BUSS1020 Quantitative Business Analysis Full Course notes (best for June final exam prep) University

BUSS1020 Quantitative Business Analysis Full Course notes (best for June final exam prep) University of Sydney

Week 1 1.1 Introduction to business analytics and statistics Business analytics is the process by which businesses use statistical methods and technologies for analysing historical data in order to gain new insight and improve strategic decision-making.

There are three branches of statistics used in business including: descriptive, inferential and predictive.

1. Descriptive: used to collect, summarise, organise and present data. Examples include the range, mean, median and variance of the data. This helps businesses understand the main features of the data.

2. Inferential: uses data from a small group to then draw conclusions about a large group. This data analysis is conducted by estimation or hypothesis testing.

3. Predictive: make predictions based on a sample of data and a model.

1.2 Vocabulary, variables and levels of measurement Key statistical terminology: ● Variables: (a.k.a. attributes) are the characteristics of an item or individual. Its data is

● ● ● ● ● ●

what is analysed using a statistical method.

Data: Observed values/outcomes of one or more variables Operational definition: Are the universally accepted meanings which are clear to all associated with an analysis.

Population: The items or individuals about which to draw a conclusion. A population is a large group of which you want to know something.

Sample: Portion of the population selected for analysis.

Parameter: A numerical measure that describes a relevant characteristic of a population.

Statistic: A numerical measure that describes a relevant characteristic of a sample.

A statistic often is used to estimate a parameter.

Parameters → population (denoted by Greek letters) Statistics → sample (denoted by Latin letters)

Population vs Sample Using a sample requires slightly different techniques to using a full population. Because it is often too difficult or expensive to obtain data about the entire population, data is collected based on a sample. Different terminology is used to indicate whether the population or a

sample is used. When calculating the standard deviation, the formula will differ depending on whether the population or a sample is being measured.

Types of variables When analysing statistics, it is important to know the data type of the variable as it strongly influences the procedure of the analysis and the choice of analysis methods used.

Variables are divided into: 1. Categorical (qualitative): have values that can only be placed into categories - e.g. 'yes' or 'no'.

2. Numerical (quantitative): have values that represent actual number quantities.

Categorical (qualitative) variables are divided into: ● Nominal variables: Represents different categories without any ranking. E.g. car models, car colours and eye colours. ● Ordinal variables: Are categories with a distinct and meaningful order, however, cannot quantify the exact difference between each rank. E.g. education level Numerical (quantitative) variables are divided into: ● Discrete variables: are those arising from a counting process, such as the number of children and defects per hour. Note: Sometimes discrete variables with many outcomes are treated like continuous variables, such as 'prices'. ● Continuous variables: are those arising from a measuring process, such as cost and financial return. Note: They can be assigned any value within a given interval.

Levels of measurement Different levels of measurement provide different amounts of detail. A variable has 4 levels of measurement including nominal, ordinal, interval, and ratio. Note: nominal provides the least detail, while ratio provides the most.

1. Nominal: used to classify and categorise. They have no real order. E.g. For employment classifications, labels would include doctor, teacher, lawyer etc.

2. Ordinal: used to classify and indicate rank and order. Typically, this is used as a scale for qualitative data. These levels cannot be compared to each other. E.g. risk in investment portfolio: 1. High risk, 2. Medium risk, 3. Low risk.

3. Interval: Measured on a numerical scale. Interval data has order, and the difference between two values is consistent and meaningful. E.g. temperature. Other characteristics of interval data include the illegitimacy of multiplication/division (e.g.

20 degrees isn’t twice as hot as 10 degrees). Interval data does not have a meaningful zero point (e.g. zero degrees celsius does not mean the absence of heat).

4. Ratio: Has the same properties as interval data but zero has true meaning. E.g. weight, volume and profit.

1.3 Data collection Sourcing data Data can be sourced from a range of methods including: ● From the activities of a business/organisation ● From the activities of an individual ● Responses from a survey ● Recording the outcomes of a designed experiment ● The results of an observational study Data is either primary or secondary: ● Primary - data analyst collects the data (within the business), e.g. transactional information to help evaluate the success of a marketing campaign. ● Secondary - data is collected from outside the business. e.g. government statistical bureaus Sources of data in business 1. Organisational data: e.g.

- Financial data - Industry data - Stock prices - Sports statistics 2. Designed experiment: e.g.

-

Consumer testing Quality testing Market testing Web page design testing

3. Survey: e.g.

-

Political polls Internet polls Customer satisfaction survey Student survey

4. Observational study: e.g.

-

Focus groups Measuring service time Measuring volume of traffic

5. Automated collection - data streams: e.g.

-

Mobile phone data

-

Click-streams Stock exchange GPS Electronic monitors Social media feeds

Streamed data results in a large volume of data referred to as ‘Big data’. This can be analysed automatically to reveal patterns or trends about customers which can lead to better decisions in business. Big data is a rapidly growing field in technology and business.

Structured & unstructured data ● Structured data is more traditional and easily accessible and hence regularly used by the business or organisation. Structured data is easily stored and analysed.

Examples are databases and excel spreadsheets

●

Unstructured data is more modern, containing irregular patterns and values that require additional interpretation. This data type takes many forms including text messages, pictures, videos, audio tracks, voice mails, emails, tweets, social media posts, and vlogs.

Data cleaning Data cleaning corrects defects (i.e. incorrect/inconsistent data) that could affect statistical results, ensuring data quality is suitable for the needs of the business. Possible irregularities in data include: - Invalid variable values (e.g. non-numerical data for a numerical value, invalid categorical values, or numerical values outside the defined range) - Coding errors (e.g. inconsistent categorical values, extraneous characters) - Integration errors (e.g. redundant columns, duplicated rows, differing column lengths, or inconsistent units of measurement).

Potential fixes to errors in the dataset: - Find the correct data - Delete the data irregularity - Delete the whole row of data - Substitute in another value

1.4 Sampling Sampling is used for the following reasons: - Often unable to access the whole population - Less time consuming and less costly - More practical (less complex) The process of sampling begins with the sampling frame (the lists of items in the population that can be sampled) e.g. customer databases. The aim is to limit bias by sampling from all different types of people in the population.

