QMBU 301 – Quantitative Methods in Business

QMBU 301 – Quantitative Methods in Business
Yalçın Akçay CASE 270 QMBU 301 – Quantitative Methods in Business

What is Statistics? We muddle through life making choices based on incomplete information QMBU 301 – Quantitative Methods in Business

What is Statistics? Most of us live comfortably with some level of uncertainty QMBU 301 – Quantitative Methods in Business

What is Statistics? What makes statistics unique is its ability to quantify uncertainty, to make it precise. This allows statisticians to make categorical statements, with complete assurance – about their level of uncertainy QMBU 301 – Quantitative Methods in Business

What is Statistics? This is not just a matter of ordering soup! Statistics also involves matters of life and death. Challenger accident 28 January 1986 QMBU 301 – Quantitative Methods in Business

What is Statistics? Finally, in discussing statistics, it is hard to avoid mentioning one other thing: The widespread mistrust of statistics in the world today. Everyone knows about “lying with statistics”, while good statistical analysis is nearly impossible to find in daily life. Statistics are like bikini; What is revealed is interesting; What is concealed is crucial. QMBU 301 – Quantitative Methods in Business

So what is statistics? Statistics is a branch of mathematics dealing with the analysis and interpretation of masses of data. QMBU 301 – Quantitative Methods in Business

What Next? QMBU 301 – Quantitative Methods in Business

Basic Definitions A population is the group of all items of interest (everything you wish to study) to a statistics practitioner A sample is a subset of the population, often randomly chosen and preferably representative of the population as a whole. A variable is an attribute, or measurement, on members of a population. An observation is a list of all variable values for a single member of a population (data) Why pick a sample at all? It would take too much time to study the entire population. It would take too much money to study the entire population. It might not be possible to identify all the members of the population. QMBU 301 – Quantitative Methods in Business

Basic Definitions A variable is numerical if meaningful arithmetic can be performed on it. Otherwise, the variable is categorical. A numerical variable is discrete if its possible values can be counted. A continuous variable is the result of an essentially continuous measurement. A categorical variable is ordinal if there is a natural ordering of its possible values. If there is no natural ordering, it is nominal. Cross-sectional data are data on a population at a distinct point in time. Time series data are collected across time. Why pick a sample at all? It would take too much time to study the entire population. It would take too much money to study the entire population. It might not be possible to identify all the members of the population. QMBU 301 – Quantitative Methods in Business

Examples Marks from an exam: Marital status: single, married, married, married, divorced, widowed, single Ratings of various aspects of a course: poor, fair, good, fair, very good, excellent, good, poor, good QMBU 301 – Quantitative Methods in Business

Describing Data: Graphs and Tables QMBU 301 – Quantitative Methods in Business

Describing Data Example: Annual salary figures and related data for 52 employees of Beta Technologies Inc. gender age number of years of relevant work experience prior to employement at Beta number of years of professional experience at Beta Technologies number of years of post-secondary education annual salary QMBU 301 – Quantitative Methods in Business

Describing Data What are some questions a human resources manager would ask? What is the average age? How much variation is there in the salary? Is there a relationship of age, gender, experience and education with salary? QMBU 301 – Quantitative Methods in Business

Frequency Tables and Histograms
A frequency table is a table containing each category or value that a variable might have and the number of times that each one occurs in the data. A histogram is a bar chart of these frequencies. QMBU 301 – Quantitative Methods in Business

Analyzing the Histogram
In statistics, the features of interest for a set of numerical data can be classified as: center – describes where, numerically, the data are centered or concentrated shape – describes how the data are spread out around the center with respect to the symmetry or skewness of the data variability – describes how the data are spread out around the center with respect to the smoothness and magnitude of the variation Together these three features describe the distribution of the data. QMBU 301 – Quantitative Methods in Business

A histogram is said to be symmetric if, when we draw a vertical line down the center of the histogram (the peak), the two sides are identical in shape and size. QMBU 301 – Quantitative Methods in Business

A skewed histogram is one with a long tail extending to either right or left. Right skewed if tail extends to the right (positive skewness) Left skewed if tail extends to the left (negative skewness) QMBU 301 – Quantitative Methods in Business

Mode is the observation with the greatest frequency. A unimodal histogram is one with a single peak. A bimodal histogram is one with two peaks, not necessarily equal in height. If you have a bimodal histogram it may indicate that data is coming from two different populations. QMBU 301 – Quantitative Methods in Business

When data are not very variable, the frequency of observations decreases steadily as you move away from the center. QMBU 301 – Quantitative Methods in Business

Describing the Relationship Between Two Variables
Suppose we are interested in the relationship between the number of years of professional experience at Beta Technologies and the annual salary of an employee. The graphical technique used to describe the relationship between two variables is the scatter plot. QMBU 301 – Quantitative Methods in Business

Analyzing Scatter Plots
No relationship between the two variables QMBU 301 – Quantitative Methods in Business

