Chapter Notes for Exam 1
Chapter Notes for Exam 1 Stat 1034
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Getting Started Ch1 Notes Material Extracted From Textbook (Brase, Charles Henry., and Corrinne Pellillo. Brase. Understandable Statistics: Concepts and Methods . 11th ed. N.p.: Cengage Learning, n.d. Print.) 1.1 Statistics: The study of how to collect, organize, analyze, and interpret numerical information from data. It is both the science of uncertainty and the technology of extracting information from data. ● Statistics helps people make decisions ● Statistics is a way to examine information ● Make inferences about populations by looking at samples ● The accuracy of a properly applied statistical procedure depends on the accuracy of the data (4). Data Individuals: are the people or objects included in the study. Variable: a characteristic of the individual to be measured or observed. Quantitative Variable: has a value or numerical measurement for which operations such as addition or averaging makes sense. Qualitative Variable: (Also called categorical variations) describes an individual by placing the individual into a category or group, such as male or female. “For instance, if we want to do a study about the people who had climbed Mt. Everest, then the individuals in the study are all people who have actually made it to the summit. One variable might be the height of the individuals (5). Other Variable Examples: Quantitative weight, age, income Qualitative gender, nationality Sources of Data Population Data: the data from every individual of interest. Sample Data: the data are from only some of the individuals of interest. Population Parameter: a numerical measure that describes an aspect of a population. Sample Statistic: is a numerical measure that describes an aspect of a sample. Data from a specific population are fixed and complete. Data from a sample has a higher chance of varying between samples. “For instance, if we have data from all other individuals who have climbed Mt. Everest, then we have population data. The proportion of males in the population of all climbers who have conquered Mt. Everest is an example of a parameter. On the other hand, if our data come from just some of the climbers, we have sample data. The proportion of male climbers in the sample is an example of a statistic… one of the important features of sample statistics is that they can vary from sample to sample, whereas population parameters are fixed for a given population” (5). ● Gathering sample data is often more realistic than population data. 4 Levels of Measurement: Nominal, Ordinal, Interval, Ratio ● Important because it helps indicate the type of arithmetic that is appropriate for data. ● Help indicate how to order data. Nominal Level: Applies to data that consists of names, labels, or categories. There are no implied criteria by which the data can be ordered from smallest to largest. Ordinal Level: Applies to data that can be arranged in order. However, differences between data values either cannot be determined or are meaningless. Interval Level: Applies to data that can be arranged in order in addition, differences between data values are meaningful. Ratio Level: Applies to data that can be arranged in order. In addition, both differences between data values are meaningful. Data at the ratio level have a true zero. True Zero/ Meaningful Zero: When zero means the absence of what it is trying to define. O degrees is not true zero because it still defines a temperature. 1.2 Random Samples ● Be wary of making hasty generalizations about a population about sample. Simple Random Sample: A simple random sample of n measurements from a population is a subset of the population selected in such a manner that every sample of size n from the population has an equal chance of being selected (13). ● Each member of a population has an equal chance of being selected. For a Simple Random Sample, every sample of the given size must also have an equal chance of being selected (13). Important Features of a Simple Random Sample 1. Every sample of specified size n from the population has an equal chance of being selected. 2. No researcher bias occurs in the items selected for the sample. 3. A random sample may not always reflect the diversity of the population of 10 cats and 10 dogs, a random sample of size 6 could consists of all cats. Easy Way to Get a Simple Random Sample ● Random number table ● Computer generator ○ Random selection does not mean haphazard selection. Procedure How to Draw a Simple Random Sample 1. Number all members of the population sequentially. 2. Use a table, calculator, or computer to select random numbers from the numbers assigned to the population members. 