There are two types of samples including non-probability samples (bad) and probability samples (good). i) Non-probability samples: doesn’t select sample members randomly. There is no guarantee that the true population will be fairly represented. This includes basing sample members on judgement (experts select the most appropriate items or people to put into the sample), self-selection (sample members choose to participate in a survey), or convenience (people are selected who are close at hand. ii) Probability samples: items are chosen randomly. Types include simple random (RNG), systematic (selected according to a random starting point but with a fixed periodic interval), stratified (frame is divided in subgroups according to their characteristics), and cluster (researcher divides the population into separate groups, called clusters).

Types of survey errors include: 1. Coverage error → a particular group is excluded from the sample frame or have little opportunity of selection 2. Non-response error → certain types of people may not give a response in the sample 3. Sampling error → Information gathered about samples within a population may be different from another sample within that same population. E.g. the average height of 10 people in sample A will be different from that of sample B. 4. Measurement error → weakness in the design of the questionnaire e.g. if the question is ambiguous or if the respondent makes an error.

Week 2 2.1 Measures of central tendency Measures of central tendency include the mean, median and mode 1. Mean: the average → uses sigma notation

Limitations: - Only useful for numerical data - Affected by outliers

2. Median: Is the middle value (has 50% above and below it) → not affected by outliers.

The median is good for asymmetric or skewed distributions and can be used for both numerical and categorical (ordinal) data.

To find the POSITION of the median (not the median itself), apply the following formula:

3. Mode: Is the value with the highest frequency in the dataset. The mode is not affected by extreme outliers, can be used for nominal and categorical data, and is the only measure that can be used for nominal categorical data.

The mean that is calculated using sigma notation (above) is referred to as the arithmetic mean which is used for numerical data types whose values are typically combined using addition or subtraction. Values combined using multiplication use the geometric mean. To calculate the geometric mean of n values, we multiple the values together and then take the nth root. The Geometric mean of 2, 2, and 16 is 4.

In order to calculate the geometric mean for a rate of return, we apply the formula:

Where Ri is the rate of return in time period i.

Make sure that the rate of return is a percentage.

In order to calculate measures of central tendency in excel use: =average(), =mode(), =median(), and =geomean(). If you use Geomean() for rates, you need to first convert the rates to factors, and then convert the answer back to a rate.

For more info: https://support.microsoft.com/en-us/office/calculate-the-average-of-a-group-of-numbers-e15 8ef61-421c-4839-8290-34d7b1e68283

2.2 Variation and shape (I) ‘Variance measures the variability (volatility) from an average. Volatility is a measure of risk, so this statistic can help determine the risk an investor might take on when purchasing a specific security.’ - Investopedia Measures of variation include: range, variance, standard deviation and the coefficient of variation.

1. Range: The difference between the largest and smallest values. It is determined heavily on the existence of outliers 2. Variance (sample): Is the typical or average squared deviation, or distance, from the mean. The formula for calculating sample variance below:

3. Standard deviation: Shows the variation around sample mean. It is the square root of the variance. Thus, the same formula used to find the variance is applied, but square root the answer.

4.

-

Coefficient of variation Measures relative variation Shows variation relative to the mean Always expressed as a percentage (%) Used to compare variability of 2 or more datasets in different units (e.g. having stocks in AUD, YEN, and pounds)

Therefore stock B has a greater investment risk.

2.3 Variation and shape (II) Further measures used to calculate variation and shape include Z-scores, Skewness and Kurtosis.

1. Z-score: The number of standard deviations from the mean.

2. Skewness (shape): measures the amount of asymmetry in the sample (zero skewness = symmetrical). When mean is less than the median, it is negatively skewed and when mean is greater than the median, it is positively skewed.

3. Kurtosis (shape): measures the relative concentration of values in the centre compared to the tails.

2.4 Variation and shape (III) ●

Quartiles

To find the POSITION of the quartiles:

Where ‘n’ is the sample size (number of observed values) ● Interquartile range (IQR) Calculated as Q3 - Q1, measures the spread in the middle 50% of data. The IQR is a measure of variability which is not influenced by extreme values/outliers. Measures that are not affected by outliers/extreme values are called ‘resistant’ or ‘robust’ measures. ● Boxplots Is a graphical display of the 5-number summary.

Distribution Shape and the Boxplot:

2.5 Descriptive measures for populations The previous measures discussed have been for samples, known as statistics. The following are for populations, known as parameters. A key distinction between parameters and statistics is that parameters are constants - e.g. having only one mean for a population.

Statistics on the other hand are random variables - e.g. each different sample taken from a population is likely to get a different sample mean.

Three key measures for population include the population mean, population variance and population standard deviation.

The formulae used for parameters (population) are similar to those used for in statistics (sample) with the differences in formulae for parameters including: ● ● ● ●

Capital 'N', replaces 'n' for the mean Capital 'N', replaces 'n-1' for the variance and standard deviation Population parameters are generally denoted using Greek letters Whereas sample statistics denoted using latin letters.

●

Empirical rule: helps us to describe a normal population a.k.a bell-shaped curve.

The empirical rule only applies to distributions that are bell-shaped.

68% of scores lie within 1 standard deviation from the mean

95% of scores of scores lie within 2 standard deviations from the mean

99.7% of scores lie within 3 standard deviations from the mean

●

Chebyshev's Rule: Applies to every data set, where regardless of how the data is 2

distributed, at least (1 − 1/𝑘 ) × 100% of the values will fall within ‘k’ standard deviations of the mean (for k>1).

For example:

Demonstrated example:

2.6 Organising and visualising data (I): Chart types It is vitally important to use the right chart types which convey the information in an appropriate way which is clear to understand. This is important as it informs business decisions. The following are common chart types divided into categorical and numerical types:

Some important questions to consider when trying to select the appropriate chart for the data include: ● ● ●

What are you trying to learn or communicate?

What are the types of comparisons you want to make?

What types of data have you collected?

2.7 Organising and visualising data (II): Principles of good graphs When trying to create effective tabular and visual summaries, it is important to avoid the following pitfalls:

1. Obscuring data Overloading information into charts can obscure data and hamper decision-making.

2. Creating False Impressions Can arise from selective summaries and improperly constructed visualisations.

3. Displaying misleading analysis (Chartjunk) Where decorative elements are added to try to enhance the graph - leading to obscuring the data or creating false impressions.

