CJUS K300 1nd Edition Exam 2 Study Guide Lectures 6 10 Lecture 6 September 17 Probability Notation Of Probability P Rules of Probability Bounding Rule all probabilities are bounded between one and zero 0 P A 1 If you come up with a negative probability or a probability higher than one re check your answer because that is not possible Must be between 0 1 Complement Rule If you know P A then you know P Not A P Not A 1 P A Mutually exclusive events are when two events cannot occur at the same time If A and B are mutually exclusive then P A AND B 0 Ex In a deck of cards pulling a diamond and a heart are mutually exclusive P heart AND diamond 0 Addition Rule either or rule for mutually exclusive events A and B are mutually exclusive P A or B P A P B Ex P Hearts or Diamonds P H P D o P H 25 o P D 25 o 25 25 5 or 5 probability Not all events are mutually exclusive Addition Rule in general form P Hearts and Ace 0 P A or B P A P B P A and B You subtract P A and B in order to not double count the event and overestimate the probability If you do not subtract A and B you would be counting it twice because you have already included A and B separately Ex P Heart or Ace P H 25 P A 08 P H A 02 31 or 31 Explanation Under the assumption that there are 52 cards in a deck o P Hearts 13 52 or 25 o P Ace 4 52 or 08 o P H A 1 52 or 02 Lecture 7 September 22 Third rule of probability Multiplication Rule general form P A and B P A x P B A Conditional Probabilities P A B the probability that A will happen given that B has already happened P B A the probability that B will happen given that A has already happened P A and B P A x P B A Independent Events When two events are statistically independent knowing that event A has already occurred does not help you at all in predicted the probability of event B occurring A good example of this is in the deck of cards Knowing the probability of drawing an Ace P Ace does not help you figure out the probability of the suit of the Ace P Heart For statistically independent event P A B P A and P B A P B The short form of the Multiplication Rule which can ONLY be used for Independent events P A and B P A x P B How do we know which events are statistically independent If P A B P A then the events must be independent Does P Ace Heart P Ace o P Ace Heart 1 13 or 08 o P Ace 4 52 or 08 o They are independent Which means that P Ace and Heart P Ace X P Heart o 4 52 x 13 52 08 x 25 02 Lecture 8 September 24 Z scores Properties of a normal curve as close to symmetrical as possible given that it would be close to impossible for it to be perfectly symmetrical Two curves can be normal with the same standard deviations but different means or the same mean and different standard deviations See power point for examples In cases of normal distribution Once you find the mean of the curve for normal curves it is in the middle of the curve 50 of the data is on the left side and 50 of the data is on the right The area between the mean and one standard deviation is 34 meaning 34 of the data points lie in between the mean middle and one standard deviation away in either direction The area between the mean and two standard deviations is 47 5 The area between the mean and 3 standard deviations is 49 5 The two important parameter are the mean and the standard deviation The mean determines where the midpoint of the distribution falls and the SD determines the spread or the width of the distribution Large SD wider more dispersed data points Small SD narrower less disperse data points Standardizing normal distribution Making the mean 0 on the graph in order to be able to compare two different sets of data Converting values to a standard score Reading the Z Table This will take some practice but gets easier 1 Calculate the z score and round to two decimal points 2 Divide the number in two sections a First number and the first decimal point second decimal point i 1 95 A 1 9 B 05 3 Find the A 1 9 on the left side of the row 4 Find B from the top of the columns Example Janes IQ score is 3 standard deviations above the mean What do we know about her IQ She is in the 99 5th percentile 50 49 5 from 3 standard deviations American IQ test x 100 s 10 Marys score was 110pts She was above the average by exactly 1 standard deviation 10 so we know that she is in the 84th percentile 50 34 Comparing 2 different tests American IQ test x 100 s 10 Mary 110 European IQ test x 350 s 36 Pierre 382 Who has the higher IQ Mary does Pierre is not a full standard deviation above the mean whereas Mary is Z X Mean S Pierre z 382 350 36 89 Mary z 110 100 10 1 If it is not a full SD above the mean you have to take it to the z table Look up the z score on the table and find the corresponding number and add it to the 50 Z 88 80 15 53 53 corresponds to 2019 so you add 20 to 50 and which makes it the 70 th percentile Lecture 9 September 29 Central Limit Theorem 1 The means from a large number of samples all of size n 100 will always form a normal distribution 2 x meaning if x 5 then 5 X Standard error of the distribution of a large number of sample means taken from the same population Confidence Interval Formula x z s n 1 Alpha levels how likely it is that you are wrong For a 99 a 01 For a 95 a 05 For 90 a 10 Example x z s n 1 Calculate a 95 confidence interval 5 5 1 96 2 5 100 1 5 5 49 We are 95 confident that the actual population mean attitude toward gun control is between 5 01 points and 5 99 points on the Gun Control Attitude Scale You must know how to express this interpretation on the exam it is almost more important than getting the numbers correct It shows you understand the answers that you have gotten Lecture 10 October 1 Student s t distribution A collection of distributions whose shape depends …