Day 13 AGENDA: DG7 --- 10 minutes

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?

Work smarter, not harder!! Create a probability distribution where the random variable of interest is number of heads occurring when tossing 4 coins. Recall: X = number of heads 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(xi) ? 0 1 2 3 4 0.0625 0.25 0.375 0.25 0.0625

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

(n)(p)(q) Find the variance for this task. (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 σ2 = ____________________________ (4)(.5)(.5) = 1 Now calculate (n)(p)(q) ______________ WOW!! **Formula: Variance of a Binomial Distribution (n)(p)(q) σ2 = ____________________________

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

X = # of twos occurring on 480 rolls of a die 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. X = # of twos occurring on 480 rolls of a die Would this be a BINOMIAL DISTRIBUTION or a NORMAL DISTRIBUTION? WHY? I DO NOT want to create a probability distribution for this!! WHY? (n)(p) = (480)(1/6) = 80 µ = __________________   σ2 = ________________ σ = _____________________ (n)(p)(q) = (480)(1/6)(5/6) = 66.67

Would this be a BINOMIAL DISTRIBUTION or a NORMAL DISTRIBUTION? 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. Would this be a BINOMIAL DISTRIBUTION or a NORMAL DISTRIBUTION? WHY? X = # of twin births occurring 8000 births µ = __________________   σ2 = ________________ σ = _____________________ (n)(p) = (8000)(0.02) = 160 (n)(p)(q) = (8000)(0.02)(0.98) = 156.8

Get out a sheet of paper to use for Ex 3. Answer #6 & #7 first.

Now finish #1 - #5 on your own. 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 p = 3/7 remains the same from restaurant to restaurant Yes, this scenario meets the criteria for a binomial distribution. 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.

Only because it’s a Binomial Distribution

Assignment: Practice Worksheet: Calculating Mean, Variance, and Standard Deviation of Binomial Distributions DG8 --- Tues