Managerial Statistics

Deze samenvatting is geschreven in collegejaar 2012-2013.


Chapter 1 – What is Statistics?

Terms
The whole population        VS               The sample
Parameter                                              Statistics

  • Population: set of all items of interest in a statistical problem (e.g. All Europeans)
  • Parameter: descriptive measure of population
  • Sample: set of data drawn from population (a sample of 20 Europeans)
  • Statistic: descriptive measure of sample
  • Variable: characteristic of interest to us

Example: opinion poll in a European election campaign

  • population: (e.g.) all adult citizens of Europe
  • parameter: (e.g.) proportion of voters for specific candidate
  • sample: 20 Europeans
  • Statistic: proportion of the sample voting for a specific candidate

The bigger the sample size, the better, because then you can take into account many different opinions. The best sample size would equal your whole population of interest. However, this is often very large and therefore too costly.

Descriptive vs. Inferential Statistics
Descriptive statistics – Deals with methods of organizing, summarizing and presenting data in convenient and informative way.
Inferential statistics – A body of methods used to draw conclusions or inferences about characteristics of populations on sample data.
Simple algorithm of solving statistical problem

  • Uncertainty involved; probability theory used to measure reliability of statements
  • Ascertain the correct technique
  • Calculate i.e. manipulate the formula with variable values
  • Interpreting the results i.e. relating it.

Statistical inference is the process of making an estimate, prediction, or decision about a population based on sample data.

  • Confidence interval
  • Significance level

Types of variable values

  1. Interval data
  2. Nominal data
  3. Ordinal data
  4. Ratio

Areas where you can apply statistics:
1. Finance Management

  • Capital budgeting
  • Stock and Bond Valuation
  • Return on Investment

2. Marketing Management; Product, price, place, promotion. Market segmentation – satisfying the demands of a particular segment.
3. Operations Management; The principal activity of every organization.
You should design a product; Plan the whole process; facility layout; Location analysis and logistics; Aggregate production plan determines the number of units to be produced over the next 6 –18 months.

Accounting

  • Managerial accounting – internal -> facilitates in determining a firm’s fixed and variable costs.
  • Financial accounting – external -> financial statements (Auditing Tax returns).

Human Resource Management
Deals with the people-related decisions made by a firm; Recruitment, training, performance appraisal, compensation, and motivation

Information systems
Refers to any process by which information is created and used.

Economics
The study of the use of scarce resources.
Opportunity costs: The income that could have been earned if they had chose to work (Students at university)
Microeconomics – The study of the behavior and operations of individual economic decision makers/ markets
Macroeconomics- concentrating on such topics as the level and growth rate in employment, income, inflation.


Chapter 2 – Graphical and Tabular Descriptive Techniques

Types of data
There are 3 types of data:
- Quantitative data– Values are real numbers; Arithmetic calculations are valid

  • interval: natural distance (e.g. shoe size)
  • ratio: absolute zero (e.g. age)

- Qualitative  data–   Categorical   data;  Values   are   the   names   of  possible   categories;  valid computation: Count of the number of observations in each category.

  • nominal: mutually exclusive categories, labeling (e.g. country of origin)
  • ordinal or ranked: natural ordering (e.g. preference for cola)

- Ranked data–   Categorical   data;  Values   must   represent  the   ranked   order   of   responses;   valid computations: those based on an ordering process.

Methods to show data
There are different methods to present your data.
- Stem-and-leaf display - the digits to the left of the decimal point is the stem, the digits to the right
of the decimal point is the leaf

  • graphical techniques often used in a preliminary analysis
  • alternative to histogram
  • shows values of original observations, whereas histogram ‘loses’ them

- Bar charts – emphasize frequency of occurrence of the different categories, used when the order in which qualitative data is presented is meaningful
- Pie   charts  –   emphasize   the   proportion   of   occurrences   of   each   category,  very   popular   tool   to  represent the proportions of appearance for nominal data

Two sorts of means

  1. The population mean, denoted as μ, this is the mean taken from all the possible items.
  2. The sample mean, denoted as  x  (x bar), this is the average taken from your sample.

E.g. population mean: the average age of all the inhabitants of the Netherlands.
Sample mean: the average age of 10,000 inhabitants of the Netherlands.

Mean = Sum of all observations / number of observations
Bell shaped histogram

  • When you draw your finding of an experiment in a histogram, it can be the case that the histogram becomes bell-shaped.
  • Many statistical techniques require that the population be bell shaped (or normally distributed)
  • It is quite logic to find a bell-shaped histogram, because in most cases, there is one large average value and the other values will lie around the average value

A histogram is symmetric if, when we draw a line down the center of the histogram, the 2 sides have identical shapes
A tail to the right – Positively skewed
A tail to the left – Negatively skewed

Ogives
Ogives are graphical representations of cumulative relative frequency distributions. See below.