Linear relationship between the two variables positive negative QMBU 301 – Quantitative Methods in Business

Non-linear relationship between the two variables Warning: Just because there is a relationship between two variables, it does not mean that there is a cause-and-effect relationship between the two QMBU 301 – Quantitative Methods in Business

Time Series Plots A scatter plot with the time series variable on the vertical axis and time itself on the horizontal axis. Useful for forecasting future values of a time series. observe trend and seasonal patterns QMBU 301 – Quantitative Methods in Business

Art and Science of Graphical Presentations
Graphical excellence: is well-designed presentations of interesting data – a matter of substance, of statistics and of design gives the viewer the greatest number of ideas in the shortest time with the least ink in the smallest space is nearly always multivariate requires telling the truth about the data Rules: The graph presents large data sets concisely and coherently The ideas and concepts the statistics practitioner wants to deliver are clearly understood by the viewer The graph encourages the viewer to compare two or more variables The display induces the viewer to adress the substance of the data and not the form of the graph There is no distortion of what the data reveal QMBU 301 – Quantitative Methods in Business

Examples Useless graph data set this small does not need a graphical display so what? no “information” QMBU 301 – Quantitative Methods in Business

Examples Chartjunk (one of the worst you will ever see) It contains very little data (a small table would suffice) The idea that the author wants to deliver is not clear (perhaps no idea) No analysis associated with the chart Viewer addresses the design rather than the content QMBU 301 – Quantitative Methods in Business

Examples The chart proveides rich details about the factors that affect the financial circumstances (more than just a graph – a short story!) QMBU 301 – Quantitative Methods in Business

Examples Size distortion – the snowman grows in width as well as height QMBU 301 – Quantitative Methods in Business

Describing Data: Summary Measures QMBU 301 – Quantitative Methods in Business

Sample Statistic or Population Parameter
A parameter is a descriptive measurement about a population. A statistic is a desciptive measurement about a sample. We use statistics from a sample to make inferences about the population parameters. Parameters are usually represented by Greek letters and statistics by Roman letter. QMBU 301 – Quantitative Methods in Business

Numerical Measures Central Tendency mean, median and mode Spread range, variance and standard deviation Association covariance and correlation Numerical measures allow the statistics practitioner to be more precise in describing characteristics of data. QMBU 301 – Quantitative Methods in Business

Measures of Central Tendency
The mean is the arithmetic average of all values of a variable. (the mean of 3, 7, 4, 9, 7 is ( )/5 = 6) sample mean population mean μ QMBU 301 – Quantitative Methods in Business

Algebraic formula: QMBU 301 – Quantitative Methods in Business

The median is the middle observation when the data are listed from smallest to largest. If there are an even number of observations then the median is the average of the two middle values 4, 9, 7, 12, 6, 5, 5, 9 4, 5, 5, 6, 7, 9, 9, 12 Mode is the most frequently occuring value. If the variables are continuous, mode is irrelevant. sort 6.5 QMBU 301 – Quantitative Methods in Business

Mean, Median, Mode: Which is Best?
The mean is generally the first selection. However, there are several circumstances when the median is better. Average = 9.08 Median = 8.5 outlier: extreme observation (either very small or very large but not both) Average = Median = 8.5 outlier QMBU 301 – Quantitative Methods in Business

When there is a relatively small number of outliers, the median usually produces a better measure of center "Should we scare the opposition by announcing our mean height or lull them by announcing our median height?" QMBU 301 – Quantitative Methods in Business

QMBU 301 – Quantitative Methods in Business

Mode is the least preferred. In some cases, there may appear to be more than one mode (e.g. bimodal data) QMBU 301 – Quantitative Methods in Business

Comparing the Mean and the Median
For most sets of data, the mean and the median will be very close to each other in value. mean median QMBU 301 – Quantitative Methods in Business

Comparing the Mean and the Median
When data are more spread out in one direction (data are skewed), the mean is pulled toward these values. ≤ median ≤ mean mean median QMBU 301 – Quantitative Methods in Business

What happens if you add a constant to your data 3, 7, 4, 9, 7  Average = 6 (3+5), (7+5), (4+5), (9+5), (7+5)  Average = ? multiply your data with a constant (3x2), (7x2), (4x2), (9x2), (7x2)  Average = ? QMBU 301 – Quantitative Methods in Business

Median splits the data in half (50th percentile) half of the data is is below the median 50% 50% 2 nd section QMBU 301 – Quantitative Methods in Business

Divide the data into four parts  quartiles 1st quartile  25% of data at or below it 2nd quartile  50% of data at or below it 3rd quartile  75% of data at or below it Interquartile range (IQR) is the difference between the 3rd quartile and the 1st quartile 25% 25% 25% 25% IQR QMBU 301 – Quantitative Methods in Business

kth percentile = k% of data are at or below this value e.g. 10th percentile of a data set is the number below which there exists 10% of the data 25th percentile = 1st quartile 75th percentile = 3rd quartile k% QMBU 301 – Quantitative Methods in Business

