Chapter 1, Statistics for Engineers and Scientists, Third Edition, Navidi Exam Questions with Answers.

DSST Principles of Statistics Exam Questions and Answers n What is Numeric data? - Correct Answers: data that can be counted or measured.. example: it answers questions like "How many fish can be counted in a net?", how many feet long is the room. Numeric data can either be discrete data or continuous data what is discrete data? - Correct Answers: A type of numeric data represents the units that can be counted. The range can be infinite or limited. For example-infinite : how many stars are in the universe example- limited: how many steps you will take in one mile. but because they are countable units. discrete can not be a fraction or deicmal what is Continuous data? - Correct Answers: A type of numeric data Can be absolutely any value in a range and can't be directly counted. ex: if you cut wedges of cheese off a 10lb wheel.. it can have fractioned or decimal weight like 2.2lbs.. anywhere from 0.01lbs-10lbs it can also be open ended, for example: how many years would it take to get the end of the universe? the answer is anything. When there is infinite number of possible values there is or there isn't a away mathmatically to assign probabilities? - Correct Answers: There isn't How do measure probability for continuous variables - Correct Answers: with the area underneath a density curve. The density curve must meet 2 conditions: 1.The curve has no negative values 2. The total area under the curve = 1. This is equal to the probability of all outcomes. What is catagorical data? - Correct Answers: Data that is not counted or measured as a number. for example, marital status, zip code, favorite food. Ordinal data - Correct Answers: catagorical data that has been organized into meaningful data for example: surveys -- like a survey asking you to rank how likely you are to vote 0-5. 0 meaning you won't vote. 5 meaning you will vote. If the average is 3 only some people are going to vote. Categories are placed in to first, second, third , dourth. What are the 4 basic levels of measurement and describe them? - Correct Answers: These levels describe how variables in a study are measured. nominal -categories that are not ranked or related (ex: types of cars: sports cars, vans, suvs. the cars can be categorized but there is no ranking system. a sports car is not better than an SUV ordinal- categories are placed into order (ex: first, second, third) The categories are placed into order of a ranking system. example the the finishers are ranked by who finished first.. interval- Places a set distance between categories of data. An example would be shirt size. There is the same size difference between a small and a medium, medium and large, large abd extra large. ** know that 0 in interval measurement is not really a 0 but a category named 0. a size 0 shirt is just a really small shirt. ratio- The level of measurement used in physical sciences and engineering and has a meaningful zero value. zero means sero as in nothing in ratio measurement. a value of 0 trees literally means there are 0 trees Define Statistics (Statistics Terms pg 1) - Correct Answers: Allows you to organize, evaluate, and interpret data. Data can be analyzed through calculation as well as graphs Define inductive statistics (Statistics Terms pg 1) - Correct Answers: enable you to make conclusions and decisions based on data Define descriptive statistics (Statistics Terms pg 1) - Correct Answers: Do NOT make conclusions or inferences based on data.. it is used to describe and analyze a group without making a decision. They provide basic summary information of the data sets, particularly, its distribution. descriptive statistics focuses on the center (median) and spread (standard deviation) in a data set. What is the mean? - Correct Answers: It was one the first descriptive statistics you will calculate is the mean of a data set. (symbolized with an X and a bar above it) AKA the **average add all the values in the data set together and then divide by the number of values to get the mean. The only problem with mean being a descriptive statistic is suseptible to outliers. outliers are values in a data set that fall well outside most of the distribution. a data set with outliers causes the mean to not be resistant What is a median? - Correct Answers: A descriptive statistic that is resistant to extreme values is the median. It is the number in a data set where half the observations fall below it and half fall above it. It is not calculated mathmatically rather It is the the middle number of a data set. It is represent as the letter M. If you have a data set with even numbers, the median is the middle two numbers averaged. ex: 4,5,6,9,11,15 6+9=15 15/2=7.5 the median would be 7.5 What is the mode? - Correct Answers: The mode is the value in a data set that occurs most often. If the data set has multiple modes, it becomes bimodel. ex: 5,3,3,4,1,3,2,5,8,5. the first mode would be 3,3,3 and the second mode would be 5,5,5 so the data set has 3 and 5 as bimodel. Define variables (Statistics Terms pg 1) - Correct Answers: a variable is a characteristic that desribes an individual. The key to a good experiment is minimizing the number of variables. Define quantitative variables (Statistics Terms pg 1) - Correct Answers: a variable that can be