1. STATISTICS AND PROBABILITY.pdf
- 1. STATISTICS AND PROBABILITY ALEMAR B. MAPINOGOS
- 2. • Statistics is the science of conducting studies to collect, organize, summarize, analyze, and draw conclusions from data. ✓Historical Note: A Scottish landowner and president of the Board of Agriculture, Sir John Sinclair introduced the word statistics into the English language in the 1798 publication of his book on a statistical account of Scotland. The word statistics is derived from the Latin word status, which is loosely defined as a statesman.
- 3. • A variable is a characteristic or attribute that can assume different values. • Data are the values (measurements or observations) that the variables can assume. • Variables whose values are determined by chance are called random variables. • A collection of data values forms a data set. Each value in the data set is called a data value or a datum.
- 4. • 2 Areas/Fields of Statistics 1. Descriptive statistics 2. Inferential statistics • Descriptive statistics consists of the collection, organization, summarization, and presentation of data. ✓In descriptive statistics the statistician tries to describe a situation. Consider the national census conducted by the U.S. government every 10 years. Results of this census give you the average age, income, and other characteristics of the U.S. population. To obtain this information, the Census Bureau must have some means to collect relevant data. Once data are collected, the bureau must organize and summarize them. Finally, the bureau needs a means of presenting the data in some meaningful form, such as charts, graphs, or tables.
- 5. • Inferential statistics consists of generalizing from samples to populations, performing estimations and hypothesis tests, determining relationships among variables, and making predictions. ✓making inferences from samples to populations ✓uses probability ✓hypothesis testing ✓relationships among variables ✓Making predictions
- 6. • Inferential statistics ✓ Here, the statistician tries to make inferences from samples to populations. Inferential statistics uses probability, i.e., the chance of an event occurring. You may be familiar with the concepts of probability through various forms of gambling. If you play cards, dice, bingo, or lotteries, you win or lose according to the laws of probability.
- 7. TRY THIS OUT! Determine whether descriptive or inferential statistics were used. a. The average jackpot for the top five lottery winners was $367.6 million. b. A study done by the American Academy of Neurology suggests that older people who had a high caloric diet more than doubled their risk of memory loss. c. Based on a survey of 9317 consumers done by the National Retail Federation, the average amount that consumers spent on Valentine’s Day in 2011 was $116. d. Scientists at the University of Oxford in England found that a good laugh significantly raises a person’s pain level tolerance.
- 8. • A population consists of all subjects (human or otherwise) that are being studied. • A sample is a group of subjects selected from a population. • An area of inferential statistics called hypothesis testing is a decision-making process for evaluating claims about a population, based on information obtained from samples.
- 9. Variables and Types of Data • Qualitative variables • Quantitative variables
- 10. Variables and Types of Data • Qualitative variables are variables that can be placed into distinct categories, according to some characteristic or attribute. ✓For example, if subjects are classified according to gender (male or female), then the variable gender is qualitative. Other examples of qualitative variables are religious preference and geographic locations.
- 11. Variables and Types of Data • Quantitative variables are numerical and can be ordered or ranked. ✓For example, the variable age is numerical, and people can be ranked in order according to the value of their ages. Other examples of quantitative variables are heights, weights, and body temperatures.
- 12. Quantitative variables can be further classified into two groups: • Discrete variables • Continuous variables
- 13. • Discrete variables assume values that can be counted. Example: 1. number of children in a family 2. number of students in a classroom 3. number of calls received by a switchboard operator each day for a month.
- 14. • Continuous variables can assume an infinite number of values between any two specific values. They are obtained by measuring. They often include fractions and decimals. ✓Temperature, for example, is a continuous variable, since the variable can assume an infinite number of values between any two given temperatures.
- 15. TRY THIS OUT! Classify each variable as a discrete variable or a continuous variable. a. The highest wind speed of a hurricane b. The weight of baggage on an airplane c. The number of pages in a statistics book d. The amount of money a person spends per year for online purchases
- 17. • Continuous variables can assume an infinite number of values between any two specific values. They are obtained by measuring. They often include fractions and decimals. ✓Temperature, for example, is a continuous variable, since the variable can assume an infinite number of values between any two given temperatures.
