6.39 Day 2: The Central Limit Theorem

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Presentation transcript:

6.39 Day 2: The Central Limit Theorem By the end of class you will be able to calculate probabilities involving sample means

A problem we know… The average teacher’s salary in North Dakota is $29,863. Assume a normal distribution with a standard deviation of $5,100. What is the probability that a random teacher’s salary is more than $40,000? .0233

A new problem…Sample Means The average teacher’s salary in North Dakota is $29,863. Assume a normal distribution with a standard deviation of $5,100. What is the probability that the mean for a sample of 80 teachers’ salaries is greater than $30,000? .4052

The Central Limit Theorem states when we can use a normal distribution to solve problems….

Central Limit Theorem As the sample size n increases without limit, the shape of the distribution of the sample means taken with replacement from a population with mean and standard deviation will approach a normal distribution with mean and standard deviation ALSO The Standard Score Must Change:

In English…. If the sample size is larger than 30, a distribution of sample means can be approximated using the normal distribution If the original population is normally distributed, then the sample means will be normally distributed for any sample size As the sample size increases, the sampling distribution of sample means approaches a normal distribution

Sampling Distribution of Sample Means Distribution formed by using the means computed from all possible random samples of a specific size taken from a population. Sampling Error: The difference between the sample measure and the corresponding population measure since samples are not perfect representations of populations.

A new problem…Sample Means The average teacher’s salary in North Dakota is $29,863. The standard deviation is $5,100. What is the probability that the mean for a sample of 80 teachers’ salaries is greater than $30,000? .4052 We can use the normal distribution here because of the large sample size….the original problem doesn’t say the population is normal…this is important to stress! Must calculate the new standard deviation using the sample size given!

At a proofreading company the average age of employees is 36 At a proofreading company the average age of employees is 36.2 years and the standard deviation is 3.7 years. If a person from the company is chosen at random, find the probability that her age will be between 36 and 37.5 years. Assume the variable is normally distributed. .1567

At a proofreading company the average age of employees is 36 At a proofreading company the average age of employees is 36.2 years and the standard deviation is 3.7 years. If a random sample of 15 proofreaders is selected find the probability that the mean age of proofreaders in the sample will be between 36 and 37.5 years. Assume the original distribution is normally distributed. .4963

(c) 2007 brainybetty.com ALL RIGHTS RESERVED. Summarizer: The most important thing about the distribution of sample means is…… I must also remember ….. And…… But the most important thing to remember about the distribution of sample means is… (c) 2007 brainybetty.com ALL RIGHTS RESERVED.