1. Sampling Distributions6
Sampling Distributions
Lesson 6.6
The Central Limit Theorem

2. The Central Limit Theorem
Determine if the sampling distribution of is approximately normal when sampling from a non-normal population.
If appropriate, use a normal distribution to calculate probabilities involving .

3. 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.
Central Limit Theorem (CLT)
Draw an SRS of size n from any population with mean µ and finite standard deviation σ. The central limit theorem (CLT) says that when n is large, the sampling distribution of the sample mean is approximately normal.

4. The Central Limit Theorem
How large a sample size n is needed for the sampling distribution of to be close to normal depends on the population distribution. A larger sample size is required if the shape of the population distribution is far from normal. In that case, the sampling distribution of will also be far from normal if the sample size is small.
Normal/Large Sample Condition
The Normal/Large Sample condition says that the distribution of will be approximately normal when either of the following is true:
The population distribution is approximately normal. This is true no matter what the sample size n is.
The sample size is large. If the population distribution is not normal, the sampling distribution of will be approximately normal in most cases if n ≥ 30.

5. LESSON APP 6.6
Keeping things cool with statistics?
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 a strongly right-skewed distribution with µ = 1 hour and σ = 1.5 hours.
As a promotion, your company will provide service to a random sample of 70 air-conditioning units free of charge. You plan to budget an average of 1.1 hours per unit for a technician to complete the work. Will this be enough time?
What is the shape of the sampling distribution of for samples of size n = 70 from this population? Justify.
Calculate the probability that the average maintenance time for 70 units exceeds 1.1 hours.
Based on your answer to the previous problem, did the company budget enough time? Explain.

6. The Central Limit Theorem
Determine if the sampling distribution of is approximately normal when sampling from a non-normal population.
If appropriate, use a normal distribution to calculate probabilities involving .