1 1 Slide Continuous Probability Distributions n A continuous random variable can assume any value in an interval on the real line or in a collection of intervals. n It is not possible to talk about the probability of the random variable assuming a particular value. n Instead, we talk about the probability of the random variable assuming a value within a given interval. x f ( x ) Normal

2 2 Slide Continuous Probability Distributions n The probability of the random variable assuming a value within some given interval from x 1 to x 2 is defined to be the area under the graph of the probability density function between x 1 and x 2. x f ( x ) Normal x1 x1x1 x1 x1 x1x1 x1 x2 x2x2 x2 x2 x2x2 x2

3 3 Slide Normal Probability Distribution n The normal probability distribution is the most important distribution for describing a continuous random variable. n It is widely used in statistical inference. x f ( x ) Normal x1 x1x1 x1 x1 x1x1 x1 x2 x2x2 x2 x2 x2x2 x2

4 4 Slide Normal Probability Distribution n It has been used in a wide variety of applications: Heights of people Heights Scientific measurements measurementsScientific Test scores scoresTest Amounts of rainfall Amounts

5 5 Slide Normal Probability Distribution n Normal Probability Density Function  = mean  = standard deviation  = e = where:

6 6 Slide The distribution is symmetric; its skewness The distribution is symmetric; its skewness measure is zero. measure is zero. The distribution is symmetric; its skewness The distribution is symmetric; its skewness measure is zero. measure is zero. Normal Probability Distribution n Characteristics x

7 7 Slide The entire family of normal probability The entire family of normal probability distributions is defined by its mean  and its distributions is defined by its mean  and its standard deviation . standard deviation . The entire family of normal probability The entire family of normal probability distributions is defined by its mean  and its distributions is defined by its mean  and its standard deviation . standard deviation . Normal Probability Distribution n Characteristics Standard Deviation  Mean  x

8 8 Slide The highest point on the normal curve is at the The highest point on the normal curve is at the mean, which is also the median and mode. mean, which is also the median and mode. The highest point on the normal curve is at the The highest point on the normal curve is at the mean, which is also the median and mode. mean, which is also the median and mode. Normal Probability Distribution n Characteristics x

9 9 Slide Normal Probability Distribution n Characteristics The mean can be any numerical value: negative, The mean can be any numerical value: negative, zero, or positive. zero, or positive. The mean can be any numerical value: negative, The mean can be any numerical value: negative, zero, or positive. zero, or positive. x

10 Slide Normal Probability Distribution n Characteristics  = 15  = 25 The standard deviation determines the width of the curve: larger values result in wider, flatter curves. The standard deviation determines the width of the curve: larger values result in wider, flatter curves. x

11 Slide Probabilities for the normal random variable are Probabilities for the normal random variable are given by areas under the curve. The total area given by areas under the curve. The total area under the curve is 1 (.5 to the left of the mean and under the curve is 1 (.5 to the left of the mean and.5 to the right)..5 to the right). Probabilities for the normal random variable are Probabilities for the normal random variable are given by areas under the curve. The total area given by areas under the curve. The total area under the curve is 1 (.5 to the left of the mean and under the curve is 1 (.5 to the left of the mean and.5 to the right)..5 to the right). Normal Probability Distribution n Characteristics.5.5 x

12 Slide Normal Probability Distribution n Characteristics of values of a normal random variable of values of a normal random variable are within of its mean. are within of its mean. of values of a normal random variable of values of a normal random variable are within of its mean. are within of its mean.68.26%68.26% +/- 1 standard deviation of values of a normal random variable of values of a normal random variable are within of its mean. are within of its mean. of values of a normal random variable of values of a normal random variable are within of its mean. are within of its mean %95.44% +/- 2 standard deviations of values of a normal random variable of values of a normal random variable are within of its mean. are within of its mean. of values of a normal random variable of values of a normal random variable are within of its mean. are within of its mean.99.72%99.72% +/- 3 standard deviations

13 Slide Normal Probability Distribution n Characteristics x  – 3   – 1   – 2   + 1   + 2   + 3  68.26% 95.44% 99.72%

14 Slide Standard Normal Probability Distribution A random variable having a normal distribution A random variable having a normal distribution with a mean of 0 and a standard deviation of 1 is with a mean of 0 and a standard deviation of 1 is said to have a standard normal probability said to have a standard normal probability distribution. distribution. A random variable having a normal distribution A random variable having a normal distribution with a mean of 0 and a standard deviation of 1 is with a mean of 0 and a standard deviation of 1 is said to have a standard normal probability said to have a standard normal probability distribution. distribution.

