8.1 Sampling Distributions LEARNING GOAL Understand the fundamental ideas of sampling distributions and how the distribution of sample means and the distribution of sample proportions are formed. Also learn the notation used to represent sample means and proportions. Page 334

Sample Means: The Basic Idea Consider the weights of the five starting players on a professional basketball team. We regard these five players as the entire population (with a mean of 242.4 pounds). Samples drawn from this population of five players can range in size from n = 1 (one player out of the five) to n = 5 (all five players). Page 334 With a sample size of n = 1: there are 5 different samples that could be selected: Each player is a sample. The mean of each sample of size n = 1 is simply the weight of the player in the sample. Slide 8.1- 2

The distribution of sample means created by this process A sampling Distribution: The distribution of sample means created by this process is an example of a sampling distribution. This term simply refers to a distribution of a sample statistic, such as a mean, taken from all possible samples of a particular size. Page 335 Figure 8.1 Sampling distribution for sample size n 1. Slide 8.1- 3

Q:What is the mean of the distribution of sample means? Page 335 This demonstrates a general rule: The mean of a distribution of sample means is the population mean. Slide 8.1- 4

With a sample size of n = 2: in which each sample consists of two different players. With five players, there are 10 different samples of size n = 2. Each sample has its own mean. Q: What is the mean of the distribution of sample means with sample size n=2? Page 335 Slide 8.1- 5

The mean of the distribution of sample means is equal to the population mean, 242.4 pounds.

Table 8.3 shows these samples and their means, and Figure Ten different samples of size n = 3 are possible in a population of five players. Table 8.3 shows these samples and their means, and Figure 8.3 shows the distribution of these sample means. Again, the mean of the distribution of sample means is equal to the population mean, 242.4 pounds. Page 336 Slide 8.1- 7

With a sample size of n = 4, only 5 different samples are possible. Table 8.4 shows these samples and their means, and Figure 8.4 shows the distribution of these sample means. Page 334 Slide 8.1- 8

To summarize, when we work with all possible samples of a Finally, for a population of five players, there is only 1 possible sample of size n = 5: the entire population. In this case, the distribution of sample means is just a single bar (Figure 8.5). Again the mean of the distribution of sample means is the population mean, 242.4 pounds. Figure 8.5 Sampling distribution for sample size n = 5. To summarize, when we work with all possible samples of a population of a given size, the mean of the distribution of sample means is always the population mean. Page 336 Slide 8.1- 9

Sample Means with Larger Populations In typical statistical applications, populations are huge and it is impractical or expensive to survey every individual in the population; consequently, we rarely know the true population mean, μ. Therefore, it makes sense to consider using the mean of a sample to estimate the mean of the entire population. Although a sample is easier to work with, it cannot possibly represent the entire population exactly. Therefore, we should not expect an estimate of the population mean obtained from a sample to be perfect. The error that we introduce by working with a sample is called the sampling error. Page 337 Slide 8.1- 10

Sampling Error The sampling error is the error introduced because a random sample is used to estimate a population parameter. It does not include other sources of error, such as those due to biased sampling, bad survey questions, or recording mistakes. Page 338 Slide 8.1- 11

Notation for Population and Sample Means n = sample size m = population mean x = sample mean ¯ Page 338 Slide 8.1- 12

The Distribution of Sample Means The distribution of sample means is the distribution that results when we find the means of all possible samples of a given size. The larger the sample size, the more closely this distribution approximates a normal distribution. In all cases, the mean of the distribution of sample means equals the population mean. If only one sample is available, its sample mean, x, is the best estimate for the population mean, m. Page 339 ¯ x Slide 8.1- 13

Page 339 Slide 8.1- 14

EXAMPLE Sampling Farms Texas has roughly 225,000 farms, more than any other state in the United States. The actual mean farm size is μ = 582 acres and the standard deviation is σ = 150 acres. For random samples of n = 100 farms, find the mean and standard deviation of the distribution of sample means. What is the probability of selecting a random sample of 100 farms with a mean greater than 600 acres? Page 340 Slide 8.1- 15

Notation for Population and Sample Proportions n = sample size p = population proportion p = sample proportion ˆ Page 341 Slide 8.1- 16

EXAMPLE: Sample and population Proportions. The college of Los Angeles had 2,444 students and 269 of them are left-handed. You conduct a survey of 50 students and find that 8 of them are left-handed. a . What is the population proportion of left-handed students? b. What is the sample proportion of left-handed students? c. Does your sample appear to be representative of the college?

The Distribution of Sample Proportions The distribution of sample proportions is the distribution that results when we find the proportions ( ) in all possible samples of a given size. The larger the sample size, the more closely this distribution approximates a normal distribution. In all cases, the mean of the distribution of sample proportions equals the population proportion. If only one sample is available, its sample proportion, , is the best estimate for the population proportion, p. ˆ p Page 342 ˆ p Slide 8.1- 18