expressed in terms of numeric value. Define qualitative variables (Statistics Terms pg 1) - Correct Answers: is a variable that can not be expressed in numeric value. What is the difference betwen sample and population? (Sampling/Experimenting Methods pg 2-3) - Correct Answers: A population is a the group which a study or experiment is meant to determine information about. All the members of the group be studied.. like nationwide pool, or a number of bacteria in a lab. It is impractical, expensive, and timely to study an entire population. Thats why scientists and statisticians study a sample. A sample is a group within a population which is used to detemine information about the population. A sample is a small portion picked from a population. What is sample design? - Correct Answers: The steps and method taken to choose your sample from a population. The key to choosing a sampling method is making sure that the sample is truely repsentative of enitre population are looking at. What is an example of poor sample deisgn? - Correct Answers: Voluntary response sample. I this a population as a whole is surveryed but only those who choose so are sampled. This means your sample is made up entirely of those who chose and gave effort to respond. These typically people with strong opinions. ** if your sample does not effectively reprsent the population as whole, the sample is said to be biased. The way to avoid biased is to have the sample chosen by some inpersonal inpartial means. The best samples are probability samples. What is a probability samples? - Correct Answers: In a probability sample, each member of a population has a known chance (or known probability) larger than 0 to be chosen. The probability that any given individual will be selected does not have to be identical to the probability for any other individual. What are sample and experiment methods? (Sampling/Experimenting Methods pg 2-3) - Correct Answers: These two methods are 2 different ways of producing or collection data. A study consists of observing individuala and collecting data, but no influencing the individuals, or subjects in any way. Observational studies are done using *sample populations (SAMPLING) to draw conlusions. Sample takes a portion of the population being studied and gather information on it. An expetiment* consists of imposing a treatment on an individual or group and observing the response to it. What is a census? How does this differ from Samples? (Sampling/Experimenting Methods pg 2-3) - Correct Answers: a census gathers inofrmation about every individual in the sample. Sample takes a portion of the population being studied and gather information on it. What are the types of samples? (Sampling/Experimenting Methods pg 2-3) - Correct Answers: 1. Convenience Samples: consists of sampling the people who are easiest to reach 2. Voluntary Response Samples: Are samples on which the participants choose themselves. ^^ these two types of samples are typically very biased and do not yield accurate results. *** The more RANDOM a sample, the less biased the results will be. What are the types of probabiilty samples? - Correct Answers: Simple random sample (SRS): in a study population, every set of # individuals within the population has an equal probability of being part of the chosen sample. Stratified sample: population is first categorized into groups similar individuals called strata. ex: infants, juveniles, and adults. Multistage samples: narrows down a sample from a large, diverse population; increasingly small samples are chosen. The US census and other major polls use this type of sampling. Making the sample smaller for each stage. Ex: what are the most popular restaurant in the greek islands. There lots of greek islands and hundreds of restaurants. It is not possible to visit them all. Stage 1- choose a sample of islands. Stage 2- choose a sample of towns within your sample of islands. stage 3- choose a sample of islands within those towns. The sample at each stage may be a simple random sample or a stratified random sample. What are some problems to considered when designing your survey? - Correct Answers: - undercoverage: Some groups are left out of the sample design. Example: when trying to study frogs in the rainforest, it is easier to see and study bright colored frogs rather than green or camoflauged frogs. -nonresponse: An individual chosen as part of a sample does not participate -response bias: respondents lie in response to the questions -wording (if a survery): Questions can be worded in such a way that they bias the response. For example: "60% of doctors recommend you get this blood test, will you get it?" vs "40% of doctors do not recommend you get this blood test, will you get it?" The same question just worded differently which causes bias. What are the types of random samples? (Sampling/Experimenting Methods pg 2-3) - Correct Answers: 1. Simple random samples: (SRS) In this sample the participants are chosen out of the population so that every individual has an equal chance of being chosen. 2. Stratified random samples: In this samples a population is divided in to groups with important similarities. These groups are called strata. AN SRS is conducted in each seperate strata. 