Introduction to Sampling Distributions Chapter 6 Introduction to Sampling Distributions

Chapter 6 - Chapter Outcomes After studying the material in this chapter, you should be able to: • Understand the concept of sampling error. • Determine the mean and standard deviation for the sampling distribution of the sample mean.

Chapter 6 - Chapter Outcomes (continued) After studying the material in this chapter, you should be able to: • Determine the mean and standard deviation for the sampling distribution of the sample proportion. • Understand the importance of the Central Limit Theorem. • Apply the sampling distributions for both the mean and proportion.

SAMPLING ERROR-SINGLE MEAN The difference between a value (a statistic) computed from a sample and the corresponding value (a parameter) computed from a population. Where:

Sampling Error -Parameters v. Statistics- • A parameter is a measure computed from the entire population • A statistic is a measure computed from a sample that has been selected from a population.

Sampling Error POPULATION MEAN Where:  = Population mean x = Values in the population N = Population size

Sampling Error (Example 6.1) If  = 158,972 square feet and a sample of n = 5 shopping centers yields = 155,072 square feet, then the sampling error would be:

Sampling Errors Useful Fundamental Statistical Concepts: • The size of the sampling error depends on which sample is taken. • The sampling error may be positive or negative. • There is potentially a different value for each possible sample mean.

Sampling Error A simple random sample is a sample selected in such a manner that each possible sample of a given size has an equal chance of being selected.

Sampling Error SAMPLE MEAN Where: = Sample mean x = Sample value selected from the population n = Sample size

POPULATION PROPORTION Sampling Errors POPULATION PROPORTION Where:  = Population proportion x = Number of items having the attribute N = Population size

Sampling Errors SAMPLE PROPORTION Where: p = Sample proportion x = Number of items in the sample having the attribute n = sample size

SINGLE PROPORTION SAMPLING ERROR Where:

Sampling Distributions A sampling distribution is a distribution of the possible values of a statistic for a given size sample selected from a population.

Sampling Distribution of the Mean THEOREM 6-1 If a population is normally distributed with a mean  and a standard deviation , the sampling distribution of the sample mean is also normally distributed with a mean equal to the population mean and a standard deviation equal to the population standard deviation divided by the square-root of the sample size .

Sampling Distribution of the Mean THEOREM 6-2: THE CENTRAL LIMIT THEREOM For random samples of n observations taken from a population with mean  and standard deviation , regardless of the population’s distribution, provided the sample size is sufficiently large, the distribution of the sample mean , will be normal with a mean equal to the population mean . Further, the standard deviation will equal the population standard deviation divided by the square-root of the sample size . The larger the sample size, the better the approximation to the normal distribution.

Sampling Distribution of the Mean z-VALUE FOR SAMPLING DISTRIBUTION OF where: = Sample mean = Population mean = Population standard deviation n = Sample size

Example of Calculation z-Value for the Sample Mean (Example 6-5) What is the probability that a sample of 100 automobile insurance claim files will yield an average claim of $4,527.77 or less if the average claim for the population is $4,560 with standard deviation of $600?

Sampling Distribution of a Proportion SAMPLING DISTRIBUTION OF p and where: = Population proportion = Sample proportion n = Sample size

Sampling Distribution of the Mean z-VALUE FOR PROPORTIONS where: z = Number of standard errors p is from p = Sample proportion = Standard error of the sampling distribution Mean of sample proportions

Example of Calculation z-Value for Proportion (Example 6-6) What is the probability that a sample of 500 units will contain 18% or more broken items given that observation over time has shown 15% of shipments damaged?

Key Terms • Central Limit Theorem • Finite Population Correction Factor • Parameter • Population Proportion • Sample Proportion • Sampling Distribution • Sampling Error • Simple Random Sample • Statistic • Theorem 6-1