3. Create the sample by using population members with numbers corresponding to those randomly selected. Simulation: (Also called “Dry Lab Approach”) A numerical facsimile or representation of a realworld phenomenon also uses random number tables. Sampling Techniques Random Sampling: use a simple random sample from an entire population. Stratified Sampling: divide the entire population into distinct subgroups called strata. The strata are based on specific characteristics such as age, income, education level and so on. All members of a stratum share the specific characteristic. Draw random samples from each stratum (16). Strata: groups or classes inside a population that share a common characteristic. Population must be divided into at least 2 different strat. example men and women of population. Systematic Sampling : Number all members of a population sequentially. Then from a starting point select at random, include every kth member of the population in the sample (16). A group of people were standing in a line at a concert. You decide to select every 5th individual to include in a sample. Usually easy to acquire. Systematic sampling should not be used for populations that are repetitive or cyclical in nature (16, 17). Cluster Sampling: Divide the population into preexisting segments or clusters. The clusters are often geographic. Make a random selection of clusters. Include every member of each selected cluster in the sample (16). Conducting a survey of school children in a large city, we could first randomly select five schools and then include all the children from each selected school. Multistage Sampling: Use a variety of sampling methods to create successively smaller groups at each stage. The final sample consists of clusters. The government Current Population Survey interviews about 60,000 households across the United States each month by means of multistage sample design. For the Current Population Survey, the first stage consists of selecting samples of large geographic areas that do not cross state lines. These areas are further broken down into smaller blocks, which are stratified according to ethnic and other factors. Stratified Samples of blocks are taken. Finally, housing united in each chosen block are broken into clusters of nearby housing units. A random sample of these clusters of housing units is selected, and each household in the final cluster is interviewed. Convenience Sampling: Create a sample by using data from population members that are readily available. ● Does Run risk of being severely bias. Sampling Frame: the list of individuals from which a sample is actually selected. Undercoverage: When the sample frame does not match the population. ● Results from omitting population members from the sample frame. Sampling Error: The difference between measurements from a sample and corresponding measurements from the respective population. It is caused by the fact that the sample does not perfectly represent the population. Nonsampling Error: The result of poor sample design, sloppy data collections, faulty measuring instruments, bias in questionnaires, and so on. ● Sampling errors do not represent mistakes. They are simply the consequences of using samples instead of populations. However, be alert to nonsampling errors, which may sometimes occur inadvertently (18). 1.3 Introduction to Experimental Design Planning a Statistical Study Procedure Basic Guidelines for Planning a Statistical Study 1. Identify individuals or objects of interest. 2. Specify the variables as well as the protocols for taking measurements or making observation. 3. Determine if you will use an entire population or representative sample. If using a sample, decide on a viable sampling method. 4. In your data collection plan, address issues of ethics, subject confidentiality and privacy. If you are collecting data a business, store, college, or other institution, be sure to be courteous and to obtain permission as necessary. 5. Collect the data. 6. Use appropriate descriptive statistics methods and make decisions using appropriate inferential statistical methods. 7. Finally, note any concerns you might have about your data collection methods and list any recommendations for future studies. Census: Measurements or observations from the entire population are used. Sample: Measurements or observations from part of the population are used. Experiments and Observations Observational Study: Observations and measurements of individuals are conducted in a way that doesn't change the response or the variable being measured. Experiment: A treatment is deliberately imposed on the individuals in order to observe a possible change in the response or variable being measured. Placebo Effect: Occurs when a subject receives no treatment but (incorrectly) believes he or she is in fact receiving treatment and responds favorably. Completely Randomized Experiment ● One in which a random process is used to assign each individual to one of the treatments. ● Blocking is used to help control variables. Block: A group of individuals sharing some common features that might affect