 ## Presentation on theme: "Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 18- 1."— Presentation transcript:

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 18- 3 Sampling Distribution of the Mean The goal of inferential statistics is to use a sample to make an inference about a population. A class of 50 students wants to study the average GPA at KSU. Student number 1 collects a sample of 5 student GPA's. S1={3.01, 3.28, 2.97, 3.41, 3.21}, average=3.176 Student number 2 collects a sample of 5 student GPA's. S2={2.89, 3.33, 1.97, 2.59, 3.01}, average=2.758 Student number 3 collects a sample of 5 student GPA's. S3={2.93, 2.78, 3.41, 3.17, 2.81}, average=3.02 The remaining 47 students proceed in a similar fashion. Are there differences in the variations in the single observations and the variations of the sample averages? Given 50 sample averages what might you do to estimate the true population average?

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 18- 4 Sampling Distribution of the Mean The sampling distribution of a sample statistic is the distribution of the values of the statistic created by repeated samples of n observations.

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 18- 5 Means – The “Average” of One Die Let’s start with a simulation of 10,000 tosses of a die. A histogram of the results is:

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 18- 6 Means – Averaging More Dice Looking at the average of two dice after a simulation of 10,000 tosses: The average of three dice after a simulation of 10,000 tosses looks like:

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 18- 7 Means – Averaging Still More Dice The average of 5 dice after a simulation of 10,000 tosses looks like: The average of 20 dice after a simulation of 10,000 tosses looks like:

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 18- 8 Assumptions and Conditions Most models are useful only when specific assumptions are true. There are two assumptions in the case of the model for the distribution of sample proportions: 1. The sampled values must be independent of each other. 2. The sample size, n, must be large enough. At least 25-30 observations

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 18- 9 Means – What the Simulations Show As the sample size (number of dice) gets larger, each sample average is more likely to be closer to the population mean. So, we see the shape continuing to tighten around 3.5 And, it probably does not shock you that the sampling distribution of a mean becomes Normal.

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 18- 10 The Fundamental Theorem of Statistics The sampling distribution of any mean becomes Normal as the sample size grows. All we need is for the observations to be independent and collected with randomization. We don’t even care about the shape of the population distribution! The Fundamental Theorem of Statistics is called the Central Limit Theorem (CLT).

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 18- 11 The Fundamental Theorem of Statistics (cont.) The Central Limit Theorem (CLT) The mean of a random sample has a sampling distribution whose shape can be approximated by a Normal model. The larger the sample, the better the approximation will be.

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 18- 12 The Fundamental Theorem of Statistics (cont.) The CLT says that the sampling distribution of any mean is approximately Normal. For means, it’s centered at the population mean. But what about the standard deviations?

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 18- 13 The Fundamental Theorem of Statistics (cont.) The Normal model for the sampling distribution of the mean has a standard deviation equal to where σ is the population standard deviation. The standard deviation of a sampling distribution is called the standard error.

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 18- 14 The average GPA at a particular school is m=2.89 with a standard deviation s=0.63. Find the probability that the average GPA for a sample of 35 students is greater than 3.0. Find the probability that the average GPA for a sample of 40 students is between 2.0 and 2.75. Find the probability that the average GPA for Nathan is between 2.0 and 2.75. What is the GPA for the worst 15% groups of 25 students? What is the GPA for the best 5% of groups of 40 students? Is the normal model good for predicting the GPA for a sample of 5 students? Explain.

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 18- 15 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. 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. Find the probability that it takes one student between 3 and 4 hours to learn to prepare seafood gumbo. Would it be unusual for a group of 50 students to learn to prepare seafood gumbo in less than two hours?