PROBABILITY AND SAMPLES: THE DISTRIBUTION OF SAMPLE MEANS

OVERVIEW DEFINITION: DEFINITION: Sampling error is the discrepancy, or amount of error, between a sample statistic and its corresponding population parameter. Sampling error is the discrepancy, or amount of error, between a sample statistic and its corresponding population parameter.

THE DISTRIBUTION OF SAMPLE MEAN DEFINITION: DEFINITION: The distribution of sample means is the collection of sample means for all the possible random samples of particular size (n) that can be obtained from a population. The distribution of sample means is the collection of sample means for all the possible random samples of particular size (n) that can be obtained from a population.

THE DISTRIBUTION OF SAMPLE MEAN DEFINITION: DEFINITION: A sampling distribution is a distribution of statistics obtained by selecting all the possible samples of a specific size from a population. A sampling distribution is a distribution of statistics obtained by selecting all the possible samples of a specific size from a population.

THE CENTRAL LIMIT THEOREM Central Limit Theorem : for any population with mean μ and standard deviation σ, the distribution of sample means for sample size n will have a mean of μ and a standard deviation of σ/ n, and will approach a normal distribution as n approaches infinity. Central Limit Theorem : for any population with mean μ and standard deviation σ, the distribution of sample means for sample size n will have a mean of μ and a standard deviation of σ/ n, and will approach a normal distribution as n approaches infinity.

THE SHAPE OF THE DISTRIBUTION OF SAMPLE MEANS 1- The population from which the samples are selected is a normal distribution. 1- The population from which the samples are selected is a normal distribution. 2- The number of scores (n) in each sample is relatively large, around 30 or more. 2- The number of scores (n) in each sample is relatively large, around 30 or more.

THE STANDARD ERROR OF X DEFINITION: DEFINITION: The standard deviation of the distribution of sample means is called the standard error of X. The standard error measures the standard amount of difference between X and μ that is reasonable to expect simply by chance. The standard deviation of the distribution of sample means is called the standard error of X. The standard error measures the standard amount of difference between X and μ that is reasonable to expect simply by chance. Standard error = σ X = standard distance between X and μ. Standard error = σ X = standard distance between X and μ.

THE STANDARD ERROR OF X The sample size. The sample size. DEFINITION: The law of large numbers states that the larger the sample size (n), the more probable it is that the sample mean will be close to the population mean. DEFINITION: The law of large numbers states that the larger the sample size (n), the more probable it is that the sample mean will be close to the population mean. The population standard deviation. The population standard deviation. Standard error = σ X = σ Standard error = σ X = σ n

PROBABLITY AND THE DISTRIBUTION OF SAMPLE MEANS FIGURE 7.3 FIGURE 7.3 The distribution of sample means for n=25. Samples were selected from a normal population with μ = 500 and σ = 100. The distribution of sample means for n=25. Samples were selected from a normal population with μ = 500 and σ = 100.

MORE ABOUT STANDARD ERROR 1- Sampling error. 1- Sampling error. 2- Standard error. 2- Standard error. FIGURE 7.5 FIGURE 7.5 An example of typical distribution of sample means. Each of the small boxes represents the mean obtained for one sample. An example of typical distribution of sample means. Each of the small boxes represents the mean obtained for one sample.