the treatment. Randomized Block Experiment: Individuals are first sorted into block, and then a random process is used to assign each individual in the block to one of the treatments. Control Group: This group receives a dummy treatment, enabling the researchers to control for the placebo effect. In general, a control group is used to account for the influence of other known or unknown variables that might be an underlying cause of a change in response in the experimental group. Replication of the experiment on many patients reduces the possibility that the differences in pain relief for the two groups occurred by chance alone. Double Blind Experiment: Neither the individuals in the study nor the observers know which subjects are receiving the treatment ● Control Bias Likert Scale: “Strongly Disagree, Disagree, Neutral, Agree, Strongly Agree” type of survey. Potential Survey Pitfalls: Nonresponse: Individuals either cannot be contacted or refuse to participate. Nonresponse can result in significant undercoverage of a population. Truthfulness of Response: Respondents may lie intentionally or inadvertently. Faulty Recall: Respondents may not accurately remember when or whether an event took place. Hidden Bias: The question may be worded in such a way as to elicit a specific response. The order of questions might lead to biased responses. Also, the number of responses on a Likert Scale may for responses that do not reflect the respondent’s feelings or experience. Vague Wording: Words such as “often,” “seldom,” “and “occasionally” mean different things to different people. Interviewer Influence: Factors such as tone of voice, body language, dress, gender, authority, and ethnicity of the interviewer might influence responses. Voluntary Response: Individuals with strong feelings about a subject are more likely than others to respond. Such a study is interesting but not reflective of the population. Lurking Variables: One for which no data have been collected but that nevertheless has influence on other variables in the study. Confounding Variables: Two variables are confounded when the effects of one cannot be distinguished from the effects of the other. Confounding Variables may be part of the study, or they may be outside lurking variables. Material Extracted From Textbook (Brase, Charles Henry., and Corrinne Pellillo. Brase. Understandable Statistics: Concepts and Methods . 11th ed. N.p.: Cengage Learning, n.d. Print.) 2.1 Frequency Distributions, Histograms, and Related Topics Frequency Table: Partitions data into classes or intervals of equal width and shows how many data values are in each class. The classes or intervals are constructed so that each data value falls into exactly one class. Steps 1. Decide how many classes you want. ● Usually 5 15 classes are used. ● Fewer than 5 classes risks losing too much information. ● More than 15 will insufficiently summarize data. 2. Find class width How to Find Class Width a) Compute: b) Increase the computed value to the next highest whole number. ● Even if the compute result was a whole number like 4, you would increase to 5. 3. Determine the data range for each range. Lowest Class Limit: The lowest data value that can fit in a class. Upper Class Limit: The highest data value that can fit in a class. Class Width: The difference between the lower class limit of one class and the lower class limit of each class. 4. Tally the data into different classes and find the frequency for each class. Tallying Data: Method of counting data values that call into a particular class or category. The Class Frequency for a Class: Is the number of tally marks corresponding to that class. 5. Midpoint (Also referred as Class Mark): The center of each class. the midpoint is often used as a representative value of the entire class. The midpoint is found by adding the lower and upper class limits of one class and dividing by two. Class Boundaries: There is a space between the upper limit of one class and the lower limit of the next class. Ths class halfway points of these intervals are the class boundaries. 6. Identify Class Boundaries. How to Find Class Boundaries (Integer Data) Upper Class Boundaries: Add 0.5 unit to the upper class limits to find upper class boundaries. Lower Class Boundaries: Subtract 0.5 units from the lower class limits. Relative Frequency (of a Class): Is the proportion of all data values that fall into that class. How to Find Relative Frequency of a Particular Class: 1. Make a Frequency Table. 2. For each class, compute the relative frequency, f/n, where f is the class frequency and n is the total sample size. Histograms and Relative Frequency Histograms ● Histograms and relative frequency histograms provide effective visual displays of data organized into frequency tables. How to Make a Histogram or Relative Frequency Histogram 1. Make a frequency table (including relative frequencies) with the designated number of classes. 