Sampling Distribution

• Statistical inference is concerned with generalizations and predictions.
• Based on the opinions of several people interviewed on the street, that in a forthcoming election 60% of the eligible voters in the city of Detroit favour a certain candidate.
• If we repeat the sampling, we would expect to obtain a different value for the sample mean.
• Therefore, like other random variables, the sample mean $\bar{X}$, possesses a probability distribution, which is more commonly called the sampling distribution of $\bar{X}$.
• Question: A company manufactures 100 Ohms resistors. A sample of 40 resistors from the assembly line is found to have a mean of 105 Ohms.
• How likely is the population mean (the mean of the probability density function) to be 100 Ohms?

• Answer: In questions like this, we need to make inferences about the population mean based on the sample mean.
• To do this, we need to know the probability distribution of the sample mean.
• Definition 8.10:
  
• Sampling Error: The difference between the sample statistic and the value of the corresponding population parameter.
  • For the sample mean, the sampling error $= \vert \bar{x} - \mu\vert$ . This is controllable by taking more $n$.
• Nonsampling Error: Human error. The error occurs while we collect, record or tabulate the data.
• The sampling distribution of a statistic depends on
  • the size of the population,
  • the size of the samples,
  • the method of choosing the samples.