Week 3 3.0 Introduction to basic probability Probability is used and modelled throughout business decisions. Examples include: ● New customer acquisition ● Measuring and boosting customer loyalty ● Cross-sell/up-sell ● Targeted advertisements ● Financial forecasting ● Fraud detection ● Product placement ● Churn measurement and reduction

3.1 Basic probability concepts The basic probability concepts include: ● ● ● ●

Probability → Chance that an uncertain event will occur (between 0 and 1) Impossible event → No chance of occurring (probability = 0) Certain event → definitely will occur (probability = 1) Uncertain event → Refers to the unpredictability of future events, influencing present decisions. They can’t be predicted as they have not yet occurred.

There are 3 common approaches for assessing the probability of an uncertain event which include: 1. ‘A priori’ → obtains probabilities through deductive reasoning based on prior knowledge of the process. It’s formula is shown below:

For example, if there are 10 countries in the cricket world cup, all ranked from 1 to 10, what is the likelihood of Australia versus a country in the top 3 rankings for the first round (given Australia was ranked 5th)?

Probability of versing top 3 = x/t = number of countries in top 3/total number of opponents.

Therefore, x/t = 3/9 (or ⅓) This demonstrates the logic used given prior knowledge that Australia is not in the top 3 in this approach when assessing the probability of an uncertain event.

2. Empirical → is based on observed data. Observations from the past are used to estimate the likelihood of an event. Formula for the empirical is:

For example, if I am about to invest in a share, I might want to observe how many times the share has gone up in a certain amount of days. If a Tesla share goes up 60 times in 100 days, then my probability of occurrence via the empirical method would be 60/100 (or ⅗).

3. Subjective → based on expert judgement/experience. There is no past data to rely on.

For example, what is the probability that Australia will win the next Ashes test series?

The sample space is the set of all possible outcomes from an experiment. For example, the sample space for rolling a dice is all possible outcomes (i.e. 1 - 6).

Events are subsets of the sample space. Each possible outcome of an experiment is called an event. Events are divided into i) simple events, and ii) joint events.

Simple events are described by having only one characteristic. E.g. whether someone is male or female?

Joint events (a.k.a. an intersection of events) are described by two or more characteristics.

E.g. Male or female, pet or no pet?

The complement of an event refers to the subset of the outcomes not included in an event.

E.g. the complement of a red traffic light would be yellow and green. The complement is demonstrated with an apostrophe. The complement formula is: P(A') = 1 - P(A) Two ways to visualise the outcomes of an event include contingency tables and decision trees. i) contingency table:

ii) decision tree:

3.2 Basic probability ➔ Marginal probability (a.k.a simple probability): Refers to the probability of a simple event (single characteristics). ➔ Joint probability: Refers to the probability of a joint event (multiple variables) Mutually exclusive: events that cannot occur at the same time

➔ Collectively exhaustive: are events that must occur. Together they cover the entire sample space.

The general addition rule is found by adding the marginal probabilities of events A and B and subtracting their joint probability.

If events X, Y and Z are mutually exclusive and collectively exhaustive, then their probabilities sum up to 1 (i.e. P(X) + P(Y) + P(Z) = 1).

3.3 Conditional probability Conditional probability is the probability of one event, given the knowledge that another event has occurred.

For example: Find the probability that given a car has AC also has a GPS i.e. P(GPS|AC)

AC

GPS

No GPS

Total

35%

55%

90%

No AC

5%

5%

10%

Total

40%

60%

100%

= P(GPS and AC)/P(AC)

When the occurrence of one event does not affect the probability of the occurrence of another event, the events are said to be independent. The independence formula is below:

The multiplication rule is used to find a joint probability of two events using a conditional probability of one event and marginal probability of the event being conditioned on.

The formula is shown below:

3.4 Bayes' theorem The Bayes’ theorem is used to revise previously calculated probability based on new information. It serves as an extension of conditional probability where it allows us to reverse the conditioning between two events or variables. For example, if we know what P(A|B) is, we are able to find P(B|A).

There are both the simple and general form to the Bayes theorem.

Simple form:

General form:

Applied example:

3.5 Counting rules Counting rules are rules for counting the number of possible discrete outcomes of an experiment. There are 5 counting rules.

1. Counting rule 1: The number of outcomes is equal to Kn e.g. 63 = 216 2. Counting rule 2: Allows for the number of events to differ. E.g. if I decide to roll a dice and flip a coin, then there are 12 possible outcomes (as 6 x 2 = 12).

3. Counting rule 3: Number of ways that n items can be arranged in unique order

4. Counting rule 4: The number of ways of arranging X objects selected from n objects in order (Permutations).

5. Counting rule 5: The number of ways of selecting X objects from n objects, irrespective of order (Combinations).

3.6 Ethical Considerations Sometimes probability statements can be incorporated into advertising campaigns with the aim of strengthening the positioning of the product.

An example includes lotteries which try to inflate the likelihood of winning despite having very little odds. Another example is investing companies which focus on the positive aspects of investing, rather than the negatives.

Week 4 4.1 Discrete random variables and probability distributions Random variable - Assigns numerical values to the outcomes from an experiment.

Numerical random variables can either be discrete or continuous.

Discrete random variables can only assume a countable number of values.

When applied to probability, the distribution is only valid if the sum of the probabilities across all outcomes is 1.

An example of a discrete random variable would be to conduct an experiment where 2 coins are tossed, where X = the number of heads.

In the experiment, there are 4 possible outcomes i.e. 2 tails; a tail and a head; a head and a tail; 2 heads. Although having 4 of these outcomes, there are only three possible outcomes for the random variable i.e. 0 (0.25 probability), 1 (0.5 probability), and 2 (0.25 probability). This can then be represented in a graph as shown below:

To find the expected value (the average or mean) of the discrete random variable, we apply the following formula:

An example to illustrate this is as follows: Tossing two coins, with X = the number of heads…

Where ‘P(X=xi)’ refers to the probability of achieving the outcome X amount of times in two coin flips…

The expected value of X can be calculated as shown:

Where each possible outcome (X) is multiplied by its probability (i.e. P(X=xi)) and then summed together. In this case, the expected value E(X) is equal to 1.

In order to determine the spread of the distribution from the mean, we can calculate the standard deviation which uses the following formula:

For example, in measuring the standard deviation for the number of traffic accidents per week: Assuming that the E(X) = 1.36…

4.2 Binomial distributions Binomial probability distributions are used where a random variable ‘X’ counts the number of “events of interest” occurring from a fixed number of observations or trials (n).

