1. 1 BA 275 Quantitative Business Methods The Sampling Distribution of a Statistic The Central Limit Theorem Agenda

2. 2 Midterm Exam #1 Wednesday, 2/01/06 in class. Closed books/notes/packet/friends exam, but I will provide you The empirical rule The binomial formula The normal table Need a good night sleep and a calculator that WORKS! Topics: Describing Categorical/Numerical Data The Empirical Rule and Box plot Binomial Distribution The Normal Distribution and the Central Limit Theorem CyberStats: A-1, A-4, A-5, A-6, A-9, B-7, B-9, B-11, B-12.

3. 3 The Empirical Rule is not Enough Mean = 30,000 miles STD = 5,000 miles Q1: If the level of warranty is set at 12,000 miles, about what % of tires will be returned under the warranty? Q2: If we can accept that up to 3% of tires can be returned under warranty, what should be the warranty level?

4. 4 Normal Curve Table (p.40)

5. 5 Sampling Distribution (CyberStats) A sampling distribution describes the distribution of all possible values of a statistic over all possible random samples of a specific size that can be taken from a population.

6. 6 Central Limit Theorem (CLT) The CLT says that the mean of a sample is distributed approximately as a normal if the sample size is large enough. Like any normal distribution, this normal has a mean and a standard deviation. CLT demo

7. 7 Central Limit Theorem (CLT) The CLT applied to Means CLT demo Example 1: X ~ a normal distribution with the mean 16, and variance 25. Example 2: X ~ a distribution with the mean 8.08, and variance 38.6884. With a sample of size n = 25, can we predict the value of the sample mean?

8. 8 Example The diameter of golf balls manufactured at a large factory is expected to have a mean of 1.30 inches and a standard deviation of 0.06 inch. If many random samples of 25 balls are selected, What distribution will the sample mean follow? What will be the mean and std of such distribution? What proportion of the sample means will be between 1.31 and 1.33?

9. 9 Example The dean of a B-school claims that the average weekly income of graduates of her school one year after graduation is $600. If the dist. of weekly incomes is normal with a std of $100, Q1. What is the prob. of one randomly selected graduate has an avg weekly income of less than $550? Q2. What is the prob. of 64 randomly selected graduates have an avg weekly income of less than $550? If the dist. of weekly incomes is UNKNOWN, but the std is believed to be $100, can we still answer the above two questions?