3. Cluster samples: In this sample the population is divided into clusters. then a random sample of clusters is chosen, and all of the members of each cluster participate. What is important in an experiment? (Sampling/Experimenting Methods pg 2-3) - Correct Answers: The point of an experiment is to measure the effect of a treatment. 1. A *control which does not receive the treatment for comparison. 2. a *large number of subjects to test, so that the variables don't corrupt the conclusion. 3.*Randomization - The process of determining which subjects receive which treatment, if any, must be a random process. What is block design? (Sampling/Experimenting Methods pg 2-3) - Correct Answers: Block design refers to a type of experiment which divides subjects into **predetermined groups, which are known or suspected to **affect the results. Then groups within each block are assigned the different treatments. For example: a test for an energy drink may split subjects up by gender. What is matched paids design? (Sampling/Experimenting Methods pg 2-3) - Correct Answers: In matched pairs design, experimental subjects are matched in pairs so that two similar subjects are paired with each other, and each given a different treatment. What is a double blind experiment? (Sampling/Experimenting Methods pg 2-3) - Correct Answers: It is a experiment in which the subjects do not know what treatment they are receiving, and the people administering the treatment do not k now which patients are receiving which treatments either. This removes effects such as experimenter bias and the placebo effect. What are histograms? (Histograms pg 4-5) - Correct Answers: Histograms are a type of frequency distribution. Frequency distribution allows for you to categorize a vast amount of data. Each catagory will have a specific frequency which is the number of data in that catagory. *A BAR CHART OF FREQUENCY DISTRIBUTION What is distribution in statistics? - Correct Answers: It is the pattern of variation in a data set. This can be view visually. Distributions only describe quantitative or ordinal data sets. ** qualitative data does not have a distribution. What is range in distribution? - Correct Answers: The complete spread of values in a distribution What is center in distribution? - Correct Answers: Values near the middle of the range What is spread in distribution? - Correct Answers: How far the vakues in the distribution spread out from the center. what is frequency distribution? And what are the types? (Histograms pg 4-5) - Correct Answers: The organizatin of raw data in a table using classes and frequencies. How frequent a class of data occurs in a raw distribution of data. Types- 1. catagorical: qualitative data 2. grouped: quantitative data What is class interval in frequency distribution? (Histograms pg 4-5) - Correct Answers: The upper and lower limit for each category in a frequency distribution table what is "size of interval" in frequency distribution? (Histograms pg 4-5) - Correct Answers: The size of the interval is determined by finding the difference between the upper and lower limit for the category. What is a class mark in frequency distribution? (Histograms pg 4-5) - Correct Answers: The class mark is located at the midpoint of the class interval. You can find this midpoint by adding the upper and lower limits and dividing by 2. What are histograms composed of? (Histograms pg 4-5) - Correct Answers: a set of rectangles that have 1. Bases on the X-axis 2. Centers at the class marks 3. length equal to the class interval size 4. areas proportional to the class frequencies What is the quickest way to analyze a distribution spread? - Correct Answers: through percentiles. The #th percentile = the value where the # percent of observation fall at or below the value What are Quartiles? (Quartiles pg 5-6) - Correct Answers: quartiles represent different components of the data set. Key numbers or data to pullout to describe your data set. What is The lower quartile (LQ or Q1) ? (Quartiles pg 5-6) - Correct Answers: is a number such that at least 25% of the numbers in the data set are no larger than this number. Q1 is the number > or equal to 25% of the data. The way to find Q1 is to take the data from the the beginning to the Q2 (median of all the data) Q1 will be the median of that lower half of data. ** If you have an even number of data and there are 2 numbers in middle of the lower half- you take the average of the two numbers. Ex: your two median numbers for your q1 are 12 and 14. The avergae is 13 therefore you Q1 is 13. What is the The second quartile (Q2) ? (Quartiles pg 5-6) - Correct Answers: It is the median. The median is the middle value or the arithmetic mean of the middle values. Ex: 3,3,5,6,7,9,10,12,14 - the median is 7 * the data must be in numerical order to find the median. Q2 is the number that is > or equal to 50% of the data. or the median ** If you have an even number of data and there are 2 numbers in middle- you have two numbers for your median- take the average. Ex: your two median numbers are 15 and 16. your Q2 is 15.5. What is The upper quartile (UQ or Q3) ? (Quartiles pg 5-6) - Correct Answers: It is a number such that at least 75% of the numbers in the data set are no larger than this number. Q3 is the number > or equal to 75% of the data. To find the Q3 you take the data from Q2 (or median of all the data) all the way to the end of the data. The median of that upper half of data is Q3. ** If you