2. Place class boundaries on the horizontal axis and frequencies or relative frequencies on the vertical axis. 3. For each class of the frequency table, draw a bar whose width extends between corresponding class boundaries. For histograms , the height of each bar is the corresponding class frequency. For r elative frequency histograms , the height of each bar is the corresponding class relative frequency. “The use of class boundaries in histograms assures us that the bars of the histogram touch and that no data fall on the boundaries… For this reason, many magazines and newspapers do not use class boundaries as labels on a histogram. Instead, some use lower class boundaries as labels on a histogram. Instead, some use lower class limits as labels, with the convention that a data value falling on the class limit is included in the next higher class (class to the to the right of the limit). Another convention is to label midpoints instead of class boundaries. Determine the default convention being used before creating frequency tables and histograms on a computer” (47). Distribution Shapes ● The distribution shape of a sample should look similar to a distribution shape of a population. Mound Shaped Symmetrical: This term refers to a histogram in which both sides are (more or less) the same when the graph is folded vertically down the middle. Uniform or Rectangle: These terms refer to a histogram in which every class has equal frequency. From one point of view a uniform distribution is symmetrical with the added property that the bars are of the same height. Skewed Left or Skewed Right: These terms refer to a histogram in which one tail is stretched out longer than the other. The direction of skewness is on one side of the longer tail. So, If the longer tails on on the left, we say the histogram is skewed to the left. Bimodal: This refers to a histogram in which the two classes with the largest frequencies are separated by at least one class. The top two frequencies of these classes may have slightly different values. This type of situation sometimes indicates that we are sampling from two different populations. ● A bimodal distribution might indicate that the data are from two different populations. ● If there are gaps in the histogram between bars at either end of the graph, the data set might include outliers. Outliers: Data values that vary from other measurements in the data set. ● Can indicate possible data recording errors. ● If the outlier is unusual, one can either not include it in the data or take a second glance at the quality of data, What Histograms and Relative Frequency Tell Us? Why They are Important Histograms and relative frequency histograms show us how the data are distributed. ● Tell how data is distributed which also helps us understand qualities about the data. ● If there are possible outliers. ● Where data intervals have the highest concentration of data. ● How spread out the data are. CumulativeFrequency Tables and Ogives ● Cumulative frequencies indicate how many data values are smaller than an upper boundary. Cumulative Frequency: For a class is the sum of the frequencies for that class and all previous classes. Ogive: Graph that displays cumulative frequencies. ● Also referred to as a cumulative frequency diagram. How to Make and Ogive 1. Make a frequency table showing class boundaries and cumulative frequencies. 2. For each class, make a dot over the upper class boundary at the height of the cumulative class frequency. Connect these dots with line segments. 3. By convention, an ogive begins on the horizontal axis at the lower class boundary of the first class. What does an ogive tell us? ● How many data are less than the indicated value on the horizontal axis. ● How slowly or rapidly the data values accumulate over the range of data. The vertical scale can be changed to cumulative frequencies by the total number of data. Then we can tell what percentages of data are below values of data are below values specified on the horizontal axis. 2.2 Graphs, Circle Graphs, and TimeSeries Graphs Features of a Bar Graph: 1. Bars can be vertical or horizontal. 2. Bars are of uniform width and uniformly spaced. 3. The lengths of the bars represent values of the variable being displayed, the frequency of occurrence, or the percentage of occurrence. The same measurement scale is used for the length of each bar. 4. The graph is well annotated with title, labels for each bar, and vertical scale or actual value for the length of each bar. Pareto Chart: A bar graph in which the bar height represents frequency of an event. In addition, the bars are arranged from left to right according to decreasing height. Circle Graph/ Pie Chart: Wedges of a circle visually display proportional parts of the total population that share a common characteristic. TimeSeries Graph: Data are plotted in order of occurrence at regular intervals over a period of time. TimeSeries Data: Consist of measurements of the same variable for the same subject at regular intervals over a period of time. How to Decide Which Type of Graph to Use: Bar Graphs: are useful for quantitative or qualitative data. With Qualitative data, the frequency or percentage of occurrence can be displayed. With quantitative data, the measurement itself can be displayed, as was done in the bar graph showing life expectancy. Watch that the measurement scale consistent or that a jump scale squiggle is used. Pareto Charts: Identify the frequency of events or categories in decreasing order of frequency of occurrence. Circle Graphs: Display how a total is dispersed into several categories. The circle graph is very appropriate for qualitative data, or any data for which percentage of occurrence makes sense. Circle graphs are most effective when the number of categories or wedges is 10 or fewer. TimeSeries Graphs: Display how data change over time. It is best if the units of time are consistent in given graphs. For instance, measurements taken every day should not be mixed on the same graph with data with data taken every week. For Any Graph: Provide a title, labeled the axis, and identify units of measure. As Edward Tufte suggests in his book The Visual Display of Quantitative Information, don’t let artwork or skewed perspective cloud the clarity of the information displayed. What Do Graphs Tell Us? Appropriate graphs provide a visual summary of data that tells us: ● How data are distributed over several categories or data intervals. ● How data from two or more data sets compare. ● How data change over time. 2.3 Stem and Leaf Displays Exploratory Data Analysis Helps Us ● Useful at detecting patterns. ● Asking questions about a data set. StemandLeaf Display: Is a method of exploratory data analysis that is used to rankorder and arrange data into groups. ● Helpful to see patterns but it's also possible to recover the original data. How To Make a Stem and Leaf Display: 1. Divide the digits of each data value into two parts. The leftmost part is called the stem and the rightmost part is called the leaf. 2. Align all the sems in a vertical column from smallest to largest. Draw a vertical line to the right of all the stems. 3. Place all the leaves with the same stem in the same row as the stem, and arrange the leaves in increasing order. 4. Use a label to indicate the magnitude of the numbers in the display. We include the decimal position in the label rather than with the stems or leaves. What StemandLeaf Displays Tell Us? StemandLeaf displays give a visual display that ● Shows us all the data (or truncated data) in order from smallest to largest. ● Helps us spot extreme data values or clusters of data values. ● Displays the shape of the data distributions. ● Easy to spot outliers. Chapter 3 Averages and Variation Material Extracted From Textbook (Brase, Charles Henry., and Corrinne Pellillo. Brase. Understandable Statistics: Concepts and Methods . 11th ed. N.p.: Cengage Learning, n.d. Print.) 3.1 Measures of Central Tendency: Mode, Median, and Mean Mode: The value that occurs most frequently. Note: if the data set has not single value that occurs more frequently than the other, then that data set has not more. If a data set has two values that occur at the same frequency it can be bimodal. Median: The central value of an ordered distribution. How to find the Median 1. Order the data from smallest to largest. 2. For an odd number of data values in the distribution. Median = Middle data 3. For an even number of data values in the distribution. Mean: An average that uses the exact value of each entry. How to find the mean What Do Averages Tell Us? ● The mode tells us the single data value that occurs most frequently in the data set. The value of the mode is completely determined by the data value that occurs most frequently. If not data value occurs more frequently than all the other data values, there is not mode. The specific values of the less frequently occurring data do not change the mode. ● The median tells us the middle value of data set that has been arranged in order from smallest to largest. The median is affected by only the relative position of the data values. For instance, if a data value about the median (or above the middle two values of a data set with an even number of data) is changed to another value above the median, the median itself does not change. ○ A disadvantage to the median is that it is not sensitive to the specific size of data. ● The mean tells us the value obtained by adding up all the data and dividing by the number of data. As such, the mean can change if just one data value changes. On the other hand, if the data values change, but the sum of the data remains the same, the mean will not change. Resistant Measure: One that is not influenced by extremely high or low data values. The mean is not a resistant measure of center because we can make the mean as large as we want by changing the size of the only one data value. The median is more resistant. Trimmed Mean: The mean of the data values left after “trimming” a specific percentage of the smallest and largest data values from the data set. ● More resistant than the mean but sensitive to specific data values. How to Compute a 5% Trimmed Mean 1. Order the data from smallest to largest. 