1. The shape of a sampling distribution is independent of the statistical population so long as the sample size is sufficiently large.

2. The variance of a sampling distribution increases as the number of sampling units in the sample decreases.

Shape independence of a sampling distribution

As long as the sample size is sufficiently large, the shape is not influenced by the shape of the statistical population. Taking the mean of multiple sampling units has the effect of averaging over asymmetries in the statistical population

Variance depends on sample size in a sampling distribution

As the sample increases, the variance of the sampling distribution decreases. The larger the sample size, the more accurate the estimate of the mean and the less it varies among repeated samples

Central limit theorem

Allows us to calculate the mean and standard deviation of the sampling distribution without the need to construct it. Formal development of the two principles of sampling distributions

Shape independence with the central limit theorem

The sampling distribution tends towards a normal distribution as the sample size gets larger. The mean of a sampling distribution is the same as the mean of the statistical population.

Standard error

The standard deviation of the sampling distribution. As the number of sampling units increases, variation (and standard error) decrease.

SE=σ/√n

As long as the sample size is large enough, the sampling distribution will:

• be a normal distribution

• have a mean equal to the statistical population

• have a standard error given by SE=σ/√n, where σ is the standard deviation of the statistical

connects the statistical population and sampling distribution with the information found in a sample, allows us to link together all of the logical connections needed for making statistical inference.

Chain of inference

A sample of random ___ with mean m and standard deviation s provides information to estimate the parameters of the statistical population (mu and σ) and sampling distribution (mu and SE). By taking one sample, can make claims about other samples.

T-distribution

Looks like a normal distribution, but has a shape that depends on the sample size (thus sample size influences the t-distribution). When sample size is small, t-distribution has fatter tails to account for uncertainty in estimating the standard deviation of the statistical population. Consequence of the inference chain.

Confidence intervals

Derived from sampling distributions — describe the range over the x-axis of a sampling distribution bracketing a certain probability of where new samples may be. Centre-most probability of interest.

Conversion to standardization

t=(x-m)/SE

Calculation of confidence intervals

Describe the uncertainty in the descriptive statistics of a sample, are derived from sampling distributions.

• Step 1: find the intervals on the standradized scale

  • Software or statistical scale

• Step 2: convert to a raw scale

  • Reverses the standardization transformation

  • x_U=m+(t_U*SE) → uses either upper t-score or lower t-score

  • If p=0.95, you find the range that encompasses 95% of the data

Interpreting confidence intervals

Convey the level of uncertainty about a descriptive statistic from a sample, are an estimate (as they are calculated from the descriptive statistics of a sample), and provide a range for where values should fall most of the time — how much a descriptive statistic can vary due to sampling error alone.