The properties needed to pass as a valid binomial distribution are: 1. Must be a fixed number of trials (n) 2. The two categories are both mutually exclusive and collectively exhaustive.

I.e. for every coin flip, the binomial distribution consists of either a head or tail (can’t get both at same time, and are no other possible outcomes) 3. The probability of each event remains constant each trial (e.g. the probability of heads being 0.5 remains the same each time the coin is flipped) 4. Observations are independent (the outcome of one observation does not affect any other) The formula for binomial distribution is:

To simplify the formula consider the example of flipping a coin five times, where X = number of heads:

Now to actually calculate a binomial probability:

The following are the summary measures for binomial distribution:

Excel function for binomial distribution is BINOM.DIST

4.3 Poisson distribution The Poisson distribution is used when counting the number of times an event occurs in a given or fixed area/time/window of opportunity. A window of opportunity could be: time, area, volume, distance, or something else.

For the Poisson model to be applied: ● There should be no upper limit on the potential counts ● The window of opportunity is a continuous interval in which at least one occurrence of an event can occur.

There are 5 properties of the Poisson probability distribution including: 1. The data count is the number of times an event occurs in a window of opportunity 2. The probability that an event occurs in one window is the same for all other windows 3. The number of events in one window is independent of the number that occur in other windows 4. The probability that 2 or more events will occur in a window approaches 0 as the window becomes smaller 5. The average number of events per window of opportunity is denoted λ (lambda), where λ>0)

Example: Customers arrive at a fast food outlet at a rate of 3 per minute. Find the distribution of customer arrivals per minute.

From this, we can gather that λ = 3. Let's assume that we observe 5 customers arriving in a one minute period. The working out and answer are shown below:

Excel function for binomial distribution is POISSON.DIST

4.4 Hypergeometric distribution To understand what hypergeometric distribution is, it is useful to compare it to binomial distribution: The binomial distribution is applicable when selecting from a finite population with replacement, OR, selecting a finite sample of n from an infinite population, without replacement.

The hypergeometric distribution is applicable when selecting from a finite population without replacement.

Thus, the hypergeometric distribution allows the probabilities to change based on the results of earlier events so the probability of an event of interest is no longer constant over all observations.

There are 4 properties of Hypergeometric distribution including: 1. "n" trials in a sample taken from a finite population of size "N" 2. The sample is taken without replacement 3. The outcome of trials are dependent 4. Finding the probability of a particular number of events of interest The formula for Hypergeometric Distribution is below:

The mean of the hypergeometric distribution is:

An example to demonstrate the hypergeometric probability distribution process:

Excel function for hypergeometric distribution is HYPGEOM.DIST

4.5 Covariance and summing random variables Covariance measures the strength of the linear relationship between 2 numerical random variables x and y. ● Positive covariance: there’s a positive linear relationship between X and Y, that is, when X increases, Y also increases, on average. ● Negative covariance: the variables have a negative linear relationship, so as one variable increases the other variable on average decreases.

Note: if the two random variables are independent then their covariance MUST be zero.

Covariance formula:

Applied problem: Investment returns

Mean:

Covariance:

Covariance is useful when dealing with the sums of random variables, which is used commonly in Finance when constructing portfolios of stocks.

A portfolio return is calculated as a weighted sum of the asset returns for the assets that make up that portfolio (where the weights are the proportion of the money invested in each asset).

Where a and b are constants, with X and Y representing random variables. If applied to the context of dealing with a portfolio of two stocks such as BHP and RIO - the constants a and b represent money invested in each of the stocks, where the returns are represented by X and Y. The sum of the proportions must be equal to 1.

Where w is a constant.

Week 5 5.1 Introducing key concepts Continuous random variables can potentially take any value in a range, depending on how accurately or precisely it can be measured.

Examples include: - Company profit - Volume of cans - Temperatures - Weight of packages Discrete distributions have probabilities at fixed points (integers).

Probabilities are online defined for continue random variables when there is a range (otherwise any value would have a probability of zero as there are infinite amount to choose).

The probability density function (PDF) shows the probability of a range of values as shown in the area shaded under the curve within the selected range.

The following are examples of continuous probability distributions:

Each density curve expresses the relative likelihood of each X value. X is represented by the shaded area. The area under the entire density curve is always equal to 1. The area under a single value for X equals 0.

5.2 Normal distribution The normal distribution is also known as the ‘bell’ curve and the ‘gaussian’ curve. The normal distribution is always symmetrical at the point of the mean. The mean, median and mode are all equal to each other. Any variations from the mean is represented by how many standard deviations are from the mean, shown by sigma (σ). The random variable has an infinite range.

Some examples of normal distributions include: ● IQ test (although having a right skew) ● Positions of particles in fluid ● Sums of random variables (need a lot of variables) Most of the time normal distributions are used to provide a hypothetical figure.

Formula for normal distribution is:

2

The higher the variance (σ ), the more spread out the distribution will be i.e.

Empirical rules include: approximately 68% of the data fall between 1 standard deviation from the mean, 95% for 2 standard deviations and 99.7 for 3.

5.3 Standard normal distribution Also known as the ‘Z’ distribution, the standard normal distribution has a mean of zero and a standard deviation of 1.

To find the Z score:

Values above the mean are positive Z values and below are negative. The probability is measured by the shaded area under the curve. This must sum up to 1. A cumulative normal table can be used to find Z scores.

Normal distributions will have a symmetric curve with the area under the curve equalling to 1. The mean is equal to the median and is equal to the mode.

To find normal distribution in excel, use the ‘NORM.DIST’ function.

5.4 Assessing normality In order to determine a normal distribution, there are 7 tests including: ● Is the sample mean ≈ sample median? ● Is the empirical rule approximately satisfied? ● Is the IQR ≈ 1.33 standard deviations? ● Is the boxplot (smaller n) and histogram (larger n) close to symmetric? ● Is the histogram roughly bell-shaped? ● Is there an absence of clear extreme, outlying observations or ‘fat tails’? ● Are the sample skewness and kurtosis statistics ≅ 0?

5.5 Uniform distribution Uniform distribution (a.k.a. regular distribution) has equal density for all possible outcomes of the random variable.