2. Delete the bottom 5% of the data and the top 5% of the data. Note: If the calculations of 5% of the number of data values does not produce a whole number, round to the nearest integer. 3. Compute the mean of the remaining 90% of the data. ● Works for any other amount of percentage! Distributions and Averages ● When a data distribution is moundshaped symmetrical, the values of the mean, median, and mode are the if not almost all the same. ● For skewedleft distributions, the mean is less than the median and the median is less than the mode. ● For skewedright distributions, the mode is the smallest value, the median is the next largest, and the mean is the largest. Weighted Averages “Sometimes we wish to average numbers, but we want to assign more importance, or weight to some of the numbers. For instance, suppose your professor tells you that your grades will be based on a midterm and a final exam, each of which is based on 100 possible points. However, the final exam will be worth 60% of the grade and the midterms only 40%. How could you determine an average score that would reflect these different weights? The average you need is the weighted average” (96). 3.2 Measures of Variation “An average is an example to summarize a set of data using just one number. As some of our examples have shown, an average taken by itself may not always be very meaningful. We need a statistical crossreference that measures the spread of the data” (102). Range: Is the difference between the largest and smallest values of a data distribution. Variance and Standard Deviation “We need a measure of the distribution or spread of data around an expected value (insert symbols). Variance and standard deviation provide such measures. ● Sample standard deviation and sample variance are used to describe the spread of data about the mean ● Standard deviation and sample variance can be used for population (just make sure to use the proper symbols! Post both photos ● If the mean is rounded, the values of the standard deviation will change. What Do Measure of Variation Tell Us? Measures of Variation give information about the spread of the data. ● The range tells us thdifference between the highest data value and the lowest . It tells us about the spread of data butdoes not tell us if most of the data is or is not closer to the mean. ● The sample standard deviation is based on the difference between each data value and the ,mean of the data set. The magnitude of each data value enters into the calculation. The formula tells us to compute the difference between each data value and the mean, square each difference, add up all the squares, divide by n1, and then take one square root of the result. The standard deviation deviation gives an average of data spread out are around the mean. A smaller standard deviation indicates that the data tend to be closer to the mean. ● The variance tells us thsquare of standard deviation . As such, it is alsomeasure of data spread around the mean. Population Parameters ● The formula for the population mean is the same as the formula for the sample mean just different samples. Coefficient of Variation Coefficient of Variation: Expresses the standard deviation as a percentage of the sample or population mean. ● The numerator and denominator have the same units. ○ This helps to compare the variability of two different populations using the coefficient of variation. “The coefficient of variation can be thought of as a measure of the spread of the data relative to the average of the data” (110). Chebyshev’s Theorem ● When dealing with a symmetrical, bellshaped distribution, then one can make definite conclusions about the proportion of the data that must lie within a certain number of standard deviations on either side of the mean. ● However, this theorem can generally help identify the the data spread about the mean for all distributions (skewed, symmetric, etc). ● Chebyshev’s Theorem refers to the minimal percentage of data that must fall within the specified number of standard deviations of the mean (111). W hat does Chebyshev’s Theorem Tell Us? ● The minimum percentage of data that falls between the mean and any specified number of standard deviations on either side of the mean. ● A minimum of 88.9% of the data falls between the values 3 standard deviations below the mean and 3 standard deviations above the mean. This implies that a maximum of 11.1% of data fall beyond 3 standard deviations of the mean. Such values might be suspect outliers, particularly for a moundshaped symmetric distribution (111). ● Tells us that no matter what the data distribution, 75% of the data lies within 2 standard deviations of the mean. Thoughts About Averages ● Averages do not tell much about the way data are distributed about the mean. ● The combination of an average (such as the mean) in addition to the variance and standard deviation helps to paint a more holistic understanding of a data set. Easier Computation with Grouped Data: 3.3 Percentiles and Boxand Whisker Plots Quartiles: Special percentiles used so frequently that we want to adopt a specific procedure for their computation. How to Compute Quartiles 1. Order the data from smallest to largest. 