Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
NOTES 6-5 Central Limit Theorem Sometimes in statistics we want to know about the probability of the SAMPLE MEAN. To do this we need to find the mean of the means and the standard deviation of the means, then continue to follow the normal distribution procedures. The 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 μ and standard deviation σ will approach a normal distribution. We can use the Central Limit Theorem as the sample size n increases without limit. Large sample sizes!!! Mar 202:30 PM Example The heights of kindergarten children are approximately normally distributed with a mean of 39 and a standard deviation of 2. If one child is randomly selected, what is the probability that the child is taller than 41 inches? This is 1 child – Not the Central Limit Theorem! Let's solve this..... Apr 69:05 AM Example Suppose we have a class of 30 kindergarten children. What is the probability that the mean height of these children exceeds 41 inches? This is the Central Limit Theorem as it is asking about the probability of a sample mean! We will wait to solve this one until later.... Mar 2010:18 PM Mar 2010:19 PM 1 Properties of the Distribution of Sample Means: 1) Back to the Example from before: Suppose there is a kindergarten class of 30 children. What is the probability that the mean height exceeds 41 inches? (Remember μ=39 and σ= 2) **Do everything in one step to avoid rounding errors! 2) So really the only difference is a new formula must be used for z values: Mar 209:48 PM Conclusion: • It is not unusual for one child, selected at random from a kindergarten class, to be taller than 41 inches. (16%) Mar 2010:12 PM Example In a certain population SAT scores are normally distributed with mean = 500 and standard deviation = 100. In a sample size of 25, what is the probability that their sample mean score is between 490 and 510? • It is highly unlikely that the mean height for 30 kindergarten students exceeds 41 inches. (0%) Mar 2010:25 PM Mar 2010:27 PM 2 Example The average number of pounds of meat that a person consumes a year is 218.4 pounds. Assume the standard deviation is 25 pounds and the distribution is approximately normal. a) Find the probability that a person selected at random consumes less than 224 pounds per year. HW pg 325 326 8, 9, 10, 12, 17 b) If a sample of 40 individuals is selected, find the probability that the mean of the sample will be less than 224 pounds per year. Apr 18:22 PM Mar 309:55 PM 3