1. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Chapter 8. Some Approximations to Probability Distributions: Limit Theorems Sections 8.4: The Central Limit Theorem Jiaping Wang Department of Mathematics 04/22/2013, Monday

2. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Theorem 8.4

3. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL An Application of CLT

4. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Example 8.6 From 1976 to 2002, a mechanical golfer, Iron Byron, whose swing was modeled after that of Byron Nelson (a leading golfer in the 1940s), was used to determine whether golf balls met the Overall Distance Standard. Specifically, Iron Byron would be used to hit the golf balls. If the average distance of 24 golf balls tested exceeded 296.8 yards, then that brand would be considered nonconforming. Under these rules, suppose a manufacturer produces a new golf ball that travels an average distance of 297.5 yards with a standard deviation of 10 yards. 1.What is the probability that the ball will be determined to be nonconforming when tested? 2.Find an interval that includes the average overall distance of 24 golf balls with probability of 0.95. From 1976 to 2002, a mechanical golfer, Iron Byron, whose swing was modeled after that of Byron Nelson (a leading golfer in the 1940s), was used to determine whether golf balls met the Overall Distance Standard. Specifically, Iron Byron would be used to hit the golf balls. If the average distance of 24 golf balls tested exceeded 296.8 yards, then that brand would be considered nonconforming. Under these rules, suppose a manufacturer produces a new golf ball that travels an average distance of 297.5 yards with a standard deviation of 10 yards. 1.What is the probability that the ball will be determined to be nonconforming when tested? 2.Find an interval that includes the average overall distance of 24 golf balls with probability of 0.95.

5. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Solution

6. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Example 8.7 A certain machine that is used to fill bottles with liquid has been observed over a long period, and the variance in the amounts of fill has been found to be approximately σ 2 =1 ounce. The mean ounces of fill μ, however, depends on an adjustment that may change from day to day or from operator. If n= 36 observations on ounces of fill dispensed are to be taken on a given day (all with the same machine setting), find the probability that the sample mean will be within 0.3 ounce of the true population mean for the setting.

7. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Approximate Binomial by Normal Distribution

8. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Example 8.8 Six percent of the apples in a large shipment are damaged. Before accepting each shipment, the quality control manager of a large store randomly selects 100 apples. If four or more are damaged, the shipment is rejected. What is the probability that this shipment is rejected?

9. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Example 8.9 Candidate A believe that she can win a city election if she receives at least 55% of the votes from precinct I. Unknown to the candidate, 50% of the registered voters in the precinct favor her. If n=100 voters show up to vote at precinct I, what is the probability that candidate A will receive at least 55% of that precinct’s votes?

10. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Additional Example 1 Apply the central limit theorem to approximate P[X 1 +X 2 +…+X 20 ≤ 50], where X 1, …, X 20 are independent random variables having a common mean μ= 2 and a common standard deviation σ= 10.

11. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Additional Example 2 Let X have a binomial distribution Bin(200,0.15). Find the normal approximation to P[25 <X < 35].

12. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Additional Example 3 Roll a fair coin 100 times, use CLT to find the approximate probability that more than 60 tails shows.

13. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Homework #11 Page 405: 8.2, 8.4, 8.6 (a); Page 417: 8.12, 8.16, 8.18, 8.28. Due next Monday, 04/29/2013. Page 405: 8.2, 8.4, 8.6 (a); Page 417: 8.12, 8.16, 8.18, 8.28. Due next Monday, 04/29/2013.