2. Find the median. This is the second quartile. 3. The first quartile Q1, is then the median of the lower half of the data; that is, it is the median of the data falling below the Q2 position (and not including Q2). 4. The third quartile Q3, is the median of the upper half of the data; that is, it is the median of the data falling above the Q2, position (and not including Q2). Interquartile Range (IQR): Q3Q1 = IQR ● A useful measure of fata spread utilizing relative position. ● Indicates the spread of the middle half of the data. Box and Whisker Plots 5NumberSummary Lowest value, Q1, median (Q2), Q3, highest value. ● BoxandWhisker Plots provide another useful technique from exploratory data analysis for describing data. How to Make BoxandWhisker Plots 1. Draw a vertical scale to include the lowest and highest data values. 2. To the right of the scale, draw a box from Q1 to Q3. 3. Include a solid line through the box at the median level. 4. Draw vertical lines, called whiskers, from Q1, to the lowest value and from Q3, to the highest value. Why Boxand Whisker Plots are helpful ● They give a graphic picture of how data is spread about the median . ● The location of the middle half of the data to help you identify whether or not the distribution is skewed or symmetrical. ● Identifying o utlier. ● Indicates the values of the “5NumberSummary.” Chapter 4 Elementary Probability Theory Material Extracted From Textbook (Brase, Charles Henry., and Corrinne Pellillo. Brase. Understandable Statistics: Concepts and Methods . 11th ed. N.p.: Cengage Learning, n.d. Print.) 4.1 What is Probability? Probability is a numerical measure between 0 and 1 that describes the likelihood that an event will occur. Probabilities closer to 1 indicate that the event is more likely to occur. Probabilities closer to 0 indicate that the event is less likely to occur. Probability Assignments 1. A probability assignment based on intuition incorporates past experience, judgement, or opinion to estimate the likelihood of an event . 2. A probability assignment based on relative frequency uses the formula: Probability of event = relative frequency = f/n Where f is the frequency of the event occurrence in a sample of n observations. 3. A probability assignment based on equally likely outcomes uses the formula: Probability of event = # of outcomes favorable to event / total # of outcomes Law of Large Numbers: In the long run, as the sample size increases and increases, the relative frequencies of outcomes get closer and closer to the theoretical (or actual) probability value. ● The underlying assumption we make is that if events occurred a certain percentage of times in the past, they will occur about the same percentage of times in the future. Statistical Experiment/Statistical Observation: Can be thought of as any random activity that results in a definite outcome. Event: Is a collection of one or more outcomes of a statistical experiment or observation. Simple Event: Is one particular outcome of a statistical experiment. Sample Space: The set of all simple events. ● The sum of the probabilities of all simple events in a sample space must equal 1. Interpreting Probabilities: ● The closer the probability is to 1, the more likely the event is to occur. ○ Just because the event of a probability is high, it is not certainty that the event will occur. ● Similarly, if the likelihood of an event is low, it is possible that the event might occur. Events with low probability but big consequences are of special concern. ● Some of people’s biggest mistakes in a person’s life can result from either misjudging: a) the size of an event’s impact. b) the likelihood the event will occur. ● An event of great importance cannot be ignored even if it has a low probability of occurrence. What Does the Probability of an Event Tell Us? ● The probability of an event A tells us the likelihood that event A will occur. If the probability is 1, the event A is certain to occur. If the probability is 0, the event A will not occur. ● The probability of event A applies only in the context of conditions surrounding the sample space containing event A. ● If we know the probability of event A, then we can easily compute the probability of event not A in the context of the same sample space. P(notA)= 1P(A). Probability Related to Statistics ● If probability did not exist, then inferential statistics would not exist. ● Probability you know the overall description of the population. The central problem is to compute the likelihood of a specific outcome. ● Statistics you know only the result of a sample drawn from a statistic. 