To find the mean of a uniform distance, find the average of ‘a’ and ‘b’ i.e.:

The standard deviation of a uniform distribution is found by:

Example:

To find the probability of a specified sub-section:

More generally:

5.6 Exponential distribution

The exponential distribution has the following characteristics: ● ● ● ●

●

Positive-valued, right skewed distribution Has one parameter, lambda (𝝀) Models the length of time between two occurrences of an event (such as time between events) e.g. time between people arriving to be seated in a restaurant.

The poisson distribution counts occurrence of the random variable at a specific point of time whereas the exponential distribution considers the time between each event counted The mean and standard deviation of an exponential distribution is found by 1/𝝀

Formula:

Now to find the probability:

Example:

Excel function: EXPON.DIST Explanation here: Exponential Probability Distribution

Week 6 6.1 Introduction to sampling distributions Whenever a sample is collected and a statistic for that sample is calculated, that sample statistic is a random variable and thus has a distribution. Any statistics we calculate (like the sample mean, sample proportion, sample median and the sample variance) all have sampling distributions that are dependent on both the population distribution and the chosen sample size.

A sampling distribution is the distribution of all possible values of a sample statistic for a given sample size (n) selected from a population. It helps us know how accurate the statistic is for estimating the corresponding population parameter.

The standard deviation of a sample statistic is referred to as the standard error (as it is used to estimate a population parameter). The mean of the sample mean is always the same as the population mean. The standard deviation of the sample mean distribution will always be smaller than the population mean.

6.2 Sampling distribution of the mean Any population with mean (μ) and standard deviation (σ), then the sampling distribution of (x-bar) has:

And…

If we use the sample mean to estimate the population mean, then on average we will get the correct result. In other words, the mean of the sample mean is always equal to the population mean. When an estimator on average is equal to the parameter it is trying to estimate, this is referred to as an unbiased estimator. Thus, the sample mean is an unbiased estimator of μ.

The standard error of the sample mean measures the variability in the mean from sample to sample as different samples of the same sample size (n) from the same population will yield different sample means. The standard error of the mean is equal to the population standard deviation divided by the square root of the sample size i.e.:

As the sample size (n) increases, the standard error of the mean decreases (which is why statisticians prefer larger sample sizes).

To find the Z-score for the sampling distribution of X-bar:

If a population is normally distributed with mean (μ) and standard deviation (σ), the sampling distribution of X-bar is also normally distributed with:

As can be seen below, the normal sampling distribution has a smaller standard deviation

As x-bar is an unbiased estimator of μ, then on average the sample mean is equal to the population mean.

To calculate the lower limit of a Z-score:

And the upper limit:

In summary:

6.3 The central limit theorem (CLT) The central limit theorem (CLT) is where even if the population is not normally distributed, sample means of random samples from any population will be approximately normally distributed, as long as the sample size is large enough.

The general rule of thumb is that if a sample is at least 30, then the CLT will apply, i.e. the sample mean’s sampling distribution will be approximately normal. The larger the sample size, however, the greater the approximation.

The properties of the sampling distribution are the same as before i.e.:

6.4 Sampling distribution of the proportion For population proportions, they are denoted with 𝜋. The sample proportion (p) estimates 𝜋. So to find the sample proportion (p), we use the following formula:

P will always lie between zero and one as it is a proportion.

The average value of our sample proportion will be equal to the population proportion 𝜋, making it an unbiased estimator i.e:

To find the standard error of the proportion:

The sampling distribution of p follows a binomial distribution. By the CLT, p is approximately distributed as a normal distribution is large enough. That is…

as summarised in the image below:

If the above criteria are met then we can say that the sample proportion is approximately normal due to the CLT. This means that if we standardise it, we should approximately get a standard normal distribution. We can then work with the standard normal tables to find probabilities for the sample proportion.

To standardise p to a z value:

In summary:

and…

6.5 Sampling from finite populations The CLT and the standard errors of the mean and of the proportion are based on samples selected from infinite populations or from finite populations with replacement. However, in instances where we sample without replacement from populations that are of a finite size we should use a finite population correction (fpc) factor whenever the sample size is a significant portion of the overall population. This will reduce the standard error of the mean and the standard error of the proportion, leading to greater accuracy of these estimators.

Some key characteristics regarding the fpc: ● The fpc is used when the sample size (n) is more than 5% of the population size (N), i.e. (i.e. n/N > 0.05) AND sampling is without replacement.

● ●

The fpc is always less than 1, meaning it will always reduce the standard error (for a more precise estimate of the population parameter).

As the sample size increases to become a larger portion of the population, the standard error is reduced by a larger amount.

The formulas regarding sample from finite populations are as follows:

Week 7 7.1 Introduction to confidence intervals Confidence intervals help us estimate population parameters. This is done in two ways i.e. using a point estimate or a confidence interval. i) Point estimate → a single number ii) Confidence interval → is a range of numbers around the point estimate.

Characteristics of confidence interval estimates: ● Gives a range of values ● Considers the variation in sample statistics ● Based on observations from only 1 sample ● Provides information about possible values of unknown population parameters ● Stated in terms of “level of confidence”, e.g. 95% confidence Example of a 95% confidence interval:

As you can see, the interval estimate represented by the red line does not contain the population mean as it is too low.

The confidence level is our confidence that the interval will contain the unknown population parameter. Note, that the percentage of the confidence level is always less than 100%.

There are two methods of interpreting the confidence level include a: ● Relative frequency interpretation

-

For a confidence level of 95%, if we collect many samples of size n, and construct confidence intervals for each, then 95% will contain the true unknown parameter

● Subjectivists interpretation - Without knowing μ, there is a 95% ‘chance’ that μ is inside a particular interval (note, you can’t say there is a 95% ‘probability’ that the true mean lies within the interval, as the population mean is not random and instead a fixed number. The interval surrounding the mean can be talked about probabilistically as it is random).

General formula for a confidence interval:

Where: ➔ Point estimate = sample statistic estimating the population parameter of interest ➔ Critical value = a value based on the sampling distribution of the point estimate and the desired confidence level ➔ Standard error = the standard deviation of the point estimate

7.2 Population mean - population variance known When we know the standard deviation of the population mean for the confidence interval, then the population is normally distributed UNLESS the population is not normal and instead has a large enough sample for the CLT to apply.