Measures of Spread We need to know how spread out or dispersed the data values are relative to typical values. Understanding the variation in a set of data is of critical importance to statistics. Minimum – smallest value in the data set Maximum – largest value in the data set Range – difference between the maximum and the minimum QMBU 301 – Quantitative Methods in Business

Measures of Spread Consider the data set: 3, 7, 4, 9, 7 Let’s try to determine the average deviation of the data from the mean. (3-6), (7-6), (4-6), (9-6), (7-6)  Average = ? QMBU 301 – Quantitative Methods in Business

Measures of Spread Variance is the average of the squared deviations of the data values from the mean. 3, 7, 4, 9, 7  Average = 6 (3-6)2, (7-6)2, (4-6)2, (9-6)2, (7-6)2  Average = 4.8 Sample variance s2 Population variance 2 QMBU 301 – Quantitative Methods in Business

Measures of Spread Variance increases as there is more variability around the mean. Large deviations from the mean contribute heavily (punished) to the variance because they are squared. QMBU 301 – Quantitative Methods in Business

Measures of Spread What happens if you add a constant to your data 3, 7, 4, 9, 7  Variance = 4.8 (3+5), (7+5), (4+5), (9+5), (7+5)  Variance = ? multiply your data with a constant (3x2), (7x2), (4x2), (9x2), (7x2)  Variance = ? QMBU 301 – Quantitative Methods in Business

Measures of Spread Standard deviation is the square root of variance. Standard deviation has the same unit as the data. population standard deviation sample standard deviation QMBU 301 – Quantitative Methods in Business

Measures of Spread Standard deviation is not as intuitive or appealing as the range (immediate picture – although sometimes false –of how far the data spread out around the center) Empirical Rule: (if the histogram is bell-shaped) about 68% of all observations are within one standard deviation of the mean about 95% of all observations are within two standard deviations of the mean about 99.7% of all observations are within three standard deviations of the mean QMBU 301 – Quantitative Methods in Business

Measures of Spread If a sample of employees were grouped according to their heights, we might see an arrangement like this. QMBU 301 – Quantitative Methods in Business

Measures of Spread Drawing a smooth line around the group of employees produces a bell-shaped curve, which shows that most people have heights gathered around 67.5 inches. QMBU 301 – Quantitative Methods in Business

Measures of Spread QMBU 301 – Quantitative Methods in Business

Measures of Spread Chebysheff’s Rule The proportion of observations in any sample that lie within k standard deviations of the mean is at least Example: k=2  75% k=3  88.9% QMBU 301 – Quantitative Methods in Business

Measures of Spread Consider the following example in which monthly revenue and profit data have been collected for the last 5 years. standard deviation of monthly revenues = $100,000 standard deviation of monthly profits = $5,000 mean of monthly revenues = $400,000 mean of monthly profits = $10,000 Do you think the company recorded a loss in any month? What is the likelihood? Distribution of Monthly Dow Returns QMBU 301 – Quantitative Methods in Business

Measures of Spread Is the standard deviation of 10 a large number? Is revenue or profit more variable? standard deviation of monthly revenues = $100,000 standard deviation of monthly profits = $5,000 mean of monthly revenues = $400,000 mean of monthly profits = $10,000 The coefficient of variation of a set of observations is the standard deviation of the observations divided by their mean. QMBU 301 – Quantitative Methods in Business

Measures of Spread Interquartile range (IQR) is the difference between the 3rd quartile and the 1st quartile 25% 25% By convention outliers are group into two: Mild (potential) outliers: More than 1.5 IQR below Q1 or above Q3 Extreme (serious) outliers: More than 3 IQR below Q1 or above Q3 IQR QMBU 301 – Quantitative Methods in Business

Measures of Shape Kurtosis measures how sharply peaked a distribution is. Values close to 0 indicate normally peaked data. Negative values indicate a distribution that is flatter than normal Positive values indicate a distribution with a sharper than normal peak QMBU 301 – Quantitative Methods in Business

Measures of Shape Skewness index measures the degree of skewness of a distribution. A value close to 0 indicates symmetric data. Negative values indicate negative/left skew. Positive values indicate positive/right skew. QMBU 301 – Quantitative Methods in Business

Measures of Association
Scatter plot describes the relationship between two variables graphically. Covariance and correlation summarize the strength of the linear relationship between the two variables numerically The two variabes, say X and Y, must have the same number of observations (paired variables). QMBU 301 – Quantitative Methods in Business

Covariance Correlation QMBU 301 – Quantitative Methods in Business

The advantage that the correlation has over covariance is that the correlation is always between -1 and +1. Corr(X,Y) = -1  negative linear relationship Corr(X,Y) = +1  positive linear relationship Corr(X,Y) = 0  no linear relationship All other values of correlation are judged in relation to these three values. QMBU 301 – Quantitative Methods in Business