Note that you put last number of the class above the number on the X-as of the graph. In this example: you put the last number of class I (number 15 in the table), above number 15 on the x-axes in the graph.

Class relative frequency = Class frequency/Total number of observations
Class width = Range / Amount of Classes
Range= largest observation – smallest observation
Relative vs. absolute
When using an ogive you put the absolute cumulative frequency on the Y-axis, or the relative cumulative frequency on the Y-axis.
In both cases you can estimate how the observations are distributed (where is the mean, where is 25%, where 75%?).

The mean of an ogive
The formula for grouped data is as follows:


where mj stands for the midpoint of the j-th class, and f for the frequency of that class, and N the number of classes.
In the case that your ogive use relatives frequencies (i.e. percentages), then you do not have to divide by N.
 
Time
Data can be classified according to the time they are measured:

  • Cross-sectional data are all collected at the same time
  • Time-series data are collected at successive points in time

Time-series data are often depicted on a line chart (which is a plot of the variable over time)

Chapter 3 – Art and Science of Graphical Presentations

Graphical excellence is achieved when the following characteristics apply:

  1. The graph presents large data sets concisely and coherently
  2. The ideas and concepts the statistician wants to deliver are clearly understood by the viewer.
  3. The display induces the viewer to address the substance of the data and not the form of the graph.
  4. There is no distortion of what the data reveal

Steps to write a statistics report:

  1. State your objectives
  2. Describe the experiment
  3. Describe your results
  4. Discuss limitations of the statistical techniques

Chapter 4 – Numerical Descriptive Techniques

Measures of central location
There are three measures of central location: mean, median, mode
Mean is the average. As written before, there is a sample mean and a population mean.
Median is the value that falls in the middle in a sequential row. The median is the most appropriate measure of central location to use when the data under consideration are ranked qualitative data, rather than quantitative data.
E.g.  row: 1, 5, 7, 9, 10, 11, 12 the median is: 9.
In case of a row with even numbers, take the average of the numbers in the middle:  E.g.  row: 1, 5, 7, 9, 10, 11, 12, 14. The median is 9.5.
Modal class = the one with the largest number of observations
There may be more (e.g. two) modal classes in one distribution, that is called bimodal.  
For large data sets the modal class is much more relevant than the single-value mode, see the following example.
Example:
The average salary in a small company would be quite high, since the directors increase the mean. In that case ,if you take the modal class, you will get a better impression of the salary distribution.
For quantitative data, the mode is the least useful. For qualitative data, the mode is the most useful. If a distribution is symmetrical and unimodel, then the mean, median and mode are the same. If the distribution is non symmetrical, then the three measures differ.
Geometrical mean
If you want to measure the growth rate, you cannot use just the normal mean. You have to use this formula

Rg stands for the average growth rate; R1, R2 and Rn stand for the growth each year, and N stands for the numbers of years.
Example
A firm’s sales were $1,000,000 three years ago; Sales have grown annually by 20%, 10%, -5%.

Note that if you had taken the ‘normal’ mean, it would be (20+10-5)/3= 8% average growth, which is incorrect.
Variance
Variance is a measure of variability of a population of observations.
It is important to know what the mean of a population is. However, you also want to know what the variability is of a mean. For example: it is nice to know if a share gives an average yearly return of 6%; however, this can be in many ways. See the following:
Example 1: Philips Light Bulbs
Return Year 1: 4%
Return Year 2: 8%
Return Year 3: 6%
Return Year 4: 5%
Return Year 5: 7%
Average return: 6 %
Example 2: Russian Light Bulbs
Return Year 1: 0%
Return Year 2: 12%
Return Year 3: 6%
Return Year 4: 1%
Return Year 5: 11%
Average return: 6%
You see that the average return in both cases is 6%, but the first example is far more secure, since there is less variability and thus you are sure of a return around the 6%. The 2nd example gives sometimes really high returns, but sometimes none at all.
Formula for variance
To measure the variability of a whole population, you calculate the variance:


To measure the variability of a sample size, you calculate the variance:


Notice the 2 differences between these 2 formulas:

  1. σ2 versus S2 à a matter of notation.   
  2. Dividing by: N versus n-1 ->

Dividing by N: if you take a whole population
Dividing by n-1: if you take a sample of that population  
The formula looks difficult, but it is not in practice. Steps for calculation:

  1. You just take from each number the difference between that numbers and the average number (x-μ)
  2. You square that answer (x-μ)2
  3. You do this for each number and sum those differences up (x-μ)2  + (x-μ)2  + (x-μ)2  + etc.
  4. And you divide it by the number of observations (N, or n-1).