have an even number of data and there are 2 numbers in middle of the upper half- you take the average of the two numbers. Ex: your two median numbers for your Q3 are 16 and 17. The avergae is 16.5 therefore you Q3 is 16.5. What is the normal approximation for data? (Quartiles pg 5-6) - Correct Answers: Normal approximation for data refers to the approximationg data in a histogram based on the normal curve is the data values are converted into standard units. In other words, the area under the histogram over the various regions is approximately equal to the area of the normal curve over the same regions. The peak of the curve is the median of the data. sometimes median is represented as 0. Area under the curve = 100% What is the IQR? - Correct Answers: The interquartile range is the distance between the 1st quartile and the 3rd quarile. IQR= Q3-Q1 This can also be a quick test for outliers. anything 1.5x the IQR = an outlier. IQR=Q3-Q1 IQRx1.5 = outlier range above Q3 or below Q1 What is the 5 number summary? - Correct Answers: 5 numbers that map out the range/shape of a distribution. the 5 numbers are: -minimum value/# -Q1 -median -Q3 -Maximum value/# What is a box plot? (Box plots pg 6-8) - Correct Answers: A box plot is a graph of the five number summary of the a set of data. The five number summary consist of the minimum, the first quartile, the median (second quartile), the third quartile, and the maximum. A box is used to represent the interquartile range (IQR), or in otherwords, the range from the first quartile (q1) to the third quartile (q3). There is a line dividing the box where the median is located, and lines extend from the box to the minimum and maximumof the data set. *** Box plots can be drawn either vertically or horizontally. What are Box plots used for? (Box plots pg 6-8) - Correct Answers: Because it is a less detailed graph it is useful for comparing different data sets, i.e test scores for the morning class compared to the afternoon class. Box plots can be fast and easy way of comparing the spread, range, and medians of different data sets. What is an outlier in a box plot? (Box plots pg 6-8) - Correct Answers: An outlier is a data value that is much smaller or much parger than the other values in a data set. IQR= Q3-Q1 to test for outliers: 1. find the IQR 2. multiply the IQR by 1.5 3. Subtract Q1 by what you get in step 2 or Q1- 1.5(IQR) 4. Q3 + 1.5(IQR) 5. Any value less that the value in step 3 or more than the value in step 4 is an OUTLIER. Outliers are removed from the data set, and instead represented by a mark on the graph. When the outliers are graphed on a box plot like this, it is called a modified box plot. What is a stem plot? (Stem and Leaf Plots pg 8-9) - Correct Answers: a stem plot or stem and leaf plot is a type of display method for single variable analysis. They are formed by first ordering each value into a stem portion and a leaf portion. The leaf portiion includes final digit, and them stem portion inludes everything but the final digit, Then, the stem values are written in a vertical column, with smallest at the top, and the lowest at the bottom, and a vertical line is drawn to the right of the numbers. Finally, the leaf values are written horizontally across from their corresponding stem valuesm in increasing order from the left to right. * in your stem column ( the numerical proceeding of the order of numbers) you can not skip numbers.. so you can not go 1,3,4,5,7,8.. you must go 1,2,3,4,5,6,7,8 or 0,1,2,3,4,5,6,7,8. *Your leaf column can be left blank in one leaf. So lets say you are listing of numbers in the 10s,20s,30,40s. You have 11, 16, 24, 41, 45. so on your stem 1 would have 1, 6. 2 would have a leaf of 4. 3 would have no numbers in its leaf. 4 would have 1 and 5 in its leaf. 3 is left blank because there are no numbers in the 30s. *You must have a key for this plot! You only need one example for the key. So for 1 | 6 = 16. Stem 1 | leaf 6 =16. ** you can also do stem and leaf plots for decimal numerals. So 1.2 would equal a stem of 1 and leaf of 2. The line between the stem and leaf would represent the decimal. MAKE SURE TO PUT THIS IN YOUR KEY. What are stem and leaf plots used for? (Stem and Leaf Plots pg 8-9) - Correct Answers: it used a quick method of displaying the general shape of data, while still incorportating the actual numerical values. Because numerical values are used, the variable must be quantitative. Stem plots do not work well with large data sets or sets that contain many decimal places. They are two ways a stem plot can be modified. They are by splitting stems (cutting the stem into two parts) and by trimming (which is eliminating on decimal place at the end of each value). How do you find the mean from a sample of data ? (Mean pg 9) - Correct Answers: The mean is also known as the arithmetic mean. It is calculated by finding the sum of all "N" values and dividing by the N. So add all the numbers together and then divide by the number of numbers. What is standard deviation (SD) ? (standard deviation pg 10) - Correct Answers: -a calculation of how far the actual observations fall from the mean. * how spread out the data is *Low SD = data is closely clustered around the mean *High SD = data is spread farther from the mean. SD is used to determine if a certain data point is standard or expected (expected