To find the confidence level estimate when the population standard deviation is known:

Where… ➔ X-bar is the point estimate ➔ Z𝛼/2 is the normal distribution critical value for a probability of 𝛼/2 in each tail ➔

σ

is the standard error

The diagram below goes into more depth concerning the critical value Z𝛼/2:

Most commonly used levels of confidence:

An example:

Here is how that example can be interpreted:

In order to produce a narrower interval and thus have a higher degree of confidence, the factory could test using more samples (n).

The CONFIDENCE.NORM excel function is used for this.

7.3 Population mean - population variance unknown It is rare in real business world applications that 𝜎 is known. As a result, we need to replace the population standard deviation with the sample standard deviation, S. As S varies from sample to sample, this creates extra uncertainty. The student’s t-distribution is used instead of the normal distribution.

In the student’s t-distribution, the t𝛼/2 values depend on the number of degrees of freedom (d.f.). These are the number of observations that are free to vary after the sample mean has been calculated.

The confidence interval for μ (𝜎 unknown) Assumptions: ● Population standard deviation is unknown ● Population is normally distributed OR not normally distributed, however with a large sample size for CLT to apply Confidence interval formula (sigma unknown):

Where ta/2 is the critical value of the t distribution with n -1 “degrees of freedom” and an area of α/2 in each tail)

Example problem:

For excel, use the T.INV.2T(probability, d.f.) function

7.4 The population proportion A confidence interval estimate, for the population proportion (π), can be calculated using the sampling distribution of the sample proportion (p) which will be approximately normally distributed for sufficiently larger samples by the Central Limit Theorem. This is shown below:

Recall that the distribution of the sample proportion is approximately normal when:

With standard deviation:

And standard error:

To find the upper and lower confidence limits for p, use the following formula:

Note that p is the sample proportion, 𝑝 = 𝑋𝑛 , where X is the number of items having the characteristic and n is the sample size.

An example:

The above example indicates: ● A 95% confidence that the true percentage of employees with adaptable desks in the population is between 16.51% and 33.49%. ● Even though the interval of 16.51% and 33.49% may or may not contain the true proportion π, 95% of intervals formed from samples of size 100 will contain the true proportion.

7.5 Determining sample size Note that there is a difference between the standard error and sampling error.

Error is caused by observing a sample instead of the whole population. The required sample size can be found that obtains a desired margin of error (e), with a specified level of confidence (1-⍺).

The margin of error, e, is also called the sampling error. This refers to the amount of imprecision in the estimate of the population parameter. This is the amount added and subtracted to the point estimate to form the confidence interval.

Sample size formula (for known 𝜎)

Let’s look at an example, for when 𝜎 is known:

Note that e here is the sampling error.

When 𝜎 is unknown, it can be estimated when determining the required sample size: ● Use a value for 𝜎 that is expected to be at least as large as the true 𝜎 ● Select a pilot sample and estimate 𝜎 with the sample standard deviation, S

●

The error in terms of the student-t can be solved numerically if desired

I.e.

To determine the sample size for the proportion (without knowing 𝜎):

When π is unknown, then π can be estimated by p, with a pilot sample, or using a conservative estimate of 0.5.

Example:

Remember to always round up the calculated sample size to the nearest whole number.

7.6 Application to auditing Auditing is the collection and evaluation of evidence about information related to an economic entity (e.g. a sole business proprietor, a partnership, a corporation, or a government agency), in order to determine and report on how well the information corresponds to established criteria.

Auditors typically rely on estimation techniques using probability sampling methods over examining a complete population of information. This is more feasible for auditors as the

population of company accounts is often too large, time-consuming and costly to keep up with.

7.7 Ethical considerations ● ● ●

A confidence interval (CI) estimate (reflecting the sampling error), should always be included when reporting a point estimate.

The level of confidence and sample size should also always be reported.

An interpretation of the CI estimate should also be provided. This is so someone untrained in statistics can understand.

When media outlets publicise the results of a political poll, the sampling error is often underplayed. Instead, an ethical presentation of poll results give equal prominence to: ● Confidence levels ● Sampling error ● Sample size ● Confidence limits of the poll

Week 8 8.1 Introduction to hypothesis testing A hypothesis is a claim about population parameters such as the population mean and the population proportion. Examples of hypotheses include: ★ At least 10% of Coca Cola’s customers will purchase their new coca cola brand ★ A cereal packet’s mean weight is 500g ★ Telstra’s market share of mobile phone users is at least 50% There are two hypotheses: the null hypothesis and the alternative hypothesis.

Null hypothesis → States a default claim/assertion about a population parameter (such as μ). E.g. The average diameter of a manufactured bolt is 30mm (written as H0: μ = 30).

Alternative hypothesis → Opposes the null hypothesis in some way. E.g. The average diameter is not equal to 30mm (written as H1: μ ≠ 30), or the average diameter is less than 30mm (written as H1: μ < 30).

There are 6 steps of hypothesis testing including: Step 1 State null (H0) and alternative (H1) hypotheses: Usually, we start with the alternative hypothesis, to determine what the null hypothesis should be We begin assuming that the null hypothesis is true when we test it Hypotheses are always about population parameters, not sample statistics

Step 2 Choose level of significance (α) and sample size (n) Step 3 Determine appropriate test statistic and sampling distribution (Z or T) Step 4 Determine critical values and identify rejection and non-rejection regions. The critical value of a test statistic creates a “line in the sand” for decision making - it helps to answer the question of “how far is far enough”.

(Note the lower and upper critical values can be found using excel or table) Step 5 Collect data, compute test statistic value, and see which region it falls in. Convert the sample statistic (X̄) to the test statistic to find where it would lie on the standard normal distribution.

Step 6 Make statistical decision and state managerial conclusion in the context of the business problem Note that these steps apply to the critical value approach So how do we know when to reject the null hypothesis?

If the sample statistic is “close enough” to the stated population parameter, the null hypothesis is not rejected. If the sample mean is “far enough” from the stated population mean, the null hypothesis will be rejected.

A test statistic is calculated from the sample statistic and is compared to the critical values of the test statistic’s distribution to help us decide whether to reject the null hypothesis. If the test statistic falls into the rejection region, we reject the null hypothesis. If not, we do not reject the null hypothesis.

8.2 Types of errors ● -

Type 1 error (α) → Wrongly reject a true null hypothesis. α = P(reject null | null true) The complement of the probability of a Type I error is called the confidence coefficient (1 - α)

● -

Type 2 error (β) → Wrongly not reject a false null hypothesis β = P(not reject null | null false)

-

The complement of the probability of a Type II error is called the power of a statistical test (1 - β)

As Type I error probability (α) increases, type II error probability (β) decreases.