Here a numerical example, based on Example 1: Philips.
σ= (4-6)2 +(8-6) 2 +(6-6) 2 +(5-6) 2 +(7-6) 2 = 10     10/5= 2.
The variance of the Philips Light Bulbs is σ2 = 2 %. The variance of the Other Light Bulbs is higher, calculate yourself.

Standard deviation
The standard deviation is the square root of the variance of the measurements.
Sample  standard deviation is s = √(s2)
Population standard deviation is σ= σ√(2)
In most cases, we are interested in the standard deviation, and not in the variance. However, to calculate the standard deviation, we always have to calculate first the variance.

Remember: The higher the standard deviation, the more variability, and thus the more unsure/riskier that project!!
Coefficient of determination
Do not be afraid by such words, the calculation is quite easy.
What does the coefficient of determination mean? In our example about the stock returns, we had the same average return (6%), but a different standard deviation. Philips’ standard deviation was lower and therefore we could say that Philips was less risky.
However, how to measure the risk of a project when the average returns (the mean) also differ? Then you use the coefficient of determination:

Remember: the higher the CV, the more risky the project!
Empirical rule
The empirical rule: If the measurements are bell-shaped, the following rules apply:

Chebysheff’s  theorem
A variation on the empirical rule is Chebysheff’s theorem:
1 - (1/k2) for K>1
K stands for the number of standard deviations and can take any value above 1, e.g. 1.2, 1.8, 3, 4. If you plug K=2 in the formula, you get 0.75. This means that 75% of all observations lie within two standard deviations of the mean. With k=3, you get 0.889. This means that at least 88.9% of all observations three standard deviations.
You see that this correspond with the Empirical Rule: the empirical rule says 95% of all the values are within 2 standard deviations, and the Chebysheff’s theory says at least 75% lies within 2 standard deviations.
Percentiles
You can divide your observations in percentiles. This means, you put groups of data together in different percentages groups.
The formula to calculate the border of the different percentiles, is as follows:
L= (n+1)(p/100)

Lp is the location of the p-th percentile
Commonly used percentiles
First (lower) decile = 10th percentile = 10%
First (lower) quartile (Q1) = 25th percentile = 25%
Middle (second) quartile (Q2) = 50th percentile = 50%
Third quartile (Q3)  = 75th percentile = 75%
Ninth (upper) decile = 90th percentile = 90%
Median  = second quartile (Q2)
Interquartile range (IQR)  = Q3 - Q1
Example:
Find the 1st and 3rd quartile of the data in Example 4.1: 0, 7, 12, 5, 33, 14, 8, 0, 9 and 22 (n = 10)
Solution:
Sort the measurements: 0, 0, 5, 7, 8, 9, 12, 14, 22, 33
1st quartile

  • L25 = (10+1)x25/100 = 2.75 (location; three-quarters of distance between 2nd and 3rd ordered observations) The 25th percentile is three-quarters of the distance between the second (which is 0) and the third (which is 5) observations.
  • Three-quarter of the distance is: (5 – 0) * 0.75 = 3.75.
  • Q1 = 3.75 + 0 = 3.75.

3rd quartile

  • L75 = (10+1)x75/100 = 8.25 (location; one-quarter of distance between 8th and
  • 9th ordered observations). The 75th percentile is one-quarter of the distance between the eight (which is 14) and ninth (which is 22) observations.
  • One-quarter of the distance is: (22-14) * 0.25 = 2
  • Q3 = 2 + 14 = 16

Boxplot
A box plot is a pictorial display of the percentiles Q1, Q3, et cetera:

  • S - The smallest measurement
  • Q1 - The lower quartile
  • Q2 - The median
  • Q3 - The upper quartile
  • L - the largest measurement

Sometimes there are outliers, an outlier is an unusually large or small value in the sample. An outlier is caused by e.g. a mistake of the observator.

  • In context of box plots an outlier is defined as a value located at a distance more than 1.5´(Q3 - Q1) from the box.

