Statistical Analysis Urmia University Faculty of Economics and Management Department of Accounting & Finance Master Program in Accounting By Dr. Gholamreza Mansourfar

Evaluation Assignments 30% Project Final Exam 40%

Basic Concepts What is a variable? Continuous Category of Variables A variable is a characteristic that can take on different values for different members of the group under study. Continuous Category of Variables Discrete Independent Dependent Measurement The process of assigning numbers to characteristics according to a defined rule Not all measurement is the same The measurement scale of the dependent variable is one of the important factors in determining the appropriate statistical methods used to analyze the data of a particular research study.

Basic Concepts Nominal scale The process of classifying different objects into categories based upon some defined characteristics. Examples are sex, color of hair/eyes, ethnic background, makes of car and so on. Ordinal Scale Data categories have some logical order. Level of Measurement Scale Interval Scale Data categories have some logical order. Equal differences in the characteristic are represented by equal differences in the numbers assigned to the categories. The point zero is just another point on the scale Ratio Scale Data categories are mutually exclusive. Data categories have some logical order. Equal differences in the characteristic are represented by equal differences in the numbers assigned to the categories. The point zero reflects an absence of the characteristic.

Basic Concepts Research objectives/questions/purpose of your study. What Factors determine the most appropriate statistical techniques? Research objectives/questions/purpose of your study. Measurement scales you use in your research. Research design of your studies. Nature of your data – meeting normality and/or equality of variance assumptions for parametric tests.

Basic Concepts Table for selecting Descriptive Measures Based on Scales of Measurement CLASSIFICATION GRAPHICAL MEASURES MEASURES OF CENTRAL TENDENCY MEASURES OF DISPERSION NOMINAL Bar graphs Mode Binomial or Multinomial variance ORDINAL Histogram or Bar Median Range P25 – P75 INTERVAL Histogram with normal curve Mean Standard Deviation (s) RATIO Geometric mean Harmonic mean Variance

Basic Concepts

Basic Concepts Normal Distribution 1- The most widely known and used of all distributions is Normal Distribution. 2- Many variables in business and industries are normally distributed. 3- When large enough sample sizes are taken, many statistics are normally distributed regardless of the shape of the underlying distribution from which they are drawn. Probability Density Function of the Normal Distribution

Basic Concepts Standard Normal Distribution (z distribution) The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. Normal distributions can be transformed to standard normal distributions by the formula: where X is a score from the original normal distribution, μ is the mean of the original normal distribution, and σ is the standard deviation of original normal distribution. A z score always reflects the number of standard deviations above or below the mean a particular score is. For instance, if a person scored a 70 on a test with a mean of 50 and a standard deviation of 10, then converting the test scores to z scores, an X of 70 would be: So, a z score of 2 means the original score was 2 standard deviations above the mean. Note that the z distribution will only be a normal distribution if the original distribution (X) is normal.

Examples: What is the probability of obtaining a score greater than 700 on a GMAT test that has a mean of 494 and a standard deviation of 100? Assume GMAT scores are normally distributed. For the same GMAT examination, what is the probability of randomly drawing a score that is 550 or less? What is the probability of randomly obtained a score between 300 and 600 on the GMAT exam?

Examples: What is the probability of getting score between 350 and 450 on the same GMAT exam? A company produces lightbulbs whose lifetimes follow a normal distribution with mean 1.200 hours and standard deviation 250 hours. If a lightbulb is chosen randomly from the company's output, what is the probability that its lifetime will be between 900 and 1.300 hours'? An investment portfolio contains stocks of a large number of corporations. Over the last year the rates of return on these corporate stocks followed a normal distribution, with mean 12.2%, and standard deviation 7.2%. (a) For what proportion of these corporations was the rate of return higher than 20%? (b) For what proportion of these corporations was the rate of return negative? (c) For what proportion of these corporations was the rate of return between 5% and 15%? I am considering two alternative investments. In both cases, I am unsure about the percentage return but believe that my uncertainty can be represented by normal distributions with the means and standard deviations shown in the accompanying table. I want to make the investment that is more likely to produce a return of at least 10%. Which should I choose?

Basic Concepts Sampling Reasons for Sampling Random Versus Nonrandom Sampling Techniques of Random Sampling Techniques of Nonrandom Sampling Distribution of Sample Means: In the inferential statistics process, a researcher selects a random sample from the population, computes a statistic on the sample, and reaches conclusion about the population parameter from the statistics. In attempting to analyze the sample statistics it is essential to know the distribution of the statistic. In this section we the sample mean as the statistic. Suppose we have a population: 6 7 8 9 Let’s take all possible samples of size n = 2 from this population:

Important properties of the sampling distribution of means: The mean of the distribution of sample means is the mean of the population: 2- Standard Deviation: 3- Shape:

Non-normal Distribution of X (Population) sample size is 2 sample size is 3 sample size is 4 sample size is 8 sample size is 16 sample size is 32

Non-normal Distribution of X (Population) sample size is 2 sample size is 3 sample size is 4 sample size is 8 Central Limit Theorem: Distribution of sample means will approach a normal distribution as the sample size increases, though the distribution of population is not normal. sample size is 16 sample size is 32

Basic Concepts Z formula for Sample Means Suppose that annual percentage salary increases for the chief executive officers of all mid-size corporations are normally distributed with mean 12.2% and standard deviation 3.6%. A random sample of nine observations from this population of percentage salary increases is taken. What is the probability that the sample mean will be less than 10%? If sample size is less than 5% of the finite population size or n<5%N Finite correction factor

Basic Concepts Distribution of Sample Proportion: If be the proportion of successes in a random sample of n observations from a population in which the proportion of successes is p. Then 1- The sampling distribution of has mean p; that is: 2- The sampling distribution of has standard deviation, that is: 3- Z formula for sample proportions for n.p>5 would be: