6-3 Sampling Distributions and Estimators 275 Sampling Distribution of the Sample Mean We now consider sample means. DEFINITION The sampling distribution of the sample mean is the distribution of all possible sample means (or the distribution of the variable x), with all samples having the same sample size n taken from the same population. (The sampling distribution of the sample mean is typically represented as a probability distribution in the format of a probability histogram, formula, or table.) Proportions Sample 1 Sample 2 Sample 3 (Population Proportion is p 5 0.5) Distribution of Sample Proportions Sample Proportions Sample p 5 0.5 Sample proportions are approximately normal • • • 0.2 0.4 0.8 Sampling Procedure: Roll a die 5 times and find the proportion p of odd numbers for each sample. ˆ FIGURE 6-18 Sample Proportions from 10,000 Trials Sampling Distribution of the Sample Proportion EXAMPLE 1 Consider repeating this process: Roll a die 5 times and find the proportion of odd numbers (1 or 3 or 5). What do we know about the behavior of all sample proportions that are generated as this process continues indefinitely? YOUR TURN. Do Exercise 10 “Sampling Distribution of the Sample Proportion.” SOLUTION Figure 6-18 illustrates a process of rolling a die 5 times and finding the proportion of odd numbers. (Figure 6-18 shows results from repeating this process 10,000 times, but the true sampling distribution of the sample proportion involves repeating the process indefinitely.) Figure 6-18 shows that the sample proportions are approximately normally distributed. (Because the values of 1, 2, 3, 4, 5, 6 are all equally likely, the proportion of odd numbers in the population is 0.5, and Figure 6-18 shows that the sample proportions have a mean of 0.50.) Behavior of Sample Means 1. The distribution of sample means tends to be a normal distribution. (This will be discussed further in the following section, but the distribution tends to become closer to a normal distribution as the sample size increases.) 2. The sample means target the value of the population mean. (That is, the mean of the sample means is the population mean. The expected value of the sample mean is equal to the population mean.) Go Figure 90%: Percentage of the world’s data that have been generated in the past two years (according to Science Daily).
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