CHAPTER 9.3 Sampling Means AP Exam Tip: Notation matters. The symbols π, π, π, π, π, ππ, ππ, ππ, ππ all have specific and different meanings. Either use notation correctly or donβt use it at all. You can expect to lose credit if you use incorrect notation. MEAN AND STANDARD DEVIATION OF A SAMPLE MEAN β’ Suppose π₯ is the mean of an SRS size π from a large population with µ and π. β’ Then the mean of the sampling distribution of π₯ is ππ₯ = π π and its standard deviation is ππ₯ = π β’ Sample mean π₯ is an unbiased estimator of the population mean π β’ Values of π₯ are less spread our for larger samples π π β’ Only use ππ₯ = when the population is at least 10 times as large as the sample or π β₯ 10π Sulfur compounds such as dimethyl sulfide (DMS) are sometimes present in wine. DMS causes βoff-odorsβ in wine, so winemakers want to know the odor threshold, the lowest concentration of DMS that the human nose can detect. THIS WINE STINKS EXAMPLE 1 Extensive studies have found that the DMS odor threshold of adults follows roughly a Normal distribution with mean π = 25 micrograms per liter and standard deviation π = 7 micrograms per liter. Suppose we take an SRS of 10 adults and determine the mean odor threshold π₯ for the individuals in the sample. β’ What is the mean of the sampling distribution of π₯? β’ What is the standard deviation of the sampling distribution of π₯? SAMPLING DISTRIBUTION OF A SAMPLE MEAN FROM A NORMAL POPULATION β’ Draw an SRS of size π from a population that has the normal distribution with mean µ and standard deviation π. β’ Then the sample mean π₯ has the normal distribution with mean µ and standard deviation π . π YOUNG WOMENβS HEIGHTS The height of young women follows a Normal distribution with π = 64.5 inches and π = 2.5 inches. a. Find the probability that a randomly selected young woman is taller than 66.5 inches. b. Find the probability that the mean height of an SRS of 10 young women exceeds 66.5 inches. SAMPLING DISTRIBUTION OF THE MEAN HEIGHT π₯ FOR SRS OF 10 YOUNG WOMEN COMPARED WITH THE POPULATION DISTRIBUTION OF YOUNG WOMENβS HEIGHTS CENTRAL LIMIT THEOREM β’ Draw an SRS of size π from any population whatsoever with mean µ and finite standard deviation π. β’ When π is large, the sampling distribution of the sample π mean π₯ is close to the normal distribution π( π, ) with π mean µ and standard deviation . π π β’ Larger sample sizes are required if shape of the population is far from Normal. NORMAL CONDITION FOR SAMPLE MEANS β’ If the population distribution is Normal, then so is the sampling distribution of π₯. No matter the sample size π. β’ If the population distribution is NOT Normal, the central limit theorem tells us that the sampling distribution of π₯ will be approximately Normal in most cases if π β₯ ππ. SERVICING AIR CONDITIONERS β’ Your company has a contract to perform preventive maintenance on thousands of air-conditioning units in a large city. Based on service records from the past year, the time (in hours) that a technician requires to complete the work follows the distribution below. This distribution is strongly right-skewed, and the most likely outcomes are close to 0. β’ The mean time is π = 1 hour and the standard deviation is π = 1 hour. In the coming week, your company will service an SRS of 70 air-conditioning units in the city. You plan to budget an average of 1.1 hours per unit for a technician to complete the work. Will this be enough? β’ What is the probability that the average maintenance time π₯ for 70 units exceeds 1.1 hours? β’ Normal approximation from the central limit theorem for the average time needed to maintain an air conditioner.
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