variation) or if the data point is unusual or unexpected (statistically significant). standard deviation relates data points to the mean of the sample data set. Simply put, it the average amount away from the mean the data points are. stardard deviation is a measure of spread. It is represented by either s (sample) or lower case greek sigma symbol (population) In order to calculate SD you first need the variance SD equation simplified** How is standard deviation calculated? (standard deviation pg 10) - Correct Answers: First determine the mean/avergae of data set. standard deviation (for a population/ sigma symbol) = the sum (each data point - mean)^2/ the number of data points. Then square root to get SD. Now if you wanting to do the SD of a sample, the only thing that changes in the equation is "SD =" changes to "sample SD = " and the denominator changes from jut "n" to "n-1" (See SD equation made super easy) Basically, the equation is this: For the numerator, (you are adding up the differences then subtract the mean from that) then square what you get from that. You did this because you what this number to be an absolute number aka a positive number. For the denominator, you take the number of values minues 1. After you get the number for the numerator and denominator, you will square root the numerator/denominator to get the standard deviation. (see youtube video at 2 min mark ) **SD equation made super easy What is variance? (standard deviation pg 10) - Correct Answers: You may have noticed that the equation for standard deviation is square rooted. Without this square root it is variance that is determined, not standard deviation. Square rooting makes sure you get an absolute number or an always positive number, never negative!!. Therefore, variance can be used as another description of spread, Variance is generally only used in detemining standard deviation. It is signified by s^2 or sigma^2 as the case may be. (see youtube video at 2:30) What is probability? (probability pg 11-12) - Correct Answers: Probability is defined as the likelihood that an event will occur expressed as the ratio of the number of favorable outcomes in the set of outcomes divided by the total number of possible outcomes. Probability is always expressed as a number between 0-1. All possible outcomes have a probability of 1. All impossible outcomes have a probability of 0. The probability that an event will NOT occur is 1 - (the probability that an event WILL occur) Probability of occurence and nonoccurence together = 1 The probability of this will occur + Probability that this won't occur = 1 What are outcomes and events? (probability pg 11-12) - Correct Answers: An outcome is the result of an experiment or other situatin involiving uncertainty. An event is collection of outcomes. what are the basic rules of probability? (probability pg 11-12) - Correct Answers: Probability is denoted with "P(event)". Probability is written in terms of the probability of a specific event will occur. For example, to determine the probability the event A will occur you denote this as P(A). How do you find the probability of a simple event? (probability pg 11-12) - Correct Answers: A simple event consists of finding the probability of one event "A". For example, event A can be rolling a dice once and having an outcome of the number 3. Find the probability of rolling a 3 when you throw the die: P(event)= number of possible occurences in one trail/number of possible outcomes in one sample p(3) = number of 3s on the die/total number of numbers on the die P(3) = 1/6 ------------- Controverly, find the probability that you don't roll a 3 when you throw the die. P(NOT 3) = total number of numbers that aren't 3 on die/ total number of numbers on die P(NOT 3) = 5/6 HOWEVER** you can compute this by following the probability rule P(NOT 3) = 1 - P(3) P(NOT 3) = 1 - 1/6 = 5/6 How do you calculate probability of multiple events? - Correct Answers: It depends on the event.. see the addition rule and multiplication rule What is the Addition rule? - Correct Answers: The OR rule!!!**** The equation to determine the probability of one thing occuring or another thing occuring in the same event. For example what is the probability of rolling a 1 OR a 6 on a die? for rolling a 1 or a 6 the probability 1/6+1/6 = 2/6 ^^ this is known as the addition rule What is the multiplication rule? - Correct Answers: The AND rule!!!!**** what the probability of rolling a 1 and then a 6 Roll 1 (Px) what is the probability of rolling a 1? 1/6 Roll2 (Py) what is the probability of rolling a 6? 1/6 These events are independent.. .meaning they do not effect each other. Event 1 does not effect event 2. They do not influence each other Probability of rolling a 1 and a 6 or P(1 and 6)= Px * Py P(1 and 6) = 1/6 * 1/6 P(1 and 6) = 1/36 If one event influences the nerxt event, this is what type of event? - Correct Answers: Dependent events since 1 event influences the other. This means the probabilty of each event may change but the multipliction rule still applies. example what is the probability fo drawing a king of hearts and then a queen a queen of hearts? After the first draw, the card stays out of the deck so the probability of choosing a queen of hearts from the deck of cards next changes. Px = pulling a king of hearts Py = probability of pulling a queen of hearts after pulling out first card P(X and Y) = 1/52 x 1/51 = 1/2652 What can probability be measured in? (probability pg 11-12) - Correct Answers: fration, decimal, or percentage. ex: the probability that you will roll a 3 on a die is 1/6 = 0.17 = 17%. The range is from 0/0 - 0.0 - 0% (probability is *impossible) through 1/2 - 0.50 - 50% (probability is as likely as not) through 1 - 1.0 - 100% ( probability is certain) < 50% = unlikely >50% = likely What is conditional probability? (conditional probability pg 12-13) - Correct Answers: conditional probability looks at the probability that more than one event occurs. conditional probability deals with dependent event and is also referred to as compound probability. Conditional probability is denoted as P(B|A) which says the "probability of event B given that event A has occured." ** this is: What is the probablity of A occuring given that we know B has occured. B is a fact or a given true event. REMEMBER EVENT B HAS ALREADY OCCURED. P(A|B) = the probability of (event A and B) / all divide out by the probability of only event B. IN other words, how often is A and B occuring amongst all the posibilities? what is the probability that we would role 2 on a die (event A) given that we know the number rolled is an even number (event B)? Even numbers on die = 2,3,6 = 1/3 so P (A and B) = 33%. See link below at 0:50 for the equation for conditional probability ( Parents have two kids. We know one of this kids is a girl. What is the probability that both are girls? (conditional probability pg 12-13) - Correct Answers: P(A|B) = What is the probability (P) that both children are girls (A) given that atleast one child is a girl (B). So the numerator would be what is the probability of both a girl AND at least one being a girl. So P(A) = GG, GB, BG, BB = 1 out of 4 = 1/4. So if you have at least two girls, one of this is at least a girl soo you probability is the same. Now for the denominator. What is the probability of at least one girl out of the two children? GG, GB, BG, BB = 3/4 sooo.. P (A|4) = (1/4) / (3/4) P (A|4) = 0.25/0.75 = 0.3333 or 1/3. The probability of both being girls given at least one is girl = 1/3 What are dependent events? What are independent events? (conditional probability pg 12-13) - Correct Answers: Both are compound events Dependent events: one event does effect the likelihood the next event will occur. The likelihood of the second is affected by the outcome of the first event. Independent event: One event DOES NOT affect the likelihood the next event will occur. Example: flipping a coin and getting heads does not affect the likelihood that you roll a 6 on a die. These are two seperate independent events that do not affect the likelihood of each other. "Find the probability of rolling a prime # on a die and flipping heads on a coin. " Is this independent events or dependent events? What is the probability? (conditional probability pg 12-13) - Correct Answers: Independent events. the equation for independent events is P(A) x P(B) What is the probability (P) of rolling a prime # (event A) and flipping heads (event B)? P(A and B) = P(A) x P(B) prime # 2,3,5 = 3/6=3/6 Probability of heads = 1/2 so 1/2 x 1/2 = 1/4 1/4 is the probability of getting a prime number and heads. "Find the probability of choosing both jokers in a deck of cards" Is this independent of dependent events? What is the probability? (conditional probability pg 12-13) - Correct Answers: This is dependent events because if you chose one joker and leave it out this effects the probability of chosing a joker the next time you pull out a card. Dependent events formula: P(A) x P(B after A has happened) So probability of pulling a joker and then another joker or P(A and B) = probability of pulling a joker (A) x probability of pulling joker after a joker has already been chosen) P(A) = 2/54 = P(B after A) = 1/53 P(A and B) = 2/24 x 1/53 1/27 * 1/53 1 x 1 = 1 27 x 53 = 1431 P(A and B) = 1/1431 What are mutually exclusive events? (conditional probability pg 12-13) - Correct Answers: mutually exclusive events are two events that can NOT happen at the same time. ** How to check if you events or mutually exclusive vs independent: Is the probability of A and B equal to 0? P(A and B) = 0? Example: Probability of pulling a red card given that the card is a spade.... well all spades are BLACK...soooo the probability of pulling a red card is 0!!!! What are permutations? - Correct Answers: Combinations that must be in a specific order aka an ordered combination. It is an ordered sequence of items taken from a set of distinct items WITHOUT REPLACEMENT. What is the Permutation Formula? (permutations pg 25) - Correct Answers: nPr = n!