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**Chapter 7: Sampling Distributions**

Section 7.3 Day 2 Sample Means The Practice of Statistics, 4th edition – For AP* STARNES, YATES, MOORE

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**The Central Limit Theorem**

Most population distributions are not Normal. What is the shape of the sampling distribution of sample means when the population distribution isn’t Normal? It is a remarkable fact that as the sample size increases, the distribution of sample means changes its shape: it looks less like that of the population and more like a Normal distribution! When the sample is large enough, the distribution of sample means is very close to Normal, no matter what shape the population distribution has, as long as the population has a finite standard deviation. Definition: Note: How large a sample size n is needed for the sampling distribution to be close to Normal depends on the shape of the population distribution. More observations are required if the population distribution is far from Normal.

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**Normal Condition for Sample Means**

Example 3: The Rice University sampling distribution applet was used to create a population distribution with a very strange shape. Here is the population distribution. Describe the shape of the sampling distributions as n increases. What do you notice? Normal Condition for Sample Means The central limit theorem allows us to use Normal probability calculations to answer questions about sample means from many observations even when the population distribution is not Normal.

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Example 4: 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 whose density curve is shown below. This distribution is strongly right-skewed, and the most likely outcomes are close to 0. The mean time is μ = 1 hour and the standard deviation is σ = 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? State: We want to find the probability that if you budget an average of 1.1 hours per unit for a technician to complete the work, will it be enough. . Let = the mean mount of time (in hours) it takes the technician to do the maintenance. Plan: Since the 10% condition is met (there are more than 10(70)=700 air conditioners in the population), the sampling distribution of the mean time spent working on the 70 units has The sampling distribution of the mean time spent working is approximately N(1, 0.12) by the CLT since n = 70 ≥ 30. The problem states that the data come from an SRS. DO: normalcdf(1.1, , 1, 0.12) = Conclude: If you budget 1.1 hours per unit, there is a 20% chance the technicians will not complete the technician will not complete the work within the budgeted time.

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