Measures of linear relationship – covariance

Chapter 5 – Data Collection and Sampling

The validity of the results of a statistical analysis clearly depends on the reliability and accuracy of the data used. Three sources of data:

  1. Direct observation: obtain data by direct observation (no attempt to control for possible influencing factors) e.g. stock market data
  2. Experiments: produce data through experiments (attempt to control for relevant factors) e.g. new pill against high blood pressure ® assign 100 patients to group 1 (new pill) and 100 to group 2 (old pill)
  3. Surveys: collect information from people (income, family size, opinions, etc.) through

a. Personal interview - high response rate relative to other survey methods fewer incorrect responses when respondents misunderstand some questions, but expensive
b. Telephone interview - less expensive but lower response rate
c. Self-administered survey (questionnaire) - least expensive, but lower response rates and accuracy than interviews.
A questionnaire; has to be short, short questions, dichotomous questions, no leading questions, time permitting, start with demographic questions.
The chief motive for examining a sample rather than a population is cost, population size and possible destructive nature of sampling process. Statistical inference permits us to draw conclusions about a population parameter based on a sample that is quite small in comparison to the size of the population.
Target population – The population about which we want to draw inferences
Sampled population – The actual population from which the sample has been taken. Self-selected samples are almost always biased, because the individuals who participate in them are more keenly interested in the issue than are the other members of the population.
Three different sampling plans:
1. Simple random sample – A sample selected in such a way that every possible sample with the same number of observations is equally likely to be chosen.
2. Stratified Random sample – is obtained by separating the population into mutually exclusive sets or strata and then drawing simple random samples from each stratum For example boys and girls samples of total mixed population.
Advantage – besides acquiring information about the entire population, we can also make inferences within each stratum or compare strata.
3. Cluster sample – A simple random sample of groups or clusters of elements. Usually some sampling errors.
2 major types of errors:
1. Sampling error – Difference between the sample and the population that exist because of sampling. Increasing sample size will reduce this error
2. Non-sampling error – More serious because taking a larger sample won’t diminish the size of the error. Three types:
a. Errors in data acquisition – recording of incorrect responses.
b. Nonresponse error – response are not obtained from some members of the sample.
c. Selection bias – Occurs when the sampling plan is such tat some members of the target population can’t possibly be selected for inclusion in the sample.
Increasing sample size will not reduce this error.

Chapter 6 - Probability

Random experiment
A process that results in one of a number of possible outcomes. The actual outcome is uncertain, but experiment’s possible outcomes are known in advance; to determine the probabilities we need to define the possible outcomes first.

Sample Space and Events
Sample Space: A sample space of a random experiment is a list of all possible outcomes of the experiment. The outcomes must be mutually exclusive and collectively exhaustive

  • Mutually exclusive no 2 outcomes on the list can both occur at the same time
  • Exhaustive Each possible outcome must be included. For example when throwing a die, there will be six possible outcomes.

Simple events: The individual outcomes are called simple events. Simple events cannot be further decomposed into constituent outcomes
Event: An event (say A) is any collection of one or more simple events
3 approaches to assigning probabilities

  1. Classical approach – Attempts to deduce the probability of an outcome logically from the symmetric nature of the experiment. This method is mostly applied in statistical inference. There are n outcomes, all equally likely -> Prob(part. outcome) = 1/n
  2. Relative frequency approach – expresses an outcome’s probability as its long-run relative frequency of occurrence. There are n trials, frequency of occurrence of particular outcome is -> Prob(part. outcome) = x/n
  3. Subjective approach – based on personal evaluation of situation. The probability assigned to an outcome simply reflects the degree to which we believe that the outcome will occur.


 

Conditional probability – Bayes Law
Conditional probability is denoted as P(A|B). It means ‘the conditional probability that event A occurs, given that event B has occurred’
Formula:      

                                          
So you put the A and the B combined above the line, and the “given”-part on his one below the line.
Independent and dependent variables

  • Two events A and B are said to be independent if: P(A|B) = P(A) or P(B|A) = P(B)
  • Otherwise, the events are dependent
  • Stated differently, if the occurrence of one event does not change the likelihood of occurrence of the other event, the two events are independent

Probability tree
A handy method to show all possible outcomes of conditional probabilities.
The theory of the probability trees and Bayes’ Law is quite hard to understand. However, if you just practice a lot with it (after reading the theory), you will understand it! Take the time for it!

Chapter 7 – Random Variables and Discrete Probability

1. Random Variables
Aspect of the real world that is analyzed; or, the aspect of the random experiment that is of interest to us. A random variable assigns a numerical value to each simple event in a sample space.
2. Random experiment
A process or course of action, whose actual outcome is uncertain, but whose possible outcomes are known in advance.

3. Two types of random variables
A random variable is

  • discrete if it is countable.
  • continuous if it can assume an uncountable (always infinite) number of values.