15 Slide  0 z The letter z is used to designate the standard The letter z is used to designate the standard normal random variable. normal random variable. The letter z is used to designate the standard The letter z is used to designate the standard normal random variable. normal random variable. Standard Normal Probability Distribution

16 Slide Table of Cumulative Distribution Function Standard Normal Probability Distribution Traditional way calculating probability without using computer programs. Traditional way calculating probability without using computer programs. Make sure you understand the meaning of the table before using it. Make sure you understand the meaning of the table before using it.

17 Slide z Table is located in the Tables section of Contents on the Course Website z Table is located in the Tables section of Contents on the Course Website Standard Normal Probability Distribution

18 Slide

19 Slide

20 Slide Exercise 1 Standard Normal Probability Distribution Find P(z ≤ 1.00)

21 Slide Exercise 1: Find P(z ≤ 1.00) Standard Normal Probability Distribution Answer: P(z ≤ 1.00) =

22 Slide Exercise 2 Standard Normal Probability Distribution Find P(-0.5 ≤ z ≤ 1.25)

23 Slide Exercise 2: Find P(-0.5 ≤ z ≤ 1.25) Standard Normal Probability Distribution Answer: P(-0.5 ≤ z ≤ 1.25) = P(z ≤ 1.25) - P(z ≤ -0.5) =Area I – Area II = – = Answer: P(-0.5 ≤ z ≤ 1.25) = P(z ≤ 1.25) - P(z ≤ -0.5) =Area I – Area II = – = = - Area I Area II

24 Slide Exercise 3 Standard Normal Probability Distribution Find P(-1 ≤ z ≤ 1)

25 Slide Exercise 3: Find P(-1 ≤ z ≤ 1) Standard Normal Probability Distribution Answer: P(-1 ≤ z ≤ 1) = P(z ≤ 1) - P(z ≤ -1) =Area I – Area II = – = Answer: P(-1 ≤ z ≤ 1) = P(z ≤ 1) - P(z ≤ -1) =Area I – Area II = – = = - Area I Area II

26 Slide Exercise 4 Standard Normal Probability Distribution Find P(z ≥ 1.58)

27 Slide Exercise 4: Find P(z ≥ 1.58) Standard Normal Probability Distribution Answer: P(z ≥ 1.58) = 1 - P(z ≤ 1.58) = 1 – = = = - Area I Area II

28 Slide Exercise 5 Standard Normal Probability Distribution Find z for cumulative probability = 0.1 in the upper tail tail

29 Slide Exercise 5: Find z for cumulative probability = 0.1 in the upper tail Exercise 5: Find z for cumulative probability = 0.1 in the upper tail Standard Normal Probability Distribution Probability is 0.1 in the upper tail What is the z value here? The lower-tail probability is = 0.9

30 Slide Exercise 5: Find z for cumulative probability = 0.1 in the upper tail Exercise 5: Find z for cumulative probability = 0.1 in the upper tail Standard Normal Probability Distribution Answer: 1. Look up the cumulative probability value closest to 0.9 (1-0.1)  Use the z value found for the estimation  1.28 Answer: 1. Look up the cumulative probability value closest to 0.9 (1-0.1)  Use the z value found for the estimation  1.28 Probability values Find the closest one to is here

31 Slide n Converting to the Standard Normal Distribution Standard Normal Probability Distribution We can think of z as a measure of the number of standard deviations x is from .

32 Slide Exercise 6 Standard Normal Probability Distribution Suppose the population mean is 10 and population standard deviation is 2. What is the probability that the random variable x is between 10 and 14? Suppose the population mean is 10 and population standard deviation is 2. What is the probability that the random variable x is between 10 and 14?

33 Slide Exercise 6 Standard Normal Probability Distribution Suppose the population mean is 10 and population standard deviation is 2. What is the probability that the random variable x is between 10 and 14? Suppose the population mean is 10 and population standard deviation is 2. What is the probability that the random variable x is between 10 and 14? Answer: When x=10, z=(10-10)/2 = 0 When x=14, z=(14-10)/2 = 2 Thus, P(0 ≤ z ≤ 2) = P(z ≤ 2) - P(z ≤ 0) = – = Answer: When x=10, z=(10-10)/2 = 0 When x=14, z=(14-10)/2 = 2 Thus, P(0 ≤ z ≤ 2) = P(z ≤ 2) - P(z ≤ 0) = – =

34 Slide is used to compute the z value is used to compute the z value given a LEFT-TAIL cumulative probability. given a LEFT-TAIL cumulative probability. is used to compute the z value is used to compute the z value given a LEFT-TAIL cumulative probability. given a LEFT-TAIL cumulative probability. NORMSINVNORMSINV NORM S INV is used to compute the LEFT-TAIL is used to compute the LEFT-TAIL cumulative probability given a z value. is used to compute the LEFT-TAIL is used to compute the LEFT-TAIL cumulative probability given a z value. NORMSDISTNORMSDIST NORM S DIST Using Excel to Compute Standard Normal Probabilities n Excel has two functions for computing probabilities and z values for a standard normal distribution: (The “S” in the function names reminds us that they relate to the standard normal probability distribution.)

