1. DG7 --- 15 minutes 11 th graders psat info --- see board

2. Accel Math III Unit #1: Data Analysis Lesson #10: Calculating Mean, Variance, and Standard Deviation for a Binomial Distribution EQ: How do you calculate mean, variance, and standard deviation of a binomial distribution? How are these used to interpret the data?

3. Recall: Create a probability distribution where the random variable of interest is number of heads occurring when tossing 4 coins. HINT: You can, but you don’t have to create a sample space to complete this task. Work smarter, not harder!! How could you use Lists and binomialpdf to find each value of P(x i ) ? 0 1 2 3 4 0.0625 0.25 0.375 0.25 0.0625 X = number of heads when tossing 4 coins

4.  Find the expected value for this task. E(X) = __________________________ Now calculate (n)(p) ______________ **Formula: Mean of a Binomial Distribution 0(0.0625) + 1(0.25) +2(0.375)+3(0.25) +4(0.0625) = 2 (4)(0.5) = 2 µ = __________________ (n)(p) WOW!!

5.  Find the variance for this task. σ 2 = ____________________________ (0 – 2) 2 (0.0625) + (1 – 2) 2 (0.25) + (2 – 2) 2 (0.375) + (3 – 2) 2 (0.25) + (4 – 2) 2 (0.0625) = 1 Now calculate (n)(p)(q) ______________ (4)(.5)(.5) = 1 **Formula: Variance of a Binomial Distribution σ 2 = ____________________________ (n)(p)(q) WOW!!

6.  Find the standard deviation for this task. σ = _____________________ **Formula: Standard Deviation of a Binomial Distribution σ = _____________________

7. Ex 1. A die is rolled 480 times. Find the mean, variance, and standard deviation of the number of 2’s that will be rolled. µ = __________________ σ 2 = ________________ σ = _____________________ X = # of twos occurring on 480 rolls of a die I DO NOT want to create a probability distribution for this!! WHY? (n)(p) = (480)(1/6) = 80 (n)(p)(q) = (480)(1/6)(5/6) = 66.67 Would this be a BINOMIAL DISTRIBUTION or a NORMAL DISTRIBUTION? WHY?

8. Ex 2. The Statistical Bulletin published by Metropolitan Life Insurance Co. reported that 2% of all American births result in twins. If a random sample of 8000 births is taken, find the mean, variance, and standard deviation of the number of births that would result in twins. µ = __________________ σ 2 = ________________ σ = _____________________ X = # of twin births occurring 8000 births Would this be a BINOMIAL DISTRIBUTION or a NORMAL DISTRIBUTION? WHY? (n)(p) = (8000)(0.02) = 160 (n)(p)(q) = (8000)(0.02)(0.98) = 156.8

9.  Answer #6 & #7 first.

10. 6. Criteria needed to be Binomial Distribution: a. Fixed # of trials n = 10 restaurants b. Success --- unsatisfactory Failure --- satisfactory c. Condition of one restaurant independent of others d.p = 3/7 remains the same from restaurant to restaurant Yes, this scenario meets the criteria for a binomial distribution.

11. 7. NO, binomial distribution simply means there are only two outcomes, NOT the probabilities are 50% and 50%.  Now finish #1 - #5 on your own.  DG8 --- Mon

12. Assignment: Practice Worksheet: Calculating Mean, Variance, and Standard Deviation of Binomial Distributions