4.2 Some Probability Rules Compound Events Conditional Probability and Multiplication Rules: Independent Events: Two events are independent if the occurrence or nonoccurrence of one event does not change the probability that the other event will occur. Dependent Events: Two events are dependent if the occurrence or nonoccurrence of one event changes the probability that the other event will occur. Why Does the Independence or Dependence Matter? ● The type of the events determines the way we compute the probability of the two events happening together. Multiplication for Independent Events P(A and B) = P(A) x P(B) Multiplication for Dependent Events P(A and B) = P(A) x P(B|A) P(A and B) = P(B) x P(A|B) Conditional Probability: The notation P(A, given B) denotes the probability that event A will occur given that event B has occurred. Insert Conditional Probability Rule How to Use the Multiplication Rules 1. First determine whether A and B are independent events. If P(A) = P(A|B), then the events are independent. 2. If A and B are independent events: P(A and B) = P(A) x P(B). 3. If A and B are any events, P(A and B) = P(A) x P(B|A) or P(A and B) = P(B) x P(A|B). What does Conditional Probability Tell Us? Conditional probability of two events A and B tell us: ● The probability that event A will happen under the assumption that event B has happened (or is guaranteed to happen in the future). This probability is designated P(A|B) and is read “probability of A given event B.” Note that P(A|B) might be larger or smaller than P(A). ● The probability that event B will happen under the assumption that event A has happened. This probability is designated P(B|A). Note that P(A|B) and P(B|A) are not necessarily equal. ● If P(A|B) = P(A) or P(B|A) = P(B), then events A and B are independent. This means the occurrence of one of the events does not change the probability that the other event will occur. ● Conditional probabilities enter into the calculations that two events A and B will both happen together. P(A and B) = P(A) x P(B|A) also P(A and B) = P(A)x P(B) In the case that events A and B are independent, then the formulas for P(A and B) simplify to. P(A and B) = P(A) x P(B). ● If we know the values of P(A and B) and P(B), then we can calculate the value of P(A|B). Mutually Exclusive/ Disjoint: Two events are mutually exclusive or disjoint if they cannot occur together. In particular, events A and B are mutually exclusive if P(A and B) = 0. Addition Rule for Mutually Exclusive Events A and B P(A or B) = P(A) + P(B) General Addition Rule for any Events A and B (Not Mutually Exclusive) P(A or B) = P(A) + P(B) P(A and B) How to Use the Addition Rules 1. First determine whether A and B are mutually exclusive events. If P(A and B) = 0, then the events are mutually exclusive. 2. If A and B are mutually exclusive events, P(A or B) = P(A) + P(B). 3. If A and B are any events, P(A or B) = P(A) + P(B) P(A and B). What Does the Fact that Two Events are Mutually Exclusive Tell Us? If two events A and B are mutually exclusive, then we know the occurrence of one of the events means that the other event will not happen. In terms of calculations, this tells us: ● P(A and B) = 0 for mutually exclusive events. ● P(A or B) = P(A) =P(B) for mutually exclusive events. ● P(A|B) =0 and P(B|A) = 0 for mutually exclusive events. That is, if event B occurs, then event A will not occur, and vice versa. 4.3 Tree and Counting Techniques ● The probability formula requires that we be able to determine the number of outcomes in the sample space. ● When an outcome of an experiment is composed of a series of events, the multiplication rule gives us the total number of outcomes . Tree Diagram: A visual display of the total number of outcomes of an experiment consisting of a series of events. Helps determine the total number of outcomes and individual outcomes. Factorial Notation: Procedure: What Do Counting Rules Tell Us? Counting rules tell us the total number of outcomes created by combining a sequence of events in specified ways. ● The multiplication rule tells us the total number of possible outcomes for a sequence of events. Tree diagrams provide a visual display of all the resulting outcomes. ● The permutation rule tells us the total number of ways we can arrange in order n distinct objects into a group of size r. ● The combination rule tells us how many ways we can form n distinct objects into a group of size r. The order of the objects is irrelevant.
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