35 Slide Practice Exercise 1 to 6 by using Excel Standard Normal Probability Distribution

36 Slide Exercise 1 Standard Normal Probability Distribution Find P(z ≤ 1.00) Ans:=NORMSDIST(1)=0.8413Ans:=NORMSDIST(1)=0.8413

37 Slide Exercise 2 Standard Normal Probability Distribution Find P(-0.5 ≤ z ≤ 1.25) Answer: P(-0.5 ≤ z ≤ 1.25) = P(z ≤ 1.25) - P(z ≤ -0.5) =Area I – Area II =NORMSDIST(1.25)-NORMSDIST(-0.5) = – = Answer: P(-0.5 ≤ z ≤ 1.25) = P(z ≤ 1.25) - P(z ≤ -0.5) =Area I – Area II =NORMSDIST(1.25)-NORMSDIST(-0.5) = – =

38 Slide Exercise 3 Standard Normal Probability Distribution Find P(-1 ≤ z ≤ 1) Answer: P(-1 ≤ z ≤ 1) = P(z ≤ 1) - P(z ≤ -1) =Area I – Area II =NORMSDIST(1)-NORMSDIST(-1) = – = Answer: P(-1 ≤ z ≤ 1) = P(z ≤ 1) - P(z ≤ -1) =Area I – Area II =NORMSDIST(1)-NORMSDIST(-1) = – =

39 Slide Exercise 4 Standard Normal Probability Distribution Find P(z ≥ 1.58) Answer: P(z ≥ 1.58) = 1 - P(z ≤ 1.58) = 1 – NORMSDIST(1.58) = 1 – = Answer: P(z ≥ 1.58) = 1 - P(z ≤ 1.58) = 1 – NORMSDIST(1.58) = 1 – =0.0571

40 Slide Exercise 5 Standard Normal Probability Distribution Find z for cumulative probability = 0.1 in the upper tail tail Answer: = NORMSINV(1-0.1) or =NORMSINV(0.9) =1.28Answer: =1.28

41 Slide Exercise 6 Standard Normal Probability Distribution Suppose the population mean is 10 and population standard deviation is 2. What is the probability that the random variable x is between 10 and 14? Suppose the population mean is 10 and population standard deviation is 2. What is the probability that the random variable x is between 10 and 14? Answer: When x=10, z=(10-10)/2 = 0 When x=14, z=(14-10)/2 = 2 Thus, P(0 ≤ z ≤ 2) = P(z ≤ 2) - P(z ≤ 0) =NORMSDIST(2)-NORMSDIST(0) = – = Answer: When x=10, z=(10-10)/2 = 0 When x=14, z=(14-10)/2 = 2 Thus, P(0 ≤ z ≤ 2) = P(z ≤ 2) - P(z ≤ 0) =NORMSDIST(2)-NORMSDIST(0) = – =

42 Slide n Excel Formula View Using Excel to Compute Standard Normal Probabilities n Excel Worksheet View See 6Exercises-key.xlsx

43 Slide n Excel Formula Worksheet Using Excel to Compute Additional Standard Normal Probabilities

44 Slide n Excel Value Worksheet Using Excel to Compute Additional Standard Normal Probabilities

45 Slide n Excel Formula Worksheet Using Excel to Compute Additional Standard Normal Probabilities

46 Slide n Excel Value Worksheet Using Excel to Compute Additional Standard Normal Probabilities

47 Slide Standard Normal Probability Distribution n Example: Pep Zone Pep Zone sells auto parts and supplies including Pep Zone sells auto parts and supplies including a popular multi-grade motor oil. When the stock of this oil drops to 20 gallons, a replenishment order is placed. The store manager is concerned that sales are The store manager is concerned that sales are being lost due to stockouts while waiting for a replenishment order. Pep Zone 5w-20 Motor Oil

48 Slide It has been determined that demand during It has been determined that demand during replenishment lead-time is normally distributed with a mean of 15 gallons and a standard deviation of 6 gallons. Standard Normal Probability Distribution n Example: Pep Zone The manager would like to know the probability The manager would like to know the probability of a stockout during replenishment lead-time. In other words, what is the probability that demand during lead-time will exceed 20 gallons? P ( x > 20) = ? P ( x > 20) = ?