Discrete: you can only have the numbers (e.g.) 1, 2, 3, 4 and 5.
Continuous: you can have any number ranging from 1 to 5, e.g. 1½
Each random variable has a probability function (discrete) or probability density function (continuous) associated with it, describing the probability distribution of the random variable

4. Probability distribution
A table, formula, or graph that lists all possible values a discrete random variable can assume, together with their associated probabilities, is called a discrete probability distribution. It is a summary (formula, table or graph) of the possible outcomes of a random variable together with their associated probabilities
It always holds that following must be true:

5. Expected value
The expected value of a random variable X is the weighted average of the possible values it can assume, where the weights are the probabilities of occurrence of those values. It is the long-run average if random experiment is repeated a large number of times

6. Variance and standard deviation
The variance of a random variable X is defined in a similar manner, with 1/N being replaced by p(xi).

7. Some calculation rules to remember

  • E(c) = c
  • E(cX + d) = cE(X) + d  à if you multiply X by c and add d, then your expected value will also increase with times c and plus d.
  • E(aX + bY) = aE(X) + bE(Y)
  • E(aX - bY) = aE(X) - bE(Y)
  • E(XY) = E(X)E(Y), if X and Y are independent
  • V (c) = 0
  • V (cX + d) = c2 V(X)
  • V(aX + bY) = a2 V(X) + b2 V(Y), if X and Y are independent
  • V(aX - bY) = a2 V(X) + b2 V(Y), if X and Y are independent

8. Covariance
Covariance is a statistical measure of the strength of the linear relationship between two random variables; it measures the degree to which the two variables tend to move together

9. Discrete probability distributions
There are two discrete probability distributions and their properties will hereafter be explained. The continuous distribution will be explained in a later chapter.

  • Binomial distribution
  • Poisson distribution

10. Binomial distribution
Used in a Binomial (Bernoulli) experiment. Used when interested in a series with a given number of repeated trials, of which the observation has only 2 outcomes: “negative/positive” “success/failure” “yes/no”, etc.

  • there are n trials (n is finite and fixed)
  • all trials are independent
  • each trial results either in ‘success’ or in ‘failure’
  • the probability of success (p) is the same in all trials

Notation: X ~ Bin(n, p)
Examples:

  • number of defective items in a series of 200 produced  X is Bin(200,p)-distributed
  • number of ‘heads’ when flipping a coin 3 times  X is Bin(3,0.5)-distributed
  • number of even numbers when rolling a die 10 times  X is Bin(10,3/6)-distributed
  • number of boys in a group of 20 babies born  X is Bin(20,0.5)-distributed


The probabilities are always P and 1-P. In the chapters before there were probabilities trees in which the probabilities changed in each stages. In a binominal distribution the probabilities always remain the same.
Via this probability three, you can calculate the possibilities on the occurrence of different events. However, more quickly is this formula to calculate the chance of an event:


Where =nxC the number of ways to get your event.
Px =  the chance to get a “success” to-the-power of how many “successes” you are looking for.
(1-p)n-x the chance to get a “failure” to-the-power of “how many times you try (N)” minus “how many successes (x) you are looking for” ->
N-X = the number of failures remaining after the successes.
As you can see in the probability tree above, there are various ways of getting (e.g.) 2 times an S. To be precise, there are 3 ways of getting 2 times an S, namely SSF, SFS, FSS. If you have many many routes, it would be time consuming to draw all different ways. There is a formula to quickly calculate all possible ways for different combinations. 

11. Mean, variance and standard deviation of a binominal distribution

12. Cumulative probability of the binominal distribution
Recall the example about the production run of the catalytic converter which was defective 5% of the time. What is the probability that we find 1 or more defective products when checking 3 times the production?
We are now looking for accumulated probabilities. Here we want to know P(X≥1) = P(1) + P(2) + P(3) = 0.1354 + 0.0071 + 0.0001 = 0.1426.
It would be too time consuming to always calculate first all individual probabilities, and then adding them up together, in order to find the cumulative probability.
Luckily, in the back of the book there are tables with the cumulative distribution. These tables are also stored in your graphical calculator.
13. Poisson distribution
A Poisson experiment possesses the following properties:

  1. The number of successes that occur in any interval is independent of the number of successes that occur in any other interval.
  2. The probability that a success will occur in an interval is the same for all intervals of equal size and is proportional to the size of the interval.
  3. The probability that two or more successes will occur in an interval approaches zero as the interval becomes smaller.

Thus it is used when interested in the number of rare events that occur over a fixed amount of time or within a specified region
The Poisson random variable indicates the number of successes that occur during a given time interval or in a specified region in a Poisson experiment.

  • Notation: X ~ Poisson(m), where m is the average number of successes in this interval

Examples:

  • number of errors a typist makes per page
  • number of customers entering a shop per hour
  • number of telephone calls received by a switchboard per minute
  • number of defects in a production process per hour

The formula is as follows:

You can find “e” as a button on your calculator.
14. Expected value and variance of the Poisson distribution
E(X) = m   and V(X) = m

15. Cumulative Poisson distribution
(continued from above example)
What is the probability that one or two cars will arrive during the specified one-minute period?

