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Business Statistics Chapter 6
- 1. © 2003 Prentice-Hall, Inc. Chap 6-1 Basic Business Statistics (9th Edition) Chapter 6 The Normal Distribution
- 2. © 2003 Prentice-Hall, Inc. Chap 6-2 Chapter Topics The Normal Distribution The Standardized Normal Distribution
- 3. © 2003 Prentice-Hall, Inc. Chap 6-3 Continuous Probability Distributions Continuous Random Variable Values from interval of numbers Absence of gaps Continuous Probability Distribution Distribution of continuous random variable Most Important Continuous Probability Distribution The normal distribution
- 4. © 2003 Prentice-Hall, Inc. Chap 6-4 Typical Problems this Chapter will solve You have analyzed the earnings per family for 10 communes in Phnom Penh. You know the mean salary is $60 per family and you have a standard deviation of 10 dollars. When you look at the data in a histogram you can see is is symmetric (bell shaped – normally distributed) In a meeting a sponsor asks you to quickly estimate: - 1. The salaries for the lowest 10% of families and 2. The what percentage earn over $100.
- 5. © 2003 Prentice-Hall, Inc. Chap 6-5 Typical Problems this Chapter will solve You have developed a website to sell clothes on-line to US and European customers. However, very few customers are buying any products because it is taking so long to down load product pictures. Old data indicates you have a mean down load time of 17 seconds with 3 second standard deviation. Approximately 95% of the downloads are between 11 and 23 seconds. When you look at the data in a histogram you can see it is symmetric (bell shaped – normally distributed) The CEO of the clothes company is very angry about your slow website. You tell him you can use new software that will speed up the mean down load time by 7 seconds. If the company buys the new software provide an estimate for: - 1. The time taken for the fastest 10% of downloads and 2. The percentage of downloads that will take over 15 Seconds.
- 6. © 2003 Prentice-Hall, Inc. Chap 6-6 The Normal Distribution “Bell Shaped” Symmetrical Mean, Median and Mode are Equal Interquartile Range Equals 1.33 σ Random Variable Has Infinite Range Mean Median Mode X f(X) µ
- 7. © 2003 Prentice-Hall, Inc. Chap 6-7 The Mathematical Model ( ) ( ) ( ) ( ) 2 (1/ 2) /1 2 : density of random variable 3.14159; 2.71828 : population mean : population standard deviation : value of random variable X f X e f X X e X X µ σ πσ π µ σ − − = ≈ ≈ −∞ < < ∞
- 8. © 2003 Prentice-Hall, Inc. Chap 6-8 Many Normal Distributions Varying the Parameters σ and µ, We Obtain Different Normal Distributions There are an Infinite Number of Normal Distributions
- 9. © 2003 Prentice-Hall, Inc. Chap 6-9 The Standardized Normal Distribution When X is normally distributed with a mean and a standard deviation , follows a standardized (normalized) normal distribution with a mean 0 and a standard deviation 1. X Z µ σ − = X f(X) µ Z σ 0Zµ = 1Zσ = f(Z) µ σ
- 10. © 2003 Prentice-Hall, Inc. Chap 6-10 Z Scores mean the Normal Modal becomes the Standardized Normal Modal N(0,1) Z Scores µ-3σ µ-2σ µ-1σ µ µ+1σ µ +2σ µ +3σ •The normal model is represented by N(µ,σ2 ) Standardized Normal Model -3 -2 -1 0 1 2 3 •The Standardized Normal Model is represented by N(0,1) s yy Z )( − = yy y Z
- 11. © 2003 Prentice-Hall, Inc. Chap 6-11 Standardizing We often adjust data to find the difference of each value from a center in terms of a suitable spread. Such standardizing is a fundamental step in many statistics calculations.
- 12. © 2003 Prentice-Hall, Inc. Chap 6-12 Finding Probabilities for Z Scores When using the Standard Normal Distribution the Probability is the area under the curve! c d X f(X) ( ) ?P c X d≤ ≤ = We have generated Z Calc
- 13. © 2003 Prentice-Hall, Inc. Chap 6-13 Which Table to Use? Infinitely Many Normal Distributions Means Infinitely Many Tables to Look Up!