49 Slide Standard Normal Probability Distribution n Example: Pep Zone

50 Slide z = ( x -  )/  z = ( x -  )/  = ( )/6 = ( )/6 =.83 =.83 z = ( x -  )/  z = ( x -  )/  = ( )/6 = ( )/6 =.83 =.83 n Solving for the Stockout Probability Step 1: Convert x to the standard normal distribution. Pep Zone 5w-20 Motor Oil Step 2: Find the area under the standard normal curve to the left of z =.83. curve to the left of z =.83. Step 2: Find the area under the standard normal curve to the left of z =.83. curve to the left of z =.83. see next slide see next slide Standard Normal Probability Distribution

51 Slide n Cumulative Probability Table for the Standard Normal Distribution Pep Zone 5w-20 Motor Oil P ( z <.83) Standard Normal Probability Distribution

52 Slide P ( z >.83) = 1 – P ( z.83) = 1 – P ( z <.83) = = =.2033 =.2033 P ( z >.83) = 1 – P ( z.83) = 1 – P ( z <.83) = = =.2033 =.2033 n Solving for the Stockout Probability Step 3: Compute the area under the standard normal curve to the right of z =.83. curve to the right of z =.83. Step 3: Compute the area under the standard normal curve to the right of z =.83. curve to the right of z =.83. Pep Zone 5w-20 Motor Oil Probability of a stockout of a stockout P ( x > 20) Standard Normal Probability Distribution

53 Slide n Solving for the Stockout Probability 0.83 Area =.7967 Area = =.2033 =.2033 z Pep Zone 5w-20 Motor Oil Standard Normal Probability Distribution

54 Slide n Standard Normal Probability Distribution Standard Normal Probability Distribution If the manager of Pep Zone wants the probability If the manager of Pep Zone wants the probability of a stockout during replenishment lead-time to be no more than.05, what should the reorder point be? (Hint: Given a probability, we can use the standard normal table in an inverse fashion to find the corresponding z value.)

55 Slide n Solving for the Reorder Point Pep Zone 5w-20 Motor Oil 0 Area =.9500 Area =.0500 z z.05 Standard Normal Probability Distribution

56 Slide n Solving for the Reorder Point Pep Zone 5w-20 Motor Oil Step 1: Find the z -value that cuts off an area of.05 in the right tail of the standard normal in the right tail of the standard normal distribution. distribution. Step 1: Find the z -value that cuts off an area of.05 in the right tail of the standard normal in the right tail of the standard normal distribution. distribution. We look up the complement of the tail area ( =.95) Standard Normal Probability Distribution Excel Approach: z = NORMSINV(0.95) = 1.645

57 Slide n Solving for the Reorder Point Pep Zone 5w-20 Motor Oil Step 2: Convert z.05 to the corresponding value of x. x =  + z.05  x =  + z.05   = (6) = or 25 = or 25 x =  + z.05  x =  + z.05   = (6) = or 25 = or 25 A reorder point of 25 gallons will place the probability A reorder point of 25 gallons will place the probability of a stockout during leadtime at (slightly less than).05. of a stockout during leadtime at (slightly less than).05. Standard Normal Probability Distribution conversion function

58 Slide n Solving for the Reorder Point Pep Zone 5w-20 Motor Oil By raising the reorder point from 20 gallons to By raising the reorder point from 20 gallons to 25 gallons on hand, the probability of a stockout decreases from about.20 to.05. This is a significant decrease in the chance that Pep This is a significant decrease in the chance that Pep Zone will be out of stock and unable to meet a customer’s desire to make a purchase. Standard Normal Probability Distribution

59 Slide Using Excel to Compute Normal Probabilities n Excel has two functions for computing cumulative probabilities and x values for any normal distribution: NORMDIST is used to compute the LEFT-TAIL cumulative probability given an x value. NORMDIST is used to compute the LEFT-TAIL cumulative probability given an x value. NORMINV is used to compute the x value given a LEFT-TAIL cumulative probability. NORMINV is used to compute the x value given a LEFT-TAIL cumulative probability.

60 Slide Using Excel to Compute Normal Probabilities Syntax: n NORMDIST(x, mu, sigma, cumulative) n mu: population mean n sigma: population standard deviation n cumulative: n true—probability is returned n false – height of the bell-shaped curve is returned n NORMINV(p, mu, sigma) n p: probability

61 Slide n Excel Formula Worksheet Using Excel to Compute Normal Probabilities Pep Zone 5w-20 Motor Oil Lower tail probability: NORMDIST(20, 15, 6, TRUE) Upper tail probability: 1-NORMDIST(20, 15, 6, TRUE)

62 Slide n Excel Value Worksheet Using Excel to Compute Normal Probabilities Note: P( x > 20) =.2023 here using Excel, while our previous manual approach using the z table yielded.2033 due to our rounding of the z value. Pep Zone 5w-20 Motor Oil