  • now use Table 2 of Appendix B in your book.
  • Table 2 contains cumulative Poisson probabilities:
    P(X≤2)= 0.062.

16. From Binominal to Poisson
When n is large (n > 20), the binomial probability table is not available. You can transform the binominal-distribution in a Poisson distribution, by doing the following:
m = n*p.
Example
A warehouse has a policy of examining 50 sunglasses from each incoming lot, and accepting the lot only if there are no more than two defective pairs
What is the probability of a lot being accepted if, in fact, 2% of the sunglasses are defective?
Solution:

  • let X = number of defective pairs among the 50 sunglasses examined
  • then X ~ Bin(n = 50, p = .02)
  • the required table (n = 50) is not available
  • the Poisson approximation can be used (µ = 50 ´ .02 = 1)
  • P(XPoisson £ 2) = .920 (Table 2, µ = 1)
  • true binomial probability: P(XBin

Chapter 8 - Continuous probability distributions

17. Differences discrete and continuous random variables
- Discrete random variables:

  • finite number of possible values (e.g. binomial variable; x = 0, 1, 2, …, n)
  • countable infinite number of possible values (e.g. Poisson variable; x = 0,1, ...)
  • all possible values of random variable X can be listed

- Continuous random variables:

  • uncountable infinite number of possible values in an interval

example: amount of time (seconds) workers on an assembly line take to
complete a particular task -> x can take all non-negative numbers

  • all possible values cannot be listed: there is always another possible value between any two of its values
  • probability that a continuous random variable X will assume a particular value is zero: P(X = x) = 0, since there are too many other values.

18. Probability density functions
There are three types of probability density functions:

  • uniform
  • normal
  • exponential

All functions measure the area under a graph of an continuous distribution.
19. Uniform distribution
In an uniform distribution, all outcomes are equal. The graph is not bell-shaped, but a straight line.

       
Intervals of equal size within the domain of X are equally likely to contain the value that the variable X will assume. 
20. Normal distribution
Normal distribution is the most important continuous distribution in statistics:

  • many random variables can be properly modeled as normally distributed
  • many distributions can be approximated by a normal distribution
  • the normal distribution is the cornerstone distribution in statistical inference; the (asymptotic) distribution of sample means is normal

A random variable X with mean μ and variance σ² is normally distributed if its probability density function is given by: 
The graph of a normal distribution looks as follows:

 

 

  • the normal distribution is symmetrical
  • any normal distribution can be transformed into a specific normal distribution, called standard normal distribution, with mean 0 and variance 1

Example:
The time it takes to write a standard entrance exam is assumed to be normally distributed, with a mean of 60 minutes and a standard deviation of 8 minutes
What is the probability that a student will finish the exam in between 60 and 70 minutes?
Solution:
1. Since the question already states it is a normal distribution, we can draw the following graph:


2. We can solve the question right now, but then we need difficult algebra. It is easier to first standardize the Mean and the SD; by standardizing these values, you can solve the question with a table in the book.

  •  if X denotes the time taken to write the exam, we seek the probability P(60 < X < 70) where m = 60 and s = 8
  • this probability can be calculated by creating a new normal variable, the standard normal variable (Z):

Z  measures the number of standard deviation units (s) that X deviates
from its  mean (m)

Every normal variable X with some m and s, can be transformed into this Z

3. We now know that the surface from 60 to 70, corresponds to the Z-values 0 to 1.25. In the table in the back of you book, you can find the corresponding value.
Table - Part of a Z-table.

How does a table like this work?
The values in this table correspond to the surface of the corresponding Z-value.
The table takes as starting point of – (negative) infinity.
We want to know the surface from Z=0 to Z=1.25.
However, the table gives us the values from – (negative) infinity till 1.25.
We now have to look at Z=1.25 (which is 0.8944).
This means, from – (negative) infinity, to Z-1.25, are 89.44% of all values.
However, we are only interested in the values from the mean 60 (à Z=0) to value 70 (à Z=1.25).
We know from – (negative) infinity to the mean is always 50%, so therefore we subtract 50% and we are at our answer: P(60 4. Our answer is as follows:

21. More options with the normal distribution
The symmetry of the normal distribution makes it possible to calculate probabilities for negative values of Z using the table as follows
Example  

What do we know?