Note that the type 2 error is affected by the sample size, thus, to reduce the probability of making a Type II error is by increasing the sample size. There is a limited amount of resources (e.g. time and money), however, affecting how large of a sample a business can have.

Why is this significant? Let’s look at an example:

Factors influencing the probability of making a type 2 error include:

● ● ● ●

When the difference between the hypothesised parameter value and it's true value decreases When α decreases When σ increases When n decreases

8.3 Tests for the mean – one sample (I) Recall that step 3 is about determining the test statistic and sampling distribution.

(note, typo → should be ‘μ’ instead of ‘m’) In this case, we use the Z-distribution to assess how far away the sample mean is from the hypothesised value of the population mean. The assumption is that the sample mean is normally distributed, or there is a large enough sample size for the CLT to apply.

The two-tailed test is used when the alternative hypothesis has a ‘not equal to’ sign (as there is no particular direction). We will reject the mean in the null hypothesis if the sample mean is too far below it or too far above it. There are two cutoff values (critical values) defining the regions of rejection.

Now, to perform step 4, which is to determine the critical values for a specified level of significance (⍺)

Step 5 is about converting the sample statistic (X̄) to test statistic (Zstat) Step 6 is then about making the final decision as to whether or not the null hypothesis should be rejected based on whether or not it lies in the rejection region. Remember, we never accept the null hypothesis.

Let’s look at an example:

There is a connection between the two-tailed tests and confidence intervals.

These two procedures ALWAYS give the same conclusion (for hypothesis tests of the mean), because the length of the non-rejection region is equal to the width of the associated confidence interval, using the same a.

Overall, a two-tailed test is where the alternative hypothesis contains a ‘not equal to’ (≠) sign.

We will reject the population mean in the null hypothesis if the sample mean is either too far below, or too far above it. So there is a rejection region in each tail of the distribution.

The p-value approach (short for ‘probability value’) is an alternative approach to conducting a hypothesis test. The p-value is the probability of observing a test statistic equal to or more extreme in the direction of the alternative hypothesis than the observed sample value, based on the condition that the null hypothesis is true. The decision rule for the p-value approach is to reject H0 if the p-value is less than α, otherwise do not reject.

The p-value approach has 5 steps (as opposed to 6), with the removal of needing to find critical values.

8.4 Tests for the mean – one sample (II) The hypothesis test for the mean when sigma (𝜎) is unknown: ● Replace 𝜎 with sample standard deviation ● Switch to the student’s t distribution ● Degrees of freedom will be n-1 ● Assume population follows a normal distribution, or sample size (n) is large enough for CLT to apply So, when sigma is unknown, we must convert the sample statistic (X̄) to a tSTAT to test howfar it is from the population mean (μ). This is seen visually:

Let’s have a look at an example:

Now to look at the alternative approach to the critical value one, the p-value approach:

8.5 Tests for the mean – one sample (III) We have looked at hypothesis tests for two-tailed distributions, but what about one-tailed where there is one direction.

For example: ● Amazon wants a more than $50 average purchase amount per customer, i.e. μ>50 ● Coca-cola wants more than 10% of their customers to purchase a new cola, i.e. 𝜋>0.1

Note that for the lower-tail test, the rejection area for the in the lower tail is now simply 𝛼 (instead of 𝛼/2). Note that for the p-value, it only uses the one tail (i.e. the lower tail) probability.

Similarly for the upper-tail:

Now to use an example for finding the mean for an upper-tail (when 𝜎 is unknown)

Now for the p-value of this test:

Using excel for the sample example question:

8.6 Tests for the proportion - one sample When conducting hypothesis tests for proportions, we use the ZSTAT test statistic i.e.:

Which much satisfy the conditions: ● n𝜋≥ 5 AND ● n(1-𝜋)≥ 5 Example:

P-value:

Let’s look at a second example, this time for a one-tailed test:

8.7 Ethical considerations Potential pitfalls and ethical considerations related to hypothesis testing: ● Use randomly collected data (probability samples) to reduce selection biases and non-sampling error AND allow the sampling distribution theory to be used! ● Choose the level of significance, α, and the type of test (one-tail or two-tail) before data collection ● Do not employ “data snooping” to choose between one-tail and two-tail tests, or to determine α

● ●

Do not practice “data cleansing” to hide observations that do not support a stated hypothesis Report all pertinent findings including both statistical significance and practical importance

Week 9 9.0 Hypothesis testing - two-sample tests The different types of sample tests covered this module is as follows:

9.1 Population means, independent samples Populations means found through independent samples require the following conditions to be met in regards to the data sources: ● Different ● Unrelated ● Independent The goal is to test the hypothesis, or form a confidence interval, for the difference between two population means, 𝜇1 - 𝜇2 The point estimate is found by X̄1 - X̄2

There are three types of tests that can be conducted when performing two-sample tests:

This subsection covers the first blue box (where the 2 standard deviations are unknown, assumed equal).

Here are the assumptions made for that: ● Samples are randomly and independently drawn ● Populations are normally distributed, or sample size is large enough (e.g. both at least 30) for the CLT to apply ● Population variances are unknown but assumed equal

The formula for the confidence interval for μ1 - μ2 (where σ1 and σ2 unknown and assumed equal):

9.2 Separate Variance t tests This time we will take a look at when both standard deviations are unknown, and are not assumed equal i.e.:

As can be seen above, this follows 3 assumptions: ● Samples are randomly and independently drawn ● Populations are normally distributed or both sample sizes are at least 30 (or at least 15 symmetric distributions) ● Population variances are unknown and not assumed to be equal.

To find the test statistic:

To find the degree of freedom:

Let’s take a look at an example:

9.3 Population means - related samples We have looked at independent samples, but what if they are related? I.e: - Paired (repeated measurements) or matched samples - Examine the difference between paired values:

-

Eliminates variation among subjects

There are assumptions to this: ● Both populations are normally distributed or, ● There is a large enough sample of differences for the Central Limit Theorem to apply (n > 30).

To find the T-stat:

As can be seen, it follows closely to the original t-test statistic formula.

Example:

So what’s the difference between paired and matched samples?

