Warm-Up

Take two dice. Roll them together 40 times and record the sums that you got. Draw a histogram of your 40 points.

Ms. Morton will generate 50 random numbers between 1 and 20. Draw a histogram of these 50 data points with bin size 2.

CENTRAL LIMIT THEOREM

Honors Advanced AlgebraPresentation 1-8

Central Limit Theorem

Central Limit Theorem - Choose a simple random sample of size n from any population with mean µ and standard deviation σ. When n is large (at least 30), the sampling distribution of the sample mean x is approximately normal with mean µ and standard deviation .

Central Limit Theorem

Central Limit Theorem - Choose a simple random sample of size n from a large population with population parameter p having some characteristic of interest. Then the sampling distribution of the sample proportion is approximately normal with mean p and standard deviation . This approximation becomes more and more accurate as the sample size n increases, and it is generally considered valid if the population is much larger than the sample, i.e. np ≥ 10 and n(1 – p) ≥ 10..

Central Limit Theorem

Central Limit Theorem - The CLT allows us to use normal calculations to determine probabilities about sample proportions and sample means obtained from populations that are not normally distributed.

Central Limit Theorem

As we make a histogram of multiple sample means, the data approaches a normal curve.

The mean of the means is the same as the mean of the population. (

The variance of the means is equal to The standard deviation of the means is

equal to The larger the sample size, the more

certain we can be of the mean and the smaller the standard deviation.

Example

The time that an A/C technician requires to perform maintenance on an A/C unit is an exponential decay distribution. The mean time is μ = 1 hour and the standard deviation is σ = 1 hour. Your company has a contract to maintain 70 of these units in an apartment building. Is it safe to budget 1.1 hours for each unit or should you budget an average of 1.25 hours?

Problem 1

The number of flaws per square yard in a type of carpet material varies with mean 1.6 flaws per square yard and standard deviation 1.2 flaws per square yard. The population distribution cannot be Normal because a count takes only whole-number values. An inspector studies 200 square yards of the material, records the number of flaws found in each square yard, and calculates (the mean number of flaws per square yard inspected). Use the central limit theorem to find the approximate probability that the mean number of flaws exceeds 2 per square yard. Show your work.

Problem 2

In response to the increasing weight of airline passengers, the FAA in 2003 told airlines to assume that passengers average 190 pounds in the summer, including clothes and carry-on baggage. But passengers vary, and the FAA did not specify a standard deviation. A reasonable standard deviation is 35 pounds. Weights are not Normally distributed, especially when the population includes both men and women, but they are not very non-Normal. A commuter plane carries 20 passengers.

Can you calculate the probability that the total weight of the passengers on the flight exceeds 4000 pounds?

Problem 3

The number of traffic accidents per week at an intersection varies with mean 2.2 and standard deviation 1.4. The number of accidents in a week must be a whole number, so the population distribution is not Normal.

Let be the mean number of accidents per week at the intersection during a year (52 weeks). What is the standard deviation of the sample means?

What is the approximate probability that is less than 2?

What is the approximate probability that there are fewer than 100 accidents at the intersection in a year?