Covar(X,Y) = 11.88 Corr(X,Y) = 0.91 Covar(X,Y) = Corr(X,Y) = QMBU 301 – Quantitative Methods in Business

Covar(X,Y) = 85.43 Corr(X,Y) = 0.20 Covar(X,Y) = Corr(X,Y) = QMBU 301 – Quantitative Methods in Business

Dilbert... QMBU 301 – Quantitative Methods in Business

Introduction to Probability QMBU 301 – Quantitative Methods in Business

Probability Essentials
Consider the following variables: Time between arrivals of customers to a bank Number of defective items in a production batch Return from a stock Customer demand for a new product Changes in interest rates random variables QMBU 301 – Quantitative Methods in Business

Random variable associates a numerical value with each possible outcome of a random phenomenon (experiment). Probability is a number between 0 and 1 that measures the likelihood that some event will occur. The probability distribution of a random variable determines the probability that the random variable will take on each of its possible values. QMBU 301 – Quantitative Methods in Business

There are two types of random variables discrete – countable number of possible values continuous – a continuum of possible values Time between arrivals of customers to a bank Number of defective items in a production batch Return from a stock Customer demand for a new product Changes in interest rates Define a random variable for each. Discrete or continuous? QMBU 301 – Quantitative Methods in Business

The manager of a computer store has kept track of the number of computers sold per day. On the basis of this information, the manager produced the following list of the number of daily sales. number of computers sold - d Probability P(D=d) 0.1 1 0.2 2 0.3 3 4 3 2 4 1 (0.1) (0.2) (0.3) Sample space: a list of all possible outcomes of the experiment. Event: a collection or set of one or more outcomes of a sample space. QMBU 301 – Quantitative Methods in Business

If A is an event, then the complement of A, denoted by Ac is the event that A does not occur. Event A: At most 1 sold Event Ac: 2 or more sold 1 (0.1) (0.2) 2 P(A) = = 0.3 P(Ac) = = 0.7 (0.3) 3 4 (0.3) (0.1) P(Ac) = 1 – P(A) rule of complements QMBU 301 – Quantitative Methods in Business

Events are mutually exclusive if at most one them can occur (if one occurs, none of the others can occur). Event A: At most 1 sold Event B: 4 sold Event C: 2 or 3 sold 3 2 4 1 (0.1) (0.2) (0.3) One of A, B or C must occur  exhaustive QMBU 301 – Quantitative Methods in Business

Addition rule: If n events, A1, A2, ..., An are mutually exclusive then P(at least one of A1, A2, ..., An) = P(A1) + P(A2)+...+P(An) 3 2 4 1 (0.1) (0.2) (0.3) P(D ≤ 2) = P(D=0) + P(D=1) + P(D=2) A1 A2 A3 P(1<D ≤ 3) = P(D=2) + P(D=3) A1 A2 QMBU 301 – Quantitative Methods in Business

Addition rule for events that are not mutually exclusive: If two events A and B are not mutually exclusive that means there is a probability that both could occur.  P(A and B) > 0 Event A: At least 2 sold Event B: Less than or equal to 4 sold 3 2 4 1 (0.1) (0.2) (0.3) P(A or B) = P(A) + P(B) – P(A and B) QMBU 301 – Quantitative Methods in Business

1 2 3 QMBU 301 – Quantitative Methods in Business

? 1 2 3 QMBU 301 – Quantitative Methods in Business

We frequently need to know how two events are related. We would like to know the likelihood of one event given the occurence of another related event. Marketing a product: Event A: a person chosen at random buys the product Event B: a person in the sample space has seen an advirtisement for the product If a person has seen the advirtisement, what is the probability that s/he will buy the product? QMBU 301 – Quantitative Methods in Business

New information changes the probability of an event. The conditional probability of an event A given an event B is: multiplication rule QMBU 301 – Quantitative Methods in Business

A marketing research team is interested in measuring the effectiveness of an advertisement for a new product. They take a random sample of 500 people and ask them whether they have bought the new product and whether they saw an advertisement before the purchase. Here are the results: Saw advertisement Did not see advertisement Total Purchased the product 175 45 220 Did not purchase the product 100 180 280 275 225 500 QMBU 301 – Quantitative Methods in Business

The marketing research firm is interested in finding out whether seeing the advertisment (B) affects the probability that a person will buy the product (A). Saw advertisement Did not see advertisement Total Purchased the product 175 45 220 Did not purchase the product 100 180 280 275 225 500 P(A) = ? P(B) = ? P(A│B) = ? P(A) = 220/500 = 44% P(A│B) = 175/275 = 63.6% QMBU 301 – Quantitative Methods in Business