- 18. • Probability- is the likelihood of occurrence of an event. • Experiment- is an activity that is under consideration and which can be done repeatedly. Examples: 1. Drawing a card from an ordinary deck of 52 cards. 2. Tossing a coin 3. Rolling a die • Sample Space- is the set of all possible outcomes of an experiment. Examples: 1. In tossing a coin, the possible outcomes are head or tail. Hence the sample space is 𝑺 = {𝑯, 𝑻}. 2. In rolling a die, the sample space is 𝑺 = {𝟏, 𝟐, 𝟑, 𝟒, 𝟓, 𝟔}.
- 19. • Sample point- is the element of the sample space. Thus, in tossing a coin, there are two sample points: head and tail. • Event- is any subset of the sample space. A simple event is one that consists of exactly one outcome, hence it cannot be decomposed. On the other hand, an even is compound it consists of more than one outcome. • The complement of an event A with respect to the sample space S is the set of all elements of S that are not in A (denoted by A^'). • Random Variable- is a variable whose possible values are determined by chance.
- 20. RANDOM VARIABLE • It is typically represented by an uppercase letter, usually 𝑿, while its corresponding lowercase letter in this case, x, is used to represent one of its values. Example: A coin is tossed thrice. Let the variable X represent the number of heads that results from this experiment.
- 21. • Example: A coin is tossed thrice. Let the variable X represent the number of heads that results from this experiment.
- 22. • In the illustration, random variable is represented by the upper case 𝑿. The lower case 𝒙 represents the specific values. Hence, 𝒙 = 𝟑,𝒙 = 𝟐, 𝒙 = 𝟏, 𝒙 = 𝟐, 𝒙 = 𝟏, 𝒙 = 𝟏,and 𝒙 = 𝟎. • The sample space for the possible outcomes is 𝑺 = 𝑯𝑯𝑯, 𝑯𝑯𝑻, 𝑯𝑻𝑯, 𝑯𝑻𝑻, 𝑻𝑯𝑯, 𝑻𝑯𝑻, 𝑻𝑻𝑯, 𝑻𝑻𝑻 . The value of the variable 𝑿 can be 𝟎, 𝟏, 𝟐, 𝒐𝒓 𝟑. Then in this example, 𝑿 is a random variable.
- 23. • Random variables can either be DISCRETE or CONTINUOUS A discrete random variable can only take a finite (countable) number of distinct values. Distinct values mean values that are exact and can be represented by nonnegative whole numbers.
- 24. • Examples: • 𝑳𝒆𝒕 𝑿 = number of students randomly selected to be interviewed by a researcher. This is a discrete random variable because its possible values are 0, 1, or 2, and so on. • 𝑳𝒆𝒕 𝒀 = number of left-handed teachers randomly selected in a faculty room. This is a discrete random variable because its possible values are 0, 1, or 2, and so on. • 𝑳𝒆𝒕 𝒁 =number of defective light bulbs among randomly selected light bulbs. This is a discrete random variable because the number of defective light bulbs, which X can assume, are 0, 1, 2, and so on.
- 25. A continuous random variable can assume an infinite number of values in an interval between two specific values. This means they can assume values that can be represented not only by nonnegative whole numbers but also by fractions and decimals. These values are often results of measurements.
- 26. Examples: • 𝐿𝑒𝑡 𝑋 = the lengths of randomly selected shoes of senior students in centimeters. The lengths of shoes of the students can be between any two given lengths. The values can be obtained by using a measuring device, a ruler. Hence, the random variable 𝑋 is a continuous random variable. • 𝐿𝑒𝑡 𝑍 = the hourly temperatures last Sunday. • 𝐿𝑒𝑡 𝑌 = the heights of daisy plants in the backyard.