Probably About Probability p <.05. Probability. What Is Probability?

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1 Probably About p <.05 1 Inferential statistics allow us to decide if one condition of an experiment likely produced different results than another condition Inferential statistics are based on the concepts of probability Thus, probability is an essential aspect of statistics 2 What Is? Probabilities often deal with events An event is something that happens E.g. Rolling a 3 on a fair die is an event The probability of an event is given by the ratio of how often that event occurs and how often all events occur 3 1

2 of Events When you role a fair, 6 sided die, each of the six faces has an equal chance of coming up Thus, the probability of any single face appearing is given by 1 (how often that event occurs) divided by 6 (the total number of events) 4 1 Die (6 events) Event 5 Mean of 2 Dice (36 events) Mean f p(mean) / 36 = / 36 = / 36 = / 36 = / 36 = / 36 = / 36 = / 36 = / 36 = / 36 = / 36 = Event 6 2

3 Mean of 3 Dice (216 events) Mean f p(mean) / 216 = / 216 = / 216 = / 216 = / 216 = / 216 = / 216 = / 216 = / 216 = / 216 = / 216 = / 216 = / 216 = / 216 = / 216 = Event / 216 = 46 7 Mean of 4 Dice (1296 events) Mean f p(mean) Event 8 Mean of 8 Dice (1,679,616 events) Event 9 3

4 p(m=5) # of Dice p = 1/ = 3/ = 10/ = 35/ = 6147/1,679,616 p(m 5) # of Dice p Continuous Variables and When the variables / events are continuous rather than discrete, we can no longer simply count the occurrence of the event or count the total number of events The continuous nature of the variable implies that there is an infinite number of values that the variable can take on 11 of Continuous Events Rather than counting, the probability is determined by areas under a probability curve p(a X b) = area under curve between a and b / total area under curve a b 12 4

5 Unit Normal Curve The unit normal curve is frequently used in statistics to determine probabilities It is a normal (or Gaussian) curve with a standard deviation of 1 The area under the unit normal curve is 1 Many variables have approximately normal distributions 13 Unit Normal Curve 34.13% 13.59% 2.28% 14 Determining Probabilities with the Unit Normal Distribution Weights are normally distributed What is the probability of a randomly selected female s weight being larger than 190 pounds? The mean weight of females is 133 pounds with a standard deviation of 22 pounds 15 5

6 Determining Probabilities with the Unit Normal Distribution Convert the raw score to a z-score: z = ( ) / 22 = 2.59 Use Table A in Appendix B of your text (or any other similar table) to find the area above a z- score of lb, z=2.59 z-score of Determining Probabilities with the Unit Normal Distribution The area above a z-score of 2.59 is.0048 The total area under the unit normal curve is 1 The probability is.0048 / 1 =.0048 There are approximately 5 chances in 1000 of a randomly selected female weighing over 190 pounds 190 lb, z=2.59 z-score of Problems What is the probability that a randomly selected female weighs less than 100 pounds? What is the probability that a randomly selected female weighs more than 110 pounds and less than 150 pounds? What is the probability that a randomly selected female weighs either less than 100 pounds or more than 166 pounds? Solution Solution Solution 18 6

7 Inferential Statistics Population Normal µ = 400 σ = 20 Sample T r e a t m e n t Treated sample 19 Inferential Statistics Initial sample should have 400 and s 20 How different does the treatment s sample mean have to be for it to be noticeably different? Is 415 enough? 420? 20 Psychologists typically want there to be less than a 5% chance of the treatment having no effect (being drawn from the original population) before they are willing to say that the treatment had an effect What z-score cuts off 2.5% above and below? Relative 21 7

8 Consult a table of z-scores to find an area above = z = ±1.96 Raw scores of and correspond to z = ±1.96 If treatment is less than or greater than 439.2, then it is unlikely that the sample mean came from the initial population Relative z = z = The End 23 Less Than 100 Pounds First, convert the raw score to a z-score N(133, 22), raw score = 100 z = ( ) / 22 = -1.5 Draw a unit normal distribution and put the z-score on it z-score of 8

9 Less Than 100 Pounds We want the area to the left (below) the z- score Because the unit normal distribution is symmetrical, the area to the left (below) z = -1.5 is the same as the area to the right (above) z = 1.5 z-score of Less Than 100 Pounds Problems Find the area to the right (above) z = 1.5 Consult Appendix B or this table The area corresponds to.0668 Thus, 6.68% of randomly selected females should weigh less than 100 pounds z-score of p (110 < X < 150) Determine the probability that a randomly selected female weighs less than 150 pounds Determine the probability that a randomly selected female weighs less than 110 pounds Take the difference z-score of 9

10 < 150 Pounds Convert 150 pounds to a z-score z = / 22 = 0.77 Draw the distribution and z-score Consult a table to find the area below a z-score of.77 p(< 150) = = z-score of < 110 Pounds Convert 110 pounds to a z-score z = / 22 = Draw the distribution and z-score Consult a table to find the area below a z-score of p(< 110) = z-score of p(110 < X < 150) Subtract the two: p(110 < X <150) = p(<150) p(<110) p(110 < X <150) = = Problems 30 10

11 < 100 or > 166 Convert to a z-score: z = ( ) / 22 = -1.5 Z = ( ) / 22 = 1.5 Draw the distribution and z-score Find, in a table, the area above z = 1.5 p(a < 100) = p(a > 166) = z-score of <100 or > = Problems 32 11