P(A) = P(A and B) + P(A and Bc) = P(A|B).P(B) + P(A |Bc).P(Bc) A P(A|B)=175/275 B P(B)=275/500 Total law of probability Ac P(Ac|B)=100/275 P(A) = 175/275 x 275/500 + 45/225 x 225/500 = 220 / 500 P(Ac|B)=45/225 A P(Bc)=225/500 Bc P(Ac|Bc)=45/225 Ac Probability Tree QMBU 301 – Quantitative Methods in Business

If A and B are two independent events then It follows that the joint probability of two independent events is simply the product of the probabilities of the two events. If two events are independent, are they also mutually exclusive? QMBU 301 – Quantitative Methods in Business

Bayes’ Law QMBU 301 – Quantitative Methods in Business

Researchers have developed statistical models based on financial ratios that predict whether a company will go bankrupt over the next 12 months. In test of one such model, the model correctly predicted the bankruptcy of 85% of the firms that did in fact fail, and it correctly predicted nonbankruptcy for 74% of the firms that did not fail. Suppose that we expect 8% of the firms in a particular city to fail over the next year and that the model predicts bankruptcy for a firm that you own. What is the probability that your firm will fail within the next 12 months? QMBU 301 – Quantitative Methods in Business

F: firm fails Fc: firm does not fail B: model predicts bankruptcy Bc: model predicts nonbankruptcy P(B | F) = 0.85 P(Bc | Fc) = 0.74 P(F) = 0.08 P(F | B) = ? ... ... ... ... QMBU 301 – Quantitative Methods in Business

Distribution of a Single Random Variable
A random variable is a function or rule that assigns a number to each outcome of an experiment. A probability distribution is a table, formula or graph that describes the values of a random variable and the probability associated with these values. QMBU 301 – Quantitative Methods in Business

Number of Color TVs Number of Households (thousands) 1,218 1 32,379 2 37,961 3 19,387 4 7,714 5 2,842 Total 101,501 x p(x) 0.012 1 0.319 2 0.374 3 0.191 4 0.076 5 0.028 Total 1.000 P(X2) = P(X=0) + P(X=1) + P(X=2) = 0.705 Cumulative probability is the probability that the random variable is less than or equal to some particular value. QMBU 301 – Quantitative Methods in Business

There are some summary measures that describe the characteristics of probability distributions. Expected value Variance and standard deviation QMBU 301 – Quantitative Methods in Business

Expected value (mean)of a probability distribution is the weighted sum of the possible values, weighted by their probabilities. The expected value gives the average of observed values of a random variable over a large number of observations. QMBU 301 – Quantitative Methods in Business

x p(x) x.p(x) 0.012 1 0.319 2 0.374 0.748 3 0.191 0.573 4 0.076 0.304 5 0.028 0.14 Total 1.000 2.084 QMBU 301 – Quantitative Methods in Business

Variance of a probability distribution is a weighted sum of squared deviatons of the possible values from the mean where the weights are the probabilities. QMBU 301 – Quantitative Methods in Business

x p(x) (x-2.084)2 p(x).(x-2.084)2 0.012 4.343 0.0521 1 0.319 1.175 0.3748 2 0.374 0.007 0.0026 3 0.191 0.839 0.1602 4 0.076 3.671 0.2790 5 0.028 8.503 0.2381 Total 1.000 1.1068 QMBU 301 – Quantitative Methods in Business

x p(x) 1 0.1 2 0.2 3 0.4 4 5 E(X) = 3 Var(X) = 1.2 x p(x) 1 0.3 2 0.15 3 0.1 4 5 E(X) = 3 Var(X) = 2.7 QMBU 301 – Quantitative Methods in Business

Linear functions of a random variable: number of computers sold - x Probability p(x) 0.1 1 0.2 2 0.3 3 4 Assume that the profit from a computer is $100. What is the expected total profit? E(Y)=100 E(X) What is the variance of total profit? Var(Y) = 1002 Var(X) ... ... QMBU 301 – Quantitative Methods in Business

Distribution of Two Random Variables: Joint Probability Approach
The manager of a retail store has been collecting data on weekly sales for two brands of diet soda at her store. She has the number of cans of soda sold for the past 400 days. The following table gives the number of days where each possible combination of PC sales occured on those days. Brand 2 sales (Y) 10 11 12 13 14 15 4 6 8 20 Brand 1 sales (X) QMBU 301 – Quantitative Methods in Business

We can derive the following probabilities from the observed frequencies on the previous slide. Brand 2 sales (Y) 10 11 12 13 14 15 0.01 0.015 0.03 0.035 0.02 0.025 0.05 Brand 1 sales (X) QMBU 301 – Quantitative Methods in Business

Brand 2 sales (Y) Can you calculate the probability of Brand 1 sales at any given day being equal to 13? P(X=13) = ? P(X=x) and P(Y=y) are called marginal probabilities. 10 11 12 13 14 15 0.01 0.015 0.03 0.035 0.02 0.025 0.05 Brand 1 sales (X) QMBU 301 – Quantitative Methods in Business

