6:59 0 A Oil . 84 M2L3 (1) (b) A probability distribution describing the population of 15 sample means: the sampling distribution of the sample mean Sample Mean Frequency Probability 29.5 1/15 30 1/15 30.5 2/15 31 WNN 2/15 31.5 3/15 32 2/15 NN 32.5 2/15 33 1/15 33.5 1/15 In order to find the probability distribution of the population of sample means different sample means correspond to different numbers of samples. For because the sample mean of 31 mpg comesponds to 2 out of 15 samples-the s 33) and the sample (30, 32)-the probability of obtaining a sample mean of 2/15. If we analyze all of the sample means in a similar fashion, we fin probability distribution of the population of sample means is as given in Table distribution is the sampling distribution of the sample mean. A graph of this is shown in Figure 1(b) and illustrates the accuracies of the different possib means as point estimates of the population mean. For example, whereas 3 sample means exactly equal the population mean of 31.5 mpg, other sample m from the population mean by amounts varying from.5 mpg to 2 mpg As illustrated in Example 1, one of the purposes of the sampling distributi sample mean is to tell us how accurate the sample mean is likely to be as a poi of the population mean. Because the population of six individual car m Example 1 is small, we were able (after the auto shows were over) to test al determine the values of the six car mileages, and calculate the population ma Often, however, the population of individual measurements under consider i large-either a large finite population or an infinite population. In this case, it would be impractical or impossible to determine the values population measurements and calculate the population mean. Instead, we ran a sample of individual measurements from the population and use the ne sample as the point estimate of the population mean. Moreover, although t impractical or impossible to list the many different possible sample means til a obtained if the sampled population is very large, statisticians know variou: properties about the sampling distribution of these sample means. ACTIVITY: Suppose that we will take a random sample of size n from a population having and standard deviation o. For each of the following situations, find the mean of sampling distribution of the sample mean x: a u = 10, o= 2, n =25 u = 500, 0 = .5, n = 100 C u=3, 0= .1, n=4 d u = 100, o = 1, n = 1,600 O