/(n-r)! the number of permutations = (n-1) x (n-2) x (n-3) so on and so forth 2,4,6 what is the number of permutations? there are 3 distint number so 3! ssooo : 3*2*1 = 6 number of permutatons - Correct Answers: What is randsomness and probability (Simple probability pg 14-16) - Correct Answers: In statistics, randomness doe snot mean un-predictable or haphazard, but rather it refers to a phenomenon in which individual outcomes are uncertain, but which follow a predictable and calculable distribution after many repetitions. This distribution, or long term relative frequency, is referred to as probability. Probability is always represented by a number between 0 and 1, or as a percentage. Thereforem the greatest value a probability can take is 1 (or 100%), and the lowest number it can is 0. What is a binomial coefficient? - Correct Answers: The is used when the order DOESN'T matter It is a NO REPETITION COMBINATION The binomial coefficient is the number of ways of picking unordered outcomes from possibilities, also known as a combination or combinatorial number. They count certain ways of grouping items. Binomial coefficients are sometimes ready as n choose k.. example 5 choose 2 meaning you have 5 things and have to choose 2.. how many comincations of 2 can be done with 5 things. choosing a comination of small number of things from a larger number of things. formula C(n,k)It counts the numvers of ways to form an unordered collection of "k" items chosen from a collections of "n" distinct items. n!/k!(n-k)! You need to build a cast of 5 people. You have 8 people to choose from. What is the binomial coefficient? - Correct Answers: n!= 8! (number is larger group) k!=5! the cast 8! / 5! ( 8 - 5 ) ! = 8! / 5! (3)! = 8x7x6x5x4x3x2x1/5x4x3x2x1 (6) *Things can be canceled out soo in the denominator, everything after 5 is canceled beause 5! is in the denominator. We are left with = 8x7x6/6 Soooo 6s can be canceled out too, so we are left with 8x7 = 56 What is sample space? (Simple probability pg 14-16) - Correct Answers: A sample space is the set of all possible outcomes. For example, the sample space when rolling a standard die is S = {1,2,3,4,5,6}. Each possible outcome is called an event. What is the multiplication principle? (Simple probability pg 14-16) - Correct Answers: The multiplication principle states that if two task are independent, and you can do task one x number of waysm and task two y number of ways, then the numver of ways the two can be done togther is x times y. A girl has 14 pairs of pants, 16 shirts, and 3 pairds of shoes. How many possible outfit combinatin does she have? What principle will you use to find this?(Simple probability pg 14-16) - Correct Answers: multiplication principle 14 x 16 x 3 = 672 different outfitd What is the addition rule? (Simple probability pg 14-16) - Correct Answers: Addition rule = probability of OR The addition rule states that if two events are disjoint ( they can not over lap.. SO NOTHINGS IS PART OF SET A AND B.. JUST PART OF EITHER SET A OR B. IF IT WAS NOT DISJOINT, THE PROBABILITY COULD BE PART OF BOTH SET A AND B) , or in other words if two events cannot occur at the same time (MUTUALLY EXCLUSIVE EVENTS) then P(A or B) = P(A) + P(B). This is also reffered to as a union and written as P( A and B) = P(A) + P(B). This equation is ONLY valid for disjoint/mutually exclusive events. Now if you have two events that can over lap. aka not disjoint, you use this equation: P(A or B)= P(A) + P(B) - P(A and B) There is a 0.5 probability that a teacher will give an assignment greater than 10 problems on any given day. There is a 0.1 probability that the teacher will give a quiz on any given day. Assuming that the teacher will never give a homework assignment greater than 10 problems and a quiz on the same day, what is the probability of having either a quiz or a homework assignment larger than 10 problems? What type of events are these? What type of rule would you use? (Simple probability pg 14-16) - Correct Answers: These disjoint or mutually exclusive events ( Two events that can not overlap.. you can not have a quiz and >10 question assignment in same day) so you will have to use the additon rule to get the probability. P(A)= probability of assignment with more than 10 questions P(A) = 0.5 P(B)= probability of quiz P(B) = 0.1 P(A or B) = P(A) + P(B) P(A or B) = 0.5 + 0.1 P(A or B) = 0.6 or 60% So there is a probability of 0.6 or 60% of having either a quiz or a homework assignment with more than 10 questions. If two events are not disjoint but you still wanted to get an A or B event probability, what equation would you use? Example to work through: You have 8 green squares, 9 green circles, 5 yellow squars, and 7 yellow circles. What is the probability that you get a yellow shape or a square? (Simple probability pg 14-16) - Correct Answers: You use this equation because the probability overlaps. The probability of getting a yellow shape or a square or even a yellow square. P(A or B) = P(A) + P(B) - P(A and B) P(A) = yellow shapes P(A) = 12/29 P(B) = squares P(B) = 13/29 P(A and B) = Probability of getting a yellow square P(A and B) = 5/29 P(A or B) = 12/29 + 13/29 - 5/29 P(A or B) = 20/29 What is the complement rule? (Simple probability pg 14-16) - Correct Answers: A compliment rule is the event that a specific event will NOT occur. In other words, the compliment of a certain probability is probability that something does not happen. The complement of event A is expressed as Ac. The complement ruke is expressed by the follwoing equation: P(Ac) = 1 - P(A) example: The probability that it will rain on any specific day is 0.3 or P(Rain) = 0.3. What is the probability that is won't rain on any specific day? P(that is won't rain or P(Rc) = 1 - P(R) P(Rc) = 1 - 0.3 P(Rc) = 0.7 What is the multiplication rule? (Simple probability pg 14-16) - Correct Answers: Multiplication rule is the probability of AND There are two different multiplication rules. The FIRST: is used in the case of independent events. Two events