- Slides: 24
SP 225 Lecture 10 The Central Limit Theorem
Differences Between Statistics and Parameters Sample: 3 Randomly Selected People Population: All People Parameter: 5 of 15 or 33% wear glasses Statistic: 0 of 3 or 0% wear glasses
Methods to for Better Samples o Random sampling makes samples representative o Book term: probability sample o EPSEM (Equal probability of selectino method)
EPSEM Technique o Begin with a list of all population members o Generate random numbers to identify members of the list to be selected in the sample o Do everything possible to get selected members to participate in the survey
Sampling Distributions o The shape, measure of center and measure of variation associated with many sample statistics o Unique and different from the population distribution
Three Separate Distributions o Sample distribution n Empirical and known n Intended to help learn about the population o Population distribution n Empirical and unknown n Properties estimated using statistics o Sampling distribution n Theoretical or non-empirical n Properties well-known and based on probabilities
Sampling Distribution of Mean
Central Limit Theorem o For normally distributed populations o If repeated samples of size N are drawn from a normal population with mean µ and standard deviation σ, then the sampling distribution σ of sample means will be normal with a mean of µ and a standard deviation of σ/√N
Central Limit Theorem o For any population o If repeated samples of size N are drawn from a normal population with mean µ and standard deviation , then, as N becomes large, the sampling distribution σ of sample means will be normal with a mean of µ and a standard deviation of σ/√N
The Probability Distribution o We can calculate probabilities for sample means using the sampling distribution for sample means o Similar to calculating probabilities for an individual using a population distribution o Use standard deviation of the sampling distribution instead of sampling distribution of the population
Empirical Rule for Data with a Bell -Shaped Distribution (3)
Example Problem (1) o The average GPA at a particular school is m=2. 89 with a standard deviation s=0. 63. A random sample of 25 students is collected. Find the probability that the average GPA for this sample is greater than 3. 0.
Example Problem (2) The time it takes students in a cooking school to learn to prepare seafood gumbo is a random variable with a normal distribution where the average is 3. 2 hours with a standard deviation of 1. 8 hours. n Find the probability that the average time it will take a class of 36 students to learn to prepare seafood gumbo is less than 3. 4 hours. n Find the probability that it takes one student between 3 and 4 hours to learn to prepare seafood gumbo.
Confidence Interval o Mathematical statement that says that the parameter lies within a certain range of values o The average employee of XYZ Automotive has been employed between 8 and 12 years o 95% confident that the mean length of employment at XYZ automotive is between 8 and 12 years
Probability Distribution for Population Mean o 95% confident that the mean length of employment at XYZ automotive is between 8 and 12 years (sample)
Confidence Level o Percent of confidence intervals that contain the population mean over the long run o Probability this confidence interval contains the population mean o 95% confident that the mean length of employment at XYZ automotive is between 8 and 12 years o 99% confident that the mean length of employment at XYZ automotive is between 8 and 12 years
Confidence Interval Formula
Confidence Interval Estimate
Confidence Interval Example o A random sample of 100 television programs contained an average of 2. 37 acts of physical violence per program. The standard deviation of the number of acts of violence on television is 3. At the 99% level, what is your estimate of the mean number of acts of violence for all television programs?
Alpha o Percent of confidence intervals that DO NOT contain the population mean over the long run o Probability this confidence interval DOES NOT contain the population mean o Complement of confidence level
Efficiency o Extent to which the confidence interval clusters around the mean o Width of the confidence interval o Determined by population standard deviation and sample size