1 Chapter 4. Section 4-4. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman M ARIO F. T RIOLA E IGHTH E DITION E LEMENTARY S TATISTICS Section 4-4 Mean, Variance, and Standard Deviation for the Binomial Distribution

2 Chapter 4. Section 4-4. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman For Any Discrete Probability Distribution: Formula 4-1 µ =  [x P(x)] Formula 4-3  2  = [  x 2 P(x) ] - µ 2

3 Chapter 4. Section 4-4. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman For Any Discrete Probability Distribution: Formula 4-1 µ =  [x P(x)] Formula 4-3  2  = [  x 2 P(x) ] - µ 2 Formula 4-4  = [  x 2 P(x) ] - µ 2

4 Chapter 4. Section 4-4. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman For Binomial Distributions: Formula 4-6 µ = n p Formula 4-7  2  = n p q

5 Chapter 4. Section 4-4. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Formula 4-8  = n p q For Binomial Distributions: Formula 4-6 µ = n p Formula 4-7  2  = n p q

6 Chapter 4. Section 4-4. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman We previously discovered that this scenario could be considered a binomial experiment where: n = 14 p = 0.5 q = 0.5 Using the binomial distribution formulas: Example: Find the mean and standard deviation for the number of girls in groups of 14 births.

7 Chapter 4. Section 4-4. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman We previously discovered that this scenario could be considered a binomial experiment where: n = 14 p = 0.5 q = 0.5 Using the binomial distribution formulas: µ = (14)(0.5) = 7 girls  = (14)(0.5)(0.5) = 1.9 girls (rounded) Example: Find the mean and standard deviation for the number of girls in groups of 14 births.

8 Chapter 4. Section 4-4. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Reminder Maximum usual values = µ + 2  Minimum usual values = µ - 2 

9 Chapter 4. Section 4-4. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman For this binomial distribution, µ = 50 girls  = 5 girls µ + 2  = (5) = 60 µ - 2  = (5) = 40 The usual number girls among 100 births would be from 40 to 60. So 68 girls in 100 births is an unusual result. Example: Determine whether 68 girls among 100 babies could easily occur by chance.