  • We know that the total surface has a value of 1.
  • We can see in the table that from Z=-1000 to Z=1.47 is 0.9292
  • Therefore, from Z=1.47 to infinity is 1-0.9292 = 0.0708.
  • P(Z > 1.47) = 1- 0.9292 = 0.0708

Example II
Find the probability P(-2.25 < Z < 1.85)
1. First, draw a sketch.
2. From Z= -infinity to  -2.25 equals 0.0122.
Further, from -infinity to Z=1.85 is 0.9678.
3.   Hence, P(-2.25 < Z < 1.85) = 0.9878 – 0.0122 = 0.9556.
You could already expect a value so close to 1, since almost the whole graph is colored.
Example III
Find the probability P(.65 < Z < 1.36)
1. First, draw a sketch

2. The surface from negative infinity to  Z=1.36 is 0.9131.
    The surface from negative infinity to Z=0.65 is 0.7422.
    We want to know the surface from Z=0.65 to Z=1.36, so we subtract these values from each other
3. Hence, P(.65 < Z < 1.36) = .9131 - .7422 = .1709
22. From normal to standard normal
The above three examples were values which were already Z-values. Now we go one step further: you have to calculate the Z-value yourself.
Example IV
The time to finish a XL-hamburger (X) is normally distributed with mean of 30 minutes and standard deviation of 10 minutes. What is the probability that the time will exceed 55 minutes?
Solution:
1. First,  draw a sketch
2.  To calculate the surface that exceeds 55, we have to
transform the normal distribution to a standard normal distribution. Then, we can use the table in the book.
3. When we go from a normal distribution to a standard normal distribution, the mean=30, the SD=10 and x=55, transform (using the formula) to mean=0, SD=1 and x=2.5
4. We know that the total graph has a surface of 1, and we see in the table that the surface from – negative infinity till z=2.5 is 0.9938.
5. We now can calculate the black part of the graph: 1-0.9938 = 0.0062.
6. P(X > 55) =1 - 0.9938 = .0062.
What is the probability that the time will be less than 22 minutes?
1. First, draw a sketch.

2. To calculate the surface that is less the 22, we have to transform the normal distribution to a standard normal distribution. Then, we can use the table in the book.
3. When we go from a normal distribution to a standard normal distribution, the mean=30, the SD=10 and x=55, transform (using the formula) to mean=0, SD=1 and x= -0.8
4. In the table in the book we see at z=0.8 is 0.2119, meaning from – (negative) infinity till 0.8 equals 0.2119.   
6. P(X  < 22) = 0.2119
Example about inverse calculation.
Determine the value z for which P(Z < z) = .6331.
1. First, draw a sketch.

2. We are looking for the Z-value corresponding with P-0.6331.
4. When we look in the table at 0.6331, its corresponding Z-value -> 0.34.
 
23. Exponential distribution
Exponential Probability Density distribution. An exponential random variable, X, can be used to measure the time that elapses before the first occurrence of an event, where occurrences of the event follow a Poisson distribution. Thus the exponential distribution is applied when we are interested in elapsed time intervals.   
Examples:

  • the length of time between telephone calls
  • the length of time between arrivals at a service station
  • the life-time of electronic components

When the number of occurrences of an event has a Poisson-distribution with average m per unit of time, the time between the occurrences has an exponential distribution with average 1/m time unit
The formula for the probability density function is

Expected value and variance


24. Cumulative exponential distribution
For this distribution, you do not use a table –in contrast to all other cumulative distributions. You simply use a formula:
P(X > a) e^(-lambda*alpha)
Notice in this formula, that you calculate the surface above the specified value – in contrast to all other cumulative distributions, in which the table calculate from 0 to the specified value.

Using the Graphical Calculator

You cannot say on your exam that you found the answer using the steps on your graphical calculator. However, it is very useful to check your answers with a graphical calculator.
25. Binomial

  • records show that 30% of the customers in a shoe store make their payments using a credit card
  • this morning 20 customers purchased shoes