- 14. © 2003 Prentice-Hall, Inc. Chap 6-14 Standardized Normal Distribution Z .00 .01 0.0 .5000 .5040 .5080 .5398 .5438 0.2 .5793 .5832 .5871 0.3 .6179 .6217 .6255 .5478 .02 0.1 .5478 Cumulative Standardized Normal Distribution Table (Portion) Probabilities Only One Table is 0 1Z Zµ σ= = Z = 0.12 0
- 15. © 2003 Prentice-Hall, Inc. Chap 6-15 Finding the Relative Frequency from Tables IF Z is positive Use Table E2 Page 811 If Z is negative Use Table E2 Page 810 Z> Relative Frequency Z<1.8 0.9641 Z>1.82 Z>1.01 Z Relative Frequency Z<-0.8 0.2119 Z>-0.82 Z<-1 µ z Z µ Table Value 1-(Table Value) Table Value 1-(Table Value) Z< Z> Z< Z>
- 16. © 2003 Prentice-Hall, Inc. Chap 6-16 Do Examples Chapter 6 P221 6.2 , 6.3, 6.4 Answers for 6.2 and 6.4 are in page 770
- 17. © 2003 Prentice-Hall, Inc. Chap 6-17 Standardizing Example 6.2 5 0.12 10 X Z µ σ − − = = = Normal Distribution Standardized Normal Distribution 10σ = 1Zσ = 5µ = 6.2 X Z 0Zµ = 0.12 calc
- 18. © 2003 Prentice-Hall, Inc. Chap 6-18 Example: Normal Distribution Standardized Normal Distribution 10σ = 1Zσ = 5µ = 7.1 X Z0Zµ = 0.21 2.9 5 7.1 5 .21 .21 10 10 X X Z Z µ µ σ σ − − − − = = = − = = = 2.9 0.21− .0832 ( )2.9 7.1 .1664P X≤ ≤ = .0832 calc calc
- 19. © 2003 Prentice-Hall, Inc. Chap 6-19 Z .00 .01 0.0 .5000 .5040 .5080 .5398 .5438 0.2 .5793 .5832 .5871 0.3 .6179 .6217 .6255 .5832 .02 0.1 .5478 Cumulative Standardized Normal Distribution Table (Portion) 0 1Z Zµ σ= = Z = 0.21 Example: ( )2.9 7.1 .1664P X≤ ≤ = (continued) 0
- 20. © 2003 Prentice-Hall, Inc. Chap 6-20 Z .00 .01 -0.3 .3821 .3783 .3745 .4207 .4168 -0.1.4602 .4562 .4522 0.0 .5000 .4960 .4920 .4168.02 -0.2 .4129 Cumulative Standardized Normal Distribution Table (Portion) 0 1Z Zµ σ= = Z = -0.21 Example: ( )2.9 7.1 .1664P X≤ ≤ = (continued) 0
- 21. © 2003 Prentice-Hall, Inc. Chap 6-21 Example: ( )8 .3821P X ≥ = Normal Distribution Standardized Normal Distribution 10σ = 1Zσ = 5µ = 8 X Z0Zµ = 0.30 8 5 .30 10 X Z µ σ − − = = = .3821 calc
- 22. © 2003 Prentice-Hall, Inc. Chap 6-22 Example: ( )8 .3821P X ≥ = (continued) Z .00 .01 0.0 .5000 .5040 .5080 .5398 .5438 0.2 .5793 .5832 .5871 0.3 .6179 .6217 .6255 .6179.02 0.1 .5478 Cumulative Standardized Normal Distribution Table (Portion) 0 1Z Zµ σ= = Z = 0.30 0
- 23. © 2003 Prentice-Hall, Inc. Chap 6-23 Do examples 6.6,6.8,6.10
- 24. © 2003 Prentice-Hall, Inc. Chap 6-24 More Examples of Normal Distribution A set of final exam grades was found to be normally distributed with a mean of 73 and a standard deviation of 8. What is the probability of getting a grade between and 91 on this exam? ( )2 73,8X N: ( )91 ?P X ≤ = Mean 73 Standard Deviation 8 X Value 91 Z Value 2.25 P(X<=91) 0.9877756 Probability for X <= 2.250 X Z 91 8σ = 73µ = =
- 25. © 2003 Prentice-Hall, Inc. Chap 6-25 Example Mao's height is 1.5 meters (mean heights = 1.4 meters Standard deviation = 2.45meters for males) What percentage of the male population is higher (Z>) than Mao? Assume the male heights are normally distributed Calculate Z Score (Z calc) Find Relative Frequency P(X>1.5meters)
- 26. © 2003 Prentice-Hall, Inc. Chap 6-26 What percentage of students scored between 65 and 89? From X Value 65 To X Value 89 Z Value for 65 -1 Z Value for 89 2 P(X<=65) 0.1587 P(X<=89) 0.9772 P(65<=X<=89) 0.8186 Probability for a Range ( )2 73,8X N: ( )65 89 ?P X≤ ≤ = 20 X Z 8965 -1 73µ = Examples of Normal Distribution (continued) ( )2 73,8X N:=
- 27. © 2003 Prentice-Hall, Inc. Chap 6-27 2 Types of problems you will have to solve using z Scores and relative frequencies Example 1: To find Relative Frequency In USA SAT Scores are used to grade students. The mean, µ = 500 and Standard Deviation, σ = 100 N(500,1002 ) When you are given 2 points in a range and have to find the proportion of data points (Relative Frequency) that are between two points. What proportion Relative Frequency is between 450 and 600 P(450 =< X<=600)? 1. Draw Normal Model 2. Indicate on the modal the area between 450 and 600 3. Calculate Z calc for 450 and 600 4. Calculate Relative Frequency P(450 =< X<=600)
- 28. © 2003 Prentice-Hall, Inc. Chap 6-28 .6217 Finding Z Values for Known Probabilities Z .00 0.2 0.0 .5000 .5040 .5080 0.1 .5398 .5438 .5478 0.2 .5793 .5832 .5871 .6179 .6255 .01 0.3 Cumulative Standardized Normal Distribution Table (Portion) What is Z Given Probability = 0.6217 ? .6217 0 1Z Zµ σ= = .31Z = 0
- 29. © 2003 Prentice-Hall, Inc. Chap 6-29 Recovering X Values for Known Probabilities ( ) ( )5 .30 10 8X Zµ σ= + = + = Normal Distribution Standardized Normal Distribution 10σ = 1Zσ = 5µ = ? X Z0Zµ = 0.30 .3821 .6179 This time You Know Mean and Standard Deviation but not X
- 30. © 2003 Prentice-Hall, Inc. Chap 6-30 73µ = Only 5% of the students taking the test scored higher than what grade? ( )2 73,8X N: ( )? .05P X≤ = Cumulative Percentage 95.00% Z Value 1.644853 X Value 86.15882 Find X and Z Given Cum. Pctage. 1.6450 X Z ? =86.16 (continued) More Examples of Normal Distribution ( )2 73,8X N:=
- 31. © 2003 Prentice-Hall, Inc. Chap 6-31 Do Examples 6.8, 6.10, 6.12