+ The Practice of Statistics, 4 th edition – For AP* STARNES, YATES, MOORE Chapter 7: Sampling Distributions Section 7.3 Sample Means

+ Sample proportions arise most often when we are interested in categoricalvariables. When we record quantitative variables we are interested in otherstatistics such as the median or mean or standard deviation of the variable.Sample means are among the most common statistics.Consider the mean household earnings for samples of size 100. Compare thepopulation distribution on the left with the sampling distribution on the right.What do you notice about the shape, center, and spread of each?

+ Note: These facts about the mean and standard deviation of are true no matter what shape the population distribution has. The Sampling Distribution ofWhen we choose many SRSs from a population, the sampling distribution of thesample mean is centered at the population mean µ and is less spread out than the population distribution. Here are the facts. as long as the 10% condition is satisfied: n ≤ (1/10)N. Mean and Standard Deviation of the Sampling Distribution of Sample Means

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+ Example 1: Sulfur compounds such as dimethyl sulfide (DMS) are sometimes present in wine. DMS causes “off-odors” in wine, so winemakers want to know the odor threshold, the lowest concentration of DMS that the human nose can detect. Extensive studies have found that the DMS odor threshold of adults follows roughly a Normal distribution with mean μ = 25 micrograms per liter and standard deviation σ = 7 micrograms per liter. Suppose we take an SRS of 10 adults and determine the mean odor threshold for the individuals in the sample. Since is an unbiased estimator of micrograms per liter. The standard deviation is since there are at least 10(10) = 100 adults in the population.

+ Sampling from a Normal Population Sampling Distribution of a Sample Mean from a Normal Population

+ Example 2: The height of young women follows a Normal distribution with mean μ = 64.5 inches and standard deviation σ = 2.5 inches. a) Find the probability that a randomly selected young woman is taller than 66.5 inches. Show your work. State: We want to find the probability that a randomly selected young woman is taller than 66.5 inches. Let X = the height of a randomly selected young woman. Plan: X is N(64.5, 2.5). Draw a picture. DO: OR P(X > 66.5) = Normalcdf(66.5, , 64.5, 2.5) = Conclude: The probability of choosing a young woman at random whose height exceeds 66.5 inches is about 0.21.

+ Example 2: The height of young women follows a Normal distribution with mean μ = 64.5 inches and standard deviation σ = 2.5 inches. b) Find the probability that the mean height of an SRS of 10 young women exceeds 66.5 inches. Show your work. State: We want to find the probability that the mean height of an SRS of 10 young women exceeds 66.5 inches. Let = the mean height of an SRS of 10 young women. Plan: For an SRS of 10 young women, the sampling distribution of their sample mean height will have a mean and standard deviation The 10% condition is met since there are at least 10(10) = 100 young women in the population. Do: Since the population distribution is Normal, the sampling distribution will follow an N(64.5, 0.79) distribution. OR Normalcdf(66.5, , 64.5, 0.79) = 0 Conclude: It is very unlikely (less than a 1% chance) that we would choose an SRS of 10 young women whose average height exceeds 66.5 inches.

+ The fact that averages of several observations are less variable than individual observations is important in many settings. For example, it is common practice to repeat a measurement several times and report the average of the results. Think of the results of n repeated measurements as an SRS from the population of outcomes we would get if we repeated the measurement forever. The average of the n results (the sample mean ) is less variable than a single measurement.