Paired → Comparing the same group of people before and after some training Matched → Items or people are matched according to some characteristics

9.4 Difference in population proportions So far we have compared the population means of both independent and related samples.

Now we look at the difference between two population proportions, with either a hypothesis test or confidence interval. I.e.:

Where π1 = first population, and π2 is the second The point estimate for the difference is p1 - p2.

To compute the test statistic we assume the null hypothesis is true, so we assume π1 = π2 and pool the two sample estimates.

The pooled estimate for the overall proportion is:

where X1 and X2 are the number of items of interest in samples 1 and 2.

Example:

Hypothesis test:

Confidence interval:

Week 10 10.1 Covariance and correlation The covariance measures the strength of the linear relationship between two numerical values (X & Y).

The formula for this is:

Note: correlation does not imply causation.

To find the strength of the linear correlation between variables, we use the correlation coefficient:

Features of the correlation coefficient: ● The population correlation is denoted 𝜌 (‘rho’) ● The sample correlation is denoted r ● Both 𝜌 and r have the following features: - Unit free (e.g. not measured by dollars, kg, km/h etc.)

- Range between –1 and 1 - The closer to –1, the stronger the negative linear relationship - The closer to 1, the stronger the positive linear relationship - The closer to 0, the weaker the linear relationship

Excel functions: Correlation → CORREL() Sample covariance → COVARIANCE.S()

10.2 Simple linear regression Regression analysis is used to: ● Predict the value of a variable based on the value of other variables ● Explain the impact on one variable resulting from changes in other variables For the regression analysis there are both dependent and independent variables.

- Dependent: the variable we explain (Y) - Independent: The variables (Xi) used to predict or explain the dependent variable (Y) (also known as explanatory variables, features, factors, regressors, predictors)

Simple linear regression: ● Only one independent variable, X ● Linear relationship between X and Y ● Changes in Y assumed to be related to changes in X ● The direction X → Y is implicit in SLR ● Regression does not prove causation Simple linear regression model (population):

The least squares method:

(Good example demonstration for a practical application on this sub-module) In excel:

10.3 Measures of variation To assess how well our regression model explains the movements in our dependent variable we calculate three measures of variation: ● ● ●

SST - the total sum of squares SSR - the regression sum of squares SSE - the error sum of squares

Formula shown below:

We can use the following measures of variation to assess the strength of our model: ● ●

2

𝑟 - the coefficient of determination SER - the standard error of the regression estimate 2

A higher value for 𝑟 a lower SER is preferred.

2

Coefficient of determination, 𝑟 :

Standard error of the regression (SER) estimate:

Comparing standard errors of the estimate:

Week 11 11.1 Residual analysis There are 4 assumptions of regression (L.I.N.E): Linearity → assumes the relationship between X and Y is linear Independence of errors → error values are statistically independent Normality of error → error values are normally distributed for any given value of X Equal variance (homoscedasticity) → the distribution of the errors for different values of X has equal variance.

The residual for observation i is the difference between the i th observed and predicted values i.e.:

(there is an excel demonstration on canvas if required)

11.2 Tests of the linear relationship If there is a linear relationship between Y & X then: i) reject slope is zero H0: ◻1=0 ii) Reject correlation is zero H0: ρ=0 So, if we can show that: - The sample estimate b1 and r are far enough from 0 to be statistically unlikely if the true values were zero…

- Then a linear relationship can be concluded at a particular level of significance Inferences about the slope - CLT:

Inferences About the Slope – Standard Error:

Inferences About the Slope: t-Test:

Example:

11.3 Estimating means and individual values Goal is to form intervals around Y to express uncertainty about the value of Y for a given XI.

Confidence interval (or prediction interval) estimate for an individual value of Y given a particular Xi is:

Example for an estimation of mean values:

Where working out for ‘Predicted price Yi :

Now an example for individual values:

11.4 Autocorrelation Autocorrelation is when there is a relationship between successive errors in a data set. This violates the independence of errors assumption. This subtopic shows you how to test for positive or negative autocorrelation using the Durbin-Watson statistic.

Autocorrelation refers to the correlation of the errors (residuals) over time.

When measuring autocorrelation, we use the Durbin-Watson statistic: ● Used when data are collected over time to detect if autocorrelation is present. ● Autocorrelation exists if residuals in one time period are related to residuals in another ● The existence of autocorrelation violates the independence assumption

Durbin-Watson table:

11.5 Pitfalls of regression Potential pitfalls: ● Lack of awareness of the assumptions underlying least-squares regression

● ● ● ● ●

Not knowing how to evaluate the assumptions of least-squares regression Extrapolating outside the relevant range of all common errors Not knowing what alternatives exist to least-squares regression, if an assumption is violated Thinking that every relationship should be linear Overlooking logical casuality (when a person gets too involved in the mechanics of performing a regression)

To avoid these pitfalls:

Week 12 12.1 Introduction to multiple linear regression Multiple linear regression considers more than one independent variable.

Let’s look as an example:

The coefficient of Multiple Determination (looking at multiple variables): This reports the proportion of total variation in Y explained by all X variables taken together i.e.

12.2 Adjusted r-squared and the F-test for overall significance 2

Since 𝑟 never decreases when a new X variable is added to the model, this can be unhelpful when comparing two or more models. When a new X variable is added, we lose a degree of freedom. We want to know if the new X variable added enough explanatory power 2

to offset the loss of one degree of freedom. This is where the adjusted 𝑟 comes in, letting us calculate the “net” effect of adding a new variable.

2

Adjusted adjusted 𝑟 in excel:

Now to look at tests for overall significance (or the F-test):

F-test formula:

Note that the F-value above is found through the F-stat table. It can also be found using the excel function F.INV().

12.3 Residual analysis to test assumptions and t-tests for coefficient significance Remember that the residuals highlight the errors in the regression model.

The residual formula is:

By performing this residual analysis, it allows us to test to see if it supports the underlying assumption (remember LINE).

There are many residual plots that can be made for multiple regression including:

If there are patterns in the plots, there probably is a problem.

12.4 Including categorical data using dummy variables Dummy variables: ● Can be used to model independent categorical variables ● Has two values (levels): ○ E.g. yes or no, on or off, male or female ○ Can be coded as as 1 or 0 (binary) ● If the categorical variable has more than two possible values then the number of dummy variables needed is the number of values -1 Time to look at an example:

i.e.

Now to look at an example with more than 2 levels: