# CENTRAL LIMIT THEOREM p p p specifies a

- Slides: 14

CENTRAL LIMIT THEOREM p p p specifies a theoretical distribution formulated by the selection of all possible random samples of a fixed size n a sample mean is calculated for each sample and the distribution of sample means is considered clt 1

SAMPLING DISTRIBUTION OF THE MEAN p p The mean of the sample means is equal to the mean of the population from which the samples were drawn. The variance of the distribution is s divided by the square root of n. (the standard error. ) clt 2

STANDARD ERROR Standard Deviation of the Sampling Distribution of Means sx = s/ /n clt 3

How Large is Large? If the sample is normal, normal then the sampling distribution of will also be normal, no matter what the sample size. p When the sample population is approximately symmetric, symmetric the distribution becomes approximately normal for relatively small values of n. p When the sample population is skewed, skewed the sample size must be at least 30 before the sampling distribution of becomes approximately normal. p clt 4

EXAMPLE A certain brand of tires has a mean life of 25, 000 miles with a standard deviation of 1600 miles. What is the probability that the mean life of 64 tires is less than 24, 600 miles? clt 5

Example continued The sampling distribution of the means has a mean of 25, 000 miles (the population mean) m = 25000 mi. and a standard deviation (i. e. . standard error) of: 1600/8 = 200 clt 6

Example continued Convert 24, 600 mi. to a z-score and use the normal table to determine the required probability. z = (24600 -25000)/200 = -2 P(z< -2) = 0. 0228 or 2. 28% of the sample means will be less than 24, 600 mi. clt 7

ESTIMATION OF POPULATION VALUES Point Estimates p Interval Estimates p clt 8

CONFIDENCE INTERVAL ESTIMATES for LARGE SAMPLES p p The sample has been randomly selected The population standard deviation is known or the sample size is at least 25. clt 9

Confidence Interval Estimate of the Population Mean -X: sample mean s: sample standard deviation n: sample size clt 10

EXAMPLE Estimate, with 95% confidence, the lifetime of nine volt batteries using a randomly selected sample where: -X = 49 hours s = 4 hours n = 36 clt 11

EXAMPLE continued Lower Limit: 49 - (1. 96)(4/6) 49 - (1. 3) = 47. 7 hrs Upper Limit: 49 + (1. 96)(4/6) 49 + (1. 3) = 50. 3 hrs We are 95% confident that the mean lifetime of the population of batteries is between 47. 7 and 50. 3 hours. clt 12

CONFIDENCE BOUNDS Provides a upper or lower bound for the population mean. p To find a 90% confidence bound, use the z value for a 80% CI estimate. p clt 13

Example The specifications for a certain kind of ribbon call for a mean breaking strength of 180 lbs. If five pieces of the ribbon have a mean breaking strength of 169. 5 lbs with a standard deviation of 5. 7 lbs, test to see if the ribbon meets specifications. p Find a 95% confidence interval estimate for the mean breaking strength. p clt 14