Brand 2 sales (Y) Let’s say we know that in a particular day Brand 1 sales is 13 units. What is the probability that Brand 2 sales is 11 units? P(Y=11 | X=13) =? 10 11 12 13 14 15 0.01 0.015 0.03 0.035 0.02 0.025 0.05 Brand 1 sales (X) QMBU 301 – Quantitative Methods in Business

Brand 2 sales (Y) P(Y=11 | X=13) = P(Y=11 and X=13) / P(X=13) = / = 0.13 Now, let’s calculate the expected value and variance of Brand 1 sales. 10 11 12 13 14 15 0.01 0.015 0.03 0.035 0.155 0.02 0.025 0.05 0.19 0.12 0.15 0.23 Brand 1 sales (X) ... QMBU 301 – Quantitative Methods in Business

Independent Random Variables
Two random variables being independent means any information about the values of one of them is worthless in predicting the value of the other. If two random variables are independent, their conditional distributions are equal to their marginal distributions. P(X=x | Y=y) = P(X=x) P(X=x | Y=y) = P(X=x and Y=y) / P(Y=y) P(X=x and Y=y) = P(X=x).P(Y=y) QMBU 301 – Quantitative Methods in Business

Independent Random Variables
Are Brand 1 and Brand 2 sales independent? Change the joint distribution such that they are independent... Brand 2 sales (Y) 10 11 12 13 14 15 0.01 0.015 0.03 0.035 0.155 0.02 0.025 0.05 0.19 0.12 0.15 0.23 Brand 1 sales (X) QMBU 301 – Quantitative Methods in Business

Weighted Sum of Random Variables
Summary measures of weighted sum of random variables have very important applications in financial investments. Example: Consider a portfolio of three stocks. Ri is the annual return from stock i, which is random. 20%, 50% and 30% of your money is invested in stocks 1, 2 and 3 respectively. T is the total return from your investment. QMBU 301 – Quantitative Methods in Business

Y = a1X1 + a2X anXn a1, a2, ..., an are constants (weights) X1, X2, ..., Xn are random variables. E(Y) = a1E(X1)+ a2E(X2) anE(Xn) This is true even if the n random variables are probabilistically dependent. QMBU 301 – Quantitative Methods in Business

If Xi are independent, then If all Xi are dependent, then QMBU 301 – Quantitative Methods in Business

Let us assume that there are two stocks in our portfolio with returns R1 and R2 and let T = 0.2R R2 Then E(T) = 0.2 E(R1) E(R2) if two stocks are independent Var(T) = 0.04 Var(R1) Var(R2) if two stocks are not independent Var(T) = 0.04 Var(R1) Var(R2) Cov(R1,R2) ... ... ... QMBU 301 – Quantitative Methods in Business

Sum of independent random variables: Y = X1 + X Xn E(Y) = E(X1) + E(X2) E(Xn) Var(Y) = Var(X1) + Var(X2) Var(Xn) Difference between two independent variables: Y = X1 - X2 E(Y) = E(X1) - E(X2) Var(Y) = Var(X1) + Var(X2) Var(Y) = Var(X1) + Var(X2) – 2Cov(X1,X2) QMBU 301 – Quantitative Methods in Business

Linear function of a random variable: Y = a + bX E(Y) = a + bE(X) Var(Y) = b2Var(X) QMBU 301 – Quantitative Methods in Business

Discrete vs. Continuous Random Variables
There are two types of random variables: discrete – can take on a countable number of values e.g. X = number of heads observed in an experiment that flips a coint 10 times continuous – values are uncountable e.g. X = time to write a statistics exam A probability distribution is a table, formula or graph that describes the values of a random variable and the probability associated with these values. QMBU 301 – Quantitative Methods in Business

Discrete vs. Continuous Random Variables
We cannot list all possible values of a continuous random variable. Since there is an infinite number of values, the probability of each individual value is virtually 0. Instead of assigning probabilities to each individual value, we spread the total probability of 1 over the continiuum (imagine a histogram with a large number of small intervals). QMBU 301 – Quantitative Methods in Business

Continuous Distributions
A probability density function, usually denoted by f(x), specifies the probability distribution of a continuous variable X. If the range of x is between a and b then, f(x)  0 for all x between a and b the total area under the curve between a and b is 1 the higher f(x) is, the more likely x is. QMBU 301 – Quantitative Methods in Business

f(x) probability density function (pdf) x a b QMBU 301 – Quantitative Methods in Business

f(x) probability density function (pdf) x a b Area = P(a ≤ x ≤ b) = 1 QMBU 301 – Quantitative Methods in Business

f(x) probability density function (pdf) x a c d b Area = P(c ≤ x ≤ d) QMBU 301 – Quantitative Methods in Business

f(x) ? x a b uniform distribution E(x) = ? QMBU 301 – Quantitative Methods in Business

Normal Distribution Normal distribution is the most important distribution in statistics. f(x) It has two parameters: μ – mean  – standard deviation  x μ QMBU 301 – Quantitative Methods in Business

Normal Distribution changing the parameters QMBU 301 – Quantitative Methods in Business

Normal Distribution increase μ changing the value of μ QMBU 301 – Quantitative Methods in Business

Normal Distribution decrease μ changing the value of μ QMBU 301 – Quantitative Methods in Business

Normal Distribution increase  changing the value of  QMBU 301 – Quantitative Methods in Business

Normal Distribution decrease  changing the value of  QMBU 301 – Quantitative Methods in Business

Normal Distribution X ~ N(μ,) P(a ≤ X ≤ b) = ? a b QMBU 301 – Quantitative Methods in Business

Normal Distribution X ~ N(μ,) P(a ≤ X ≤ b) = Area a b QMBU 301 – Quantitative Methods in Business

Normal Distribution X ~ N(μ,) P(X ≤ b) = Area1 a b Excel function: NORMDIST(b, μ,,1) QMBU 301 – Quantitative Methods in Business

Normal Distribution X ~ N(μ,) P(X ≤ a) = Area2 a b Excel function: NORMDIST(a, μ,,1) QMBU 301 – Quantitative Methods in Business

Normal Distribution X ~ N(μ,) P(a ≤ X ≤ b) = Area1 – Area2 a b Excel function: NORMDIST(b, μ,,1) – NORMDIST(a, μ,,1) QMBU 301 – Quantitative Methods in Business

Normal Distribution A factory produces 20 cm-long plastic pipes which have a target diameter of 5 mm. The product is designated acceptable quality if this thickness is between 4.5 mm and 5.5 mm. The production process’ current output has N(5,0.2). What is the defective rate? What if they can reduce  to 0.1 mm? QMBU 301 – Quantitative Methods in Business

Normal Distribution X within one   X within one 2  X within one 3  μ-3 μ-2 μ- μ μ+ μ+2 μ+3 QMBU 301 – Quantitative Methods in Business

Normal Distribution Let X ~ N(μ,) A linear function of X is still normally distributed. Y = aX + b Then Y~ N(aμ+b,a) QMBU 301 – Quantitative Methods in Business

Binomial Distribution
Binomial distribution is a discrete distribution that occurs when we perform a sequence of independent identical experiments (trials) with two outcomes. Consider a situation in which there are n independent, identical trials, where the probability of a success on each trial is p and the probability of a failure is 1 – p. QMBU 301 – Quantitative Methods in Business

X = random number of successes n = number of trials p = probability of success X~Binomial(n,p) E(X)=np and Var(X)=np(1-p) QMBU 301 – Quantitative Methods in Business

Binomial(n=10, p=0.2) μ=2 and 2=1.6 QMBU 301 – Quantitative Methods in Business

Binomial(n=5, p=0.3) μ=1.5 and 2=1.05 QMBU 301 – Quantitative Methods in Business

Excel function: BINOMDIST(k,n,p,cumulative) k = number of successes n = number of trials p = probability of “success” cumulative = 0 if probability distribution function 1 if cumulative distribution function QMBU 301 – Quantitative Methods in Business

Customers at a supermarket spend varying amounts. Historical data shows that the amount spent per customer is normally distributed with mean $85 and standard deviation $30. 500 customers shop in a given day. Calculate the probability that at least 30% of all customers spend at least $100. QMBU 301 – Quantitative Methods in Business

If np > 5 and n(1-p) > 5 then we can approximate the binomial distribution by the normal distribution with mean np and variance np(1-p). Binomial(40,0.4) Normal(16,3.1) QMBU 301 – Quantitative Methods in Business

Poisson Distribution Poisson distribution is a discrete distribution when we are a specified period of time. λ is the expected number of events in the interval X = number of event interested in the count of events occuring within occuring in a given interval X ~ Poisson(λ) E(X) = λ and Var(X)= λ2 QMBU 301 – Quantitative Methods in Business

Poisson Distribution Excel: POISSON(k, λ, cumulative) number of events QMBU 301 – Quantitative Methods in Business

Poisson Distribution λ = 2 E(X) = 2 and Var(X) = 4 QMBU 301 – Quantitative Methods in Business

Poisson Distribution Number of users of an automatic banking machine is Poisson distributed. The mean number of users per 5-minute interval is 1.5. P(no users in the next 5 minutes) = ? P(5 or fewer users in the next 15 minutes) = ? P(3 or more users in the next 10 minutes) = ? QMBU 301 – Quantitative Methods in Business

