Section 5.1 Toward Statistical Inference:

Parameters and Statistics:

• A parameter is a number that describes some characteristic of the population.

• In statistical practice, the value of a parameter is not known because we cannot examine the entire population.

• 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, but it can change from sample to sample.

• We often use a statistic to estimate an unknown parameter.

• Remember s and p: statistics come from samples, and parameters come from populations.

  
  

Notation for Parameters and Statistics:

• Population:
  • μ (the Greek letter mu): population mean
  • σ (Greek letter sigma): population standard deviation

• Sample:
  •̅ 𝑥 (we say x-bar): sample mean
  • s: sample standard deviation

  
  

Statistical Estimation:

• The process of statistical inference involves using information from a sample to draw conclusions about a wider population.

• Different random samples yield different statistics. We need to be able to describe the Sampling Variability of the possible values of a statistic in order to perform statistical inference.

• The sampling distribution of a statistic consists of all possible values of the statistic and the relative frequency with which each value occurs. We may plot this distribution using a histogram, just as we plotted a histogram to display the distribution of data in Chapter 1

Sampling Variability:

• Sampling variability is a term used for the fact that the value of a statistic varies in repeated random sampling

• To make sense of sampling variability, we ask, “What would happen if
  we took many samples?”

Sampling Distributions:

• 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

• In practice, it is difficult to take all possible samples of size n to obtain the actual sampling distribution of a statistic. Instead, we can use simulation to imitate the process of taking many, many samples.

  
  

Bias and Variability:

• Bias concerns the center of the sampling distribution.

  • A statistic used to estimate a parameter is unbiased if the mean of its sampling distribution is equal to the true value of the parameter being estimated

• The variability of a statistic is the spread of its sampling distribution

  • It is determined by the sampling design and the sample size n

• Statistics from larger samples have smaller spreads.

  
  

Managing Bias and Variability:

• To reduce bias, use random sampling

• To reduce the variability of a statistic from an SRS, use a larger sample

• The variability of a statistic from a random sample does not depend on the size of the population, as long as the population is at least 20 times larger than the sample.

  
  

Why Randomize?

• The process of drawing conclusions about a population on the basis of sample data is called inference

• 1. Random sampling eliminates bias in selecting samples from the list of available individuals.

• 2. The laws of probability allow trustworthy inference about the population

  • Results from random samples come with a margin of error that sets bounds on the size of the likely error

  • Larger random samples give better information about the population than smaller samples.