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Probability And Statistics
I. Overview
Probability theory.
A parameter.
A statistic.
A probability distribution.
- If the relative frequency of the values of a variable can be specified, the variable is a random variable.
- If the variable can assume only certain values in an interval, the variable is discrete.
When no gaps exist in the values, it may assume.
Objective probabilities.
Subjective probabilities.
Zero to one or 0 to 1.
That the event is certain to occur.
A probability of zero or 0.
- Summarize large amounts of data.
- Measures of central tendency and measures of dispersion are such summaries.
The median.
The cost of capital.
- Measures of central tendency are values typical of a set of data. (e.g., mean, median, and mode)
- Measures of dispersion indicate the variation within a set of numbers. (e.g., variance, standard deviation, interquartile range, and range)
- Subtracting the mean from each value.
- Squaring each difference.
- Adding the squared differences.
- Dividing the sum by the number of data items.
Standard deviation.
Inferential statistics.
It is important because measuring the entire population is usually inefficient.
Statistical sampling.
The central limit theorem.
- It is the most important and useful of all probability distributions.
- It has a symmetrical bell-shaped curve centered about the mean.
- 95.5% lies within two standard deviations.
A standard normal distribution.
To permit the use of standard normal distribution tables.
- Determine objectives.
- Define the population.
- Determine acceptable sampling risk (precision or confidence interval).
- Calculate the sample size using standard tables or formulas.
- Selecting the sampling approach.
- Take the sample.
- Evaluate the results.
The sum of the products of the probability of each outcome and its payoff.
Note: It is the long-term average pay-off for repeated trials.
II. Basic Concepts
- It is an extension of sensitivity analysis.
- It is used to estimate the likelihood of possible outcomes of a decision.
- It answers what-if questions about the effects of changes in the variables included in a decision model.
Conditional probability.
A joint probability is a probability that both or more events will occur.
Events are independent when one event has no effect on the probability of a second event, or the conditional probability of each event is its unconditional probability.
Mutually exclusive events.
Nonparametric statistic.
The mode, the median, and the mean.
The standard deviation.
The confidence level.
Based on the auditor’s judgment after determining the allowable audit risk.
Maximum likelihood.
- It is an interval around the sample statistic that is expected to contain the true value of the population.
- It is a variable in the sample size formula and must be estimated before calculating the sample size.
- The estimated precision interval is based upon a point estimate of the population value and the tolerable rate (for attribute sampling) or the tolerable misstatement (for variables sampling) determined by materiality consideration.
- The achieved (computed) precision interval is a function of population size and standard deviation, sample size, and specified confidence level.
- It is under the performer or auditor’s control.
- It is interdependent with reliability.
When the sample distribution is normal or, in other words, when the shape of the sampling distribution of the means approaches the normal distribution as the sample size increases.
It equals the standard deviation of the population (σ)divided by the square root of the sample size (σ÷√n).
Decision-making under risk.
Decision-making under uncertainty.
Subjective probabilities of occurrence for the multiple outcomes and then proceed as though under conditions of risk.
Because objective probability can not be determined.
- They neither seek nor avoid risk.
- Choose investments for which the expected monetary values equal their subjectively perceived utility value.
- The decision maker has a linear utility function.
- The monetary amounts and utils have a constant or directly proportional relationship.
- They are indifferent to risk.
- Avoid risk.
- They prefer a certain return on investments to the risk involved in other investments with potentially significant gains and large losses.
- The utility function for them increases at a decreasing rate.
- The utility of a gain is lower than the disutility of a loss of the same absolute amount.
- Risk chosen.
- They prefer investments that have the potential for large gains although large losses may be possible.
- The utility function for them increases at an increasing rate (risker investments are appealing).
To detect performance trends away from normal operations.
- It is a graphic aid for monitoring the status of any process subject to random variations.
- It consists of three horizontal lines plotted on a horizontal time scale.
- The vertical scale represents the appropriate quantitative measure.
- The center line represents the average or means value for the process being controlled.
- The other two lines are the upper control limit and the lower control limit.
- The processes are measured periodically, and the values are plotted on the chart.
III. Probability Distributions
Linear programing.
Gantt charting.
This is used to project a firm’s sales and profits.
It can be modeled using a formula or graph that provides the probability for every possible outcome.
Between 0 (no chance) to 1 (certainty).
Must be equal to 1.
Continuous distributions describe ranges in which any possible value has a probability of occurrence, but discrete distributions assign probabilities only to a finite number of values with a range.
Poisson distribution.
Each event has only two possible outcomes.
One in which sampling occurs without replacement.
An exponential distribution.
- t is used when small samples of less than 30 are examined.
- The underlying population is assumed to be normal.
- The population variance is not given.
- It is bell-shaped and symmetric like a normal distribution but flatters with more variation.
Chi-square distribution.
The Chi-square statistic equals the sample variance, multiplied by its degree of freedom, and divided by the hypothesized population variance.
The chi-square can be applied to nominal data.
- The curve is symmetrical.
- It is bell-shaped.
IV. Hypothesis Testing
Sample data are generated, and the hypothesis is formulated.
The hypothesis is rejected or not rejected based on the sample measurement.
V. Expected Value
- A tool is most likely to be used to determine the best course of action under conditions of uncertainty.
- It is a rational means of selecting the best option for a decision involving risk.
- It is equal to the sum of the product of the probability of an outcome and its probability.
- It is the long-term average payoff for repeat trials.
Program evaluation and review technique (PERT).
- It may be used to demonstrate correlations.
- Each observation is represented by a dot with X and a dot with Y coordinates.
- The linearity and slope of these observations are related to the coefficient of correlation.
- The sum of the probabilities of the events is equal to one.
- All of the events are mutually exclusive.
- All of the events are included in the decision.
- The branches emanate from a node from left to right.
Payoff table.
- The possible decision for each decision point.
- Events that might follow from each decision.
- Probabilities of these events.
- Quantified outcomes of the events.
Queuing theory.