Exponential Distribution
Most common probability distribution used to model time between arrivals of consecutive events (interarrival times) is the exponential distribution. f(x) = λe-λx and F(x) = P(X≤x) = 1 – e-λx λ is the mean number of events per unit time interval Excel: EXPONDIST(x, λ, cumulative) QMBU 301 – Quantitative Methods in Business

f(x) λ = 1 x QMBU 301 – Quantitative Methods in Business

f(x) λ = 0.5 x QMBU 301 – Quantitative Methods in Business

f(x) λ = 2 x QMBU 301 – Quantitative Methods in Business

Suppose that customer interarrival times to a store are exponentially distributed. On average time between customer arrivals is 15 minutes. Assume that a customer has just arrived. Find the probability that another customer arrives within the next 15 minutes. Assume that you have waited for 15 minutes but the next customer has not arrived. What is the probability that you will have to wait another 15 minutes for his/her arrival? QMBU 301 – Quantitative Methods in Business

Sampling and Estimation
Parameters describe populations. Parameters are almost always unknown. We take a random sample of a population to obtain the necessary data. We calculate one or more statistics from the data. An inference is a statement about a parameter of a population. Sampling error is the inevitable result of basing an inference on a random sample rather than on the entire population. QMBU 301 – Quantitative Methods in Business

Key Terms in Sampling Point estimate is a single numeric value, a “best guess” of a population parameter, based on the data in a sample. The estimation error is the difference between the point estimate and the true value of the population parameter being estimated. The sampling distribution of any point estimate is the distribution of the point estimates we would see from all possible samples (of a given sample size) from the population. QMBU 301 – Quantitative Methods in Business

Key Terms in Sampling An interval estimate (or confidence interval) is an interval around the point estimate, where we strongly believe the true value of the population parameter lies. The standard error of an estimate is the standard deviation of the sampling distribution of the estimate. An unbiased estimate is a point estimate such that the mean of its sampling distribution is equal to the true value of the population parameter. QMBU 301 – Quantitative Methods in Business

Central Limit Theorem For any population distribution with mean μ and standard deviation , the sampling distribution of the sample mean is approximately normally distributed with mean μ and standard deviation μ QMBU 301 – Quantitative Methods in Business

Sampling Distributions
The sampling distribution of the population mean: μ Z QMBU 301 – Quantitative Methods in Business

What is the probability that the mean of a random sample of 100 weights drawn from a population will be within 2kg of the true population mean weight, if the std for the population is estimated to be 15 kgs ? QMBU 301 – Quantitative Methods in Business

The sampling distribution of the population mean when the population standard deviation is unknown: replace  with s the sampling distribution is not normal the sampling distribution is t-distribution with degrees of freedom of n – 1. t-value indicates the number of standard errors by which a sample mean differs from a population mean. QMBU 301 – Quantitative Methods in Business

standard normal t with 3 df QMBU 301 – Quantitative Methods in Business

t with 5 df QMBU 301 – Quantitative Methods in Business

t with 30 df QMBU 301 – Quantitative Methods in Business

Confidence Interval for the Mean
Confidence interval for the population mean is: 90% CI  t-multiple = for n = 30 95% CI  t-multiple = for n = 30 99% CI  t-multiple = for n = 30 QMBU 301 – Quantitative Methods in Business

Confidence Interval for the Mean
To find the 100(1-)% CI for the mean in EXCEL: TINV(,n-1) use time_data! QMBU 301 – Quantitative Methods in Business

Hypothesis Testing Formulate the hypothesis being tested (called the null hypothesis, H0) and state the alternative hypothesis (the one concluded if null hypothesis is rejected, H1) Collect a random sample from the population and compute the appropriate sample test statistic Assume the null hypothesis is true and determine the sampling distribution Compute the probability that a value of the sample statistic at least as large as the one observed could have been drawn from this sampling distribution If this probability is high, do not reject the null hypothesis; if this probability is low, the null hypothesis is discredited and can be rejected with a small chance of error QMBU 301 – Quantitative Methods in Business

Hypothesis Testing Do not reject H0 Reject H0 H0 True Correct decision Type I Error: probability α H0 False Type II Error: probability β Alpha is known as the significance level of the test The p-value is the probability of getting at least as extreme a sample result as the one actually observed if H0 is true If p < α  H0 is rejected QMBU 301 – Quantitative Methods in Business

Hypothesis Testing H0 represents status quo or the current belief in situation H1 represents a research claim or specific inference you would like to prove If you reject H0, you have statistical proof that alternative hypothesis is correct If you do not reject H0, then you have failed to prove H1, however, does not mean that you have proven H0 H0 always refers to a specified value of the population parameter, not a sample statistic The statement of H0 always contains an equal sign regarding the specified value of the population parameter The statement of H1 never contains an equal sign regarding the specified value of the population parameter QMBU 301 – Quantitative Methods in Business

Hypothesis Testing Oxford Cereal Company wants to determine whether the cereal-filling process is working properly, i.e., whether the mean fill per box throughout the entire packaging process remains at the specified 368 grams. To evaluate the 368-gram requirement, a random sample of 25 boxes is taken. QMBU 301 – Quantitative Methods in Business

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