Chapter 6 Part 1 Quiz 5 Review PDF

Quiz 5 Review Question 1 Question: Suppose the average score on the first exam is 60, and the variance is 5. If a professor decides to add 10 points to each student’s grade, calculate the new average score and the new variance. BUSINESS STATISTICS |...

Quiz 5 Review Question 1 Question: Suppose the average score on the first exam is 60, and the variance is 5. If a professor decides to add 10 points to each student’s grade, calculate the new average score and the new variance. BUSINESS STATISTICS | Jaggia, Kelly 3-1  Quiz 5 Review Question 1 Answer: Suppose the average score on the first exam is 60, and the variance is 5. If a professor decides to add 10 points to each student’s grade, calculate the new average score and the new variance. New average: New variance: BUSINESS STATISTICS | Jaggia, Kelly 3-2  Quiz 5 Review Question 2 Question: The number of homes that a realtor sells over a one- month period has the probability distribution shown in the table. Calculate the variance. Var(X) = BUSINESS STATISTICS | Jaggia, Kelly 3-3  Quiz 5 Review Question 2 Answer: The number of homes that a realtor sells over a one- month period has the probability distribution shown in the table. Calculate the variance. Var(X) = BUSINESS STATISTICS | Jaggia, Kelly 3-4  6 Continuous Probability Distributions Business Statistics: Communicating with Numbers, 5e By Sanjiv Jaggia and Alison Kelly 9/26/24 McGraw-Hill/Irwin Copyright © 2016 by The McGraw-Hill Companies, Inc. All  Chapter 6 Learning Objectives (LOs) LO 6.1 Describe a continuous random variable LO 6.2 Calculate and interpret probabilities for a random variable that follows continuous uniform distribution LO 6.3 Explain the characteristics of the normal distribution LO 6.4 Calculate and interpret probabilities for a random variable that follows the normal distribution BUSINESS STATISTICS | Jaggia, Kelly 3-6 6.1 Continuous Random Variables and the Uniform Distribution LO 6.1 Describe a continuous random variable Two Types of Random Variable  Discrete Random Variable - Assuming a countable number of distinct value - ex) Values on the roll of dice: 1, 2,…, 6  Continuous Random Variable - Assuming uncountable values within any interval - Cannot describe the values with a list - ex) time (30.1 minutes, or 30.100000000001 minutes) Example: Integers are discrete, real numbers are continuous BUSINESS STATISTICS | Jaggia, Kelly 3-7  6.1 Continuous Random Variables and the Uniform Distribution LO 6.1 Describe a continuous random variable For a continuous random variable X, the probability density function denoted by f(x) will be used - for all possible values x of X - The area under f(x) over all values x of X equals 1 - is defined as the area under f(x) between a and b - - BUSINESS STATISTICS | Jaggia, Kelly 3-8 6.1 Continuous Random Variables and the Uniform Distribution LO 6.1 Describe a continuous random variable BUSINESS STATISTICS | Jaggia, Kelly 3-9  6.1 Continuous Random Variables and the Uniform Distribution LO 6.2 Calculate and interpret probabilities for random variable that follows continuous uniform distribution One of the simplest continuous probability distributions is called the continuous uniform distribution A random variable X follows the continuous uniform distribution, if its probability density function (a: lower limit, b: upper limit) is The underlying random variable has an equally likely chance of assuming a value within a specified range [a, b] BUSINESS STATISTICS | Jaggia, Kelly 3-10  6.1 Continuous Random Variables and the Uniform Distribution LO 6.2 Calculate and interpret probabilities for random variable that follows continuous uniform distribution The probability density function does not directly represent probability - The area under the curve or line represents probability - Area of a rectangle: base times height - BUSINESS STATISTICS | Jaggia, Kelly 3-11  6.1 Continuous Random Variables and the Uniform Distribution LO 6.2 Calculate and interpret probabilities for random variable that follows continuous uniform distribution For uniformly distributed continuous random variables (a: lower limit, b: upper limit), Expected Value Variance Standard Deviation BUSINESS STATISTICS | Jaggia, Kelly 3-12  Example Questions Q1: The amount of gasoline solid daily at a service station is uniformly distributed with a minimum of 2,000 gallons and a maximum of 5,000 gallons. , a. What is the probability that the service station will sell at least 4,000 gallons? b. Calculate the expected value and the standard deviation. BUSINESS STATISTICS | Jaggia, Kelly 3-13  Example Questions A1: a. What is the probability that the service station will sell at least 4,000 gallons? b. Calculate the expected value and the standard deviation. BUSINESS STATISTICS | Jaggia, Kelly 3-14  Example Questions Q2: A manager of a local drugstore is projecting next month’s sales for a particular cosmetic line. She knows from historical data that sales follow a continuous uniform distribution with a lower limit of $2,500 and upper limit of $5,000. , a. Calculate the expected value and standard deviation. b. What is the probability that sales exceed $4,000? c. What is the probability that sales are between $3,200 and $3,800? BUSINESS STATISTICS | Jaggia, Kelly 3-15  Example Questions A2: A manager of a local drugstore is projecting next month’s sales for a particular cosmetic line. She knows from historical data that sales follow a continuous uniform distribution with a lower limit of $2,500 and upper limit of $5,000. a. Calculate the expected value and standard deviation. BUSINESS STATISTICS | Jaggia, Kelly 3-16  Example Questions A2: A manager of a local drugstore is projecting next month’s sales for a particular cosmetic line. She knows from historical data that sales follow a continuous uniform distribution with a lower limit of $2,500 and upper limit of $5,000. b. What is the probability that sales exceed $4,000? BUSINESS STATISTICS | Jaggia, Kelly 3-17  Example Questions A2: A manager of a local drugstore is projecting next month’s sales for a particular cosmetic line. She knows from historical data that sales follow a continuous uniform distribution with a lower limit of $2,500 and upper limit of $5,000. c. What is the probability that sales are between $3,200 and $3,800? BUSINESS STATISTICS | Jaggia, Kelly 3-18  6.2 The Normal Distribution LO 6.3 Explain the characteristics of the normal distribution The normal probability distribution (or normal distribution) is the familiar bell-shaped distribution - Also known as Gaussian distribution - The most extensively used probability distribution in statistical work - Serves as the cornerstone of statistical inference Examples of a random variable following the normal distribution - Salary of employees in a tech firm - Scores on the SAT exam - Heights or weights of newborn babies BUSINESS STATISTICS | Jaggia, Kelly 3-19  6.2 The Normal Distribution LO 6.3 Explain the characteristics of the normal distribution Three Characteristics of the Normal Distribution  The normal distribution is bell-shaped and symmetric around its mean - One side of the mean is the mirror image of the other side - The mean, median and mode are all equal  The normal distribution is described by two parameters (: population mean and : population variance)  The normal distribution is asymptotic - Tails get closer and closer to X axis but never touch it - A normal random variable X: BUSINESS STATISTICS | Jaggia, Kelly 3-20  6.2 The Normal Distribution LO 6.3 Explain the characteristics of the normal distribution The probability density function f(x) of a normal random variable - Referred to as the normal curve or the bell curve - Bell-shaped and symmetric around its mean - : population mean, : population standard deviation : the base of natural logarithm () - Asymptotic BUSINESS STATISTICS | Jaggia, Kelly 3-21  6.2 The Normal Distribution LO 6.3 Explain the characteristics of the normal distribution For normal distributions, increasing the mean shifts curve to the right BUSINESS STATISTICS | Jaggia, Kelly 3-22  6.2 The Normal Distribution LO 6.3 Explain the characteristics of the normal distribution Example: ages of employees in Industries A, B and C BUSINESS STATISTICS | Jaggia, Kelly 3-23  6.2 The Normal Distribution LO 6.3 Explain the characteristics of the normal distribution For normal distributions, increasing the standard deviation “flattens” curve BUSINESS STATISTICS | Jaggia, Kelly 3-24  6.2 The Normal Distribution LO 6.3 Explain the characteristics of the normal distribution Example: ages of employees in Industries A, B and C BUSINESS STATISTICS | Jaggia, Kelly 3-25  6.2 The Normal Distribution LO 6.3 Explain the characteristics of the normal distribution The standard normal random variable Z is a normal random variable with E(Z) = 0 (mean = 0) and SD(Z) = 1 (standard deviation = 1) - (Symmetric around its mean of 0) Any normal distribution can be converted to a standard normal distribution BUSINESS STATISTICS | Jaggia, Kelly 3-26  Next Class (10/03) Quiz 6 (Chapter 5 & 6) The Normal Distribution (Ch. 6) Homework Assignment 3 was posted on Canvas (Due: 10/04, 11:59pm) BUSINESS STATISTICS | Jaggia, Kelly 3-27