are independent if the outcome of one event does NOT affect the outcome of the other. The equation is P(A and B) = P(A) x P(B). The SECOND: When two events are not independent (so dependent events), the equation is P(A and B) = P(A) x P(B after/given A) The probability that a given student will forget their homework on any given day is P(H)= 0.1. The probability that the same student will forget a writing implement on any given day is P(W)= 0.6. Assuming the two events are independent, what is the probabilitu of a student forgeting both their homework and a writing implement? (Simple probability pg 14-16) - Correct Answers: These events are independent. P(H and W) = P(H) x P(W) P(H and W) = 0.1 x 0.6 P(H and W) = 0.06 What are Z scores? (Z-scores pg16-18) - Correct Answers: Also called standard units because z scores are used to analyze data , not in terms of numerical value, but in terms of distance from the mean. The units of Z scores are standard deviations. This means the Z score of 1 would descibe a value which is 1 standard deviation away from the mean. Z scores are generally used in determining percentiles and probabilities. Z scores can only be used when data follows a normal distribution. Basically tells you how many standard deviations above or below the mean that a particular data value is Z scores lets us analyze things that are not on the same scale or different types of data as long as they are normally distributed. What is the formula for Z scores? (Z-scores pg16-18) - Correct Answers: Z scores are calculated using z= x - mean/SD if x is positive, it is above the mean. If it is negative, it is below the mean. Find the Z score for the following problem. Mean (u looking symbol)= 24 standard deviation (sigma/or cursive o looking symbol)= 5 What is the Z score for 20? - Correct Answers: Z= 20-24/5 Z= -4/5 = -0.8 FInd and compare the Z scores for this problem. Kim got a 1200 on the SAT Jason got a on the ACT SAT SD: 200 ACT: 4.8 SAT scores are between 400-1600. The mean = 1000 ACT scores are between 6-36. The mean = 21 - Correct Answers: z = x- mean/SD Find the Z score for both SAT Z = /200 Z= 200/200 Z=1 ACT z= 25-21/4.8 z= 4/4.8 Z= 0.83 Soo Jason did better than Kim What is the Empirical Rule? (Empirical Rule pg 19-20) - Correct Answers: Epirical Rule is a way of analyzing normal distribution in terms of standard deviation. It is also called the 68-95-99.7 rule because it states that approx 65% of the observations will fall within two standard deviations (2 sigma symbol.. or cursive o)of the mean (u looking symbol) of a normal distribution, and 99.5% of the observations will fall within three standard deviations (3 sigma symbol) of the mean (u looking symbol) of a normal distribution. Empirical Rule example: A data set follows normal distribution, and has a mean of 50 and a SD of 5. What percent of the data is between the values x = 45 and x = 55? (empirical rule pg 19-20) - Correct Answers: Empirical rule states that 65% if the observations will fall within standard deviation of the mean of a normal distribution mean - SD = 50 - 5 = 45 mean + SD = 50 + 5 = 55 Therefore 45 and 55 are the two values withing standard deviation of the mean, and 65% of the observations will lie within this range. What is correlation? - Correct Answers: What is regression? - Correct Answers: What are the two types of variables? - Correct Answers: Explanatory (or independent) and Respone (or dependent). Define Explanatory variables (Statistics Terms pg 1) - Correct Answers: Sometimes referred to independent variables. This variable explains the change in a response variable. In a graph, the explanatory variable is graphed along the x-axis. These explain things or cause change in response variables. Define Response variable (Statistics Terms pg 1) - Correct Answers: Also known as a dependent variable. This is a variable that is influenced by the other variable. In a graph, a response variable is grapher along the y-axis. Response variables measures the study's outcome. The explanatory varriable acts on the response variable. A study is examing how heart rate responds to exercise. The harder you exercise, the more your heart rate incrases. In this study, what is the explanatory variable and what is the response variable? - Correct Answers: Exercise is influency the heart rate. So exercise intesity is the explanatory/independent variable. Heart rate is being effected so heart rate is the response/dependent variable.. A _________________ plot is a way to examine explanatory and response variables. - Correct Answers: Scatter plot In a scatter plot always plot the explanatory variables on the horizontal or X-axis. The response variable is plotted on the vertical or y-axis. On a scatter plot, what is a positive association? - Correct Answers: The scatter plot is sloping up. A positive association is when the explanatory variable increases while the response variable also increases. aka positive correlation. On a scatter plot, what is a negative association? - Correct Answers: Downward slope on a scatter plot. When the response variable decreases while the explanatory variable in creases. By looking at a scatter plot. how do you know there isn't a relationship between the two variables? - Correct Answers: The plots will not have a trench and will be scattered all of the plot with no matter. No correlation.