First, notice that X is Bin(n = 20, p = .30)-distributed.
Question 1: what is the probability that at maximum 12 customers used a credit card?
1. Go to you Stat(istics) menu
2. Go to Dist(ributions)
3. Go to Bin(ominal)
4. Go to BCD à (Binominal Cumulative Distribution)
5. Data: Variable
x: 12 à this is means “all X-values from 0 till 12)
n: 20
p: 0.3
6. Execute
7. P(x) = 0.99872.
Interpret: this means that there 99.9% chance that at most 12 out of 20 customers will use a credit card.
Question 2: what is the probability that at least 12 customers used a credit card?
Now, we are looking for 12 and MORE customers (so 12 till 20). This is a problem, since our GC can only calculate values starting from 0. Therefore, we first have to calculate P(x≤11), which gives us all values from 0 till 11. Then we will subtract this value from 1, which gives us the value of P(x≥12).
1. Go to you Stat(istics) menu
2. Go to Dist(ributions)
3. Go to Bin(ominal)
4. Go to BCD à (Binominal Cumulative Distribution)
5. Data: Variable
x: 11 à this is means “all X-values from 0 till 12)
n: 20
p: 0.3  
6. Execute
7. P(x) = 0.99486 à P(x≤11)
8. P(x≥12) = 1 - P(x≤11) =  1 - 0.99486 = 0.00514
Question 3: what is the probability that at least 3 but not more than 6 customers used a credit card?
P(3 ≤ X ≤ 6)
= P(3) + P(4) + P(5) + P(6)
= P(X ≤ 6) - P(X ≤ 2)
= .608 - .035 = .573
Question 4: what is the probability that exactly 14 customers did use a credit card?
Now we want an exact number, not 0 till 14, but just 14.
1. Go to you Stat(istics) menu
2. Go to Dist(ributions)
3. Go to Bin(ominal)
4. Go to BPD à
www.joho.nl 19
Managerial Statistics part 2
5. Data: Variable
x: 14
n: 20
p: 0.3
6. Execute
7. P(14) =0.00218
Notice that the calculator did all calculations for you. So it execute the formula
26. Poisson

  • cars arrive at a tollbooth at a rate of 360 cars per hour
  • what is the probability that one or two cars will arrive during a specified one-minute period?

Solution:

  • let X denote the number of arrivals per one-minute period
  • 360 cars arrive in an hour, so 360/6 = 6 cars arrive in a minute
  • the mean number of arrivals per minute is  µ = 360/60 = 6  cars
  • it follows that X ~ Poisson(6)

1. Go to you Stat(istics) menu
2. Go to Dist(ributions)
3. Go to Poisson
4. Go to Pcd à (Poisson Cumulative Distribution)
9. Data: Variable
x: 2 -> this is means “all X-values from 0 till 2)
m: 6 -> six arrivals in a minute
10. Execute
11. P(X≤2)= 0.062.
27. Normal
Find the probability P(Z > 1.47)
1. Go to you Stat(istics) menu
2. Go to Dist(ributions)
3. Go to Norm(al)
4. Go to Ncd -> (Normal Cumulative Distribution)
5. Lower: 1.47  -> you want to know all values above 1.47, up to infinity.
Upper:  10000 -> just fill in a large number: infinity
SD: 1 -> when using a standard normal distribution, SD is always 1
Mean: 0 -> when using a standard normal distribution, mean is always 0.
6. Execute
7.  P(Z > 1.47) = 0.07078
Find the probability P(-2.25 < Z < 1.85)
1. Go to you Stat(istics) menu
2. Go to Dist(ributions)
3. Go to Norm(al)
4. Go to Ncd -> (Normal Cumulative Distribution)
8. Lower: -2.25  -> you want to know all values from -2.25 to 1.85
Upper:  1.85 -> this is your maximum value
SD: 1 -> when using a standard normal distribution, SD is always 1
Mean: 0 -> when using a standard normal distribution, mean is always 0.
9. Execute
10.  P(-2.25 < Z < 1.85) = 0.95561
The time (X) to eat an XL-hamburger is normally distributed with mean of 30 minutes and standard deviation of 10 minutes. What is the probability that the time will exceed 55 minutes?
1. Go to you Stat(istics) menu.
2. Go to Dist(ributions).
3. Go to Norm(al).
4. Go to Ncd -> (Normal Cumulative Distribution).
11. Lower: 55  -> you want to know all values from -2.25 to 1.85.
Upper:  10000 -> this is your maximum value.
SD: 10 -> when using a normal distribution, use the given SD.
Mean: 30 -> when using a normal distribution, use the given mean.
12. Execute
13.  P(55 ≥ X ) = 0.0062096
What is the probability that the time will be less than 22 minutes?
1. Go to you Stat(istics) menu.
2. Go to Dist(ributions).
3. Go to Norm(al).
4. Go to Ncd -> (Normal Cumulative Distribution).
14. Lower: -1000  -> you want to know all values till 22.
Upper:  22 -> this is your maximum value.
SD: 10 -> when using a normal distribution, use the given SD.
Mean: 30 -> when using a normal distribution, use the given mean.
15. Execute
16.  P( X < 22 ) = 0.21185
Determine the value z for which P(Z < z) = .6331.
1. Go to you Stat(istics) menu.
2. Go to Dist(ributions).
3. Go to Norm(al).
4. Go to Inv(erse) -> you are going to make an inverse calculation
5. Area: 0.6331
SD: 1 -> when using a standard normal distribution, SD is always 1
Mean: 0 -> when using a standard normal distribution, mean is always 0.
6. Execute
7. The corresponding Z-value is 0.34.

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