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 Why understanding probability is important?  What is normal curve  How to compute and interpret z scores. 2

 The chance of winning a lotter  The chance to get a head on one flip of a coin  Determine the degree of confidence to state a finding 3

 Percentages Under the Normal Curve  Almost 100% of the scores fall between (-3SD, +3SD)  Around 34% of the scores fall between (0, 1SD) Are all distributions normal? 4

The distance betweencontainsRange (if mean=100, SD=10) Mean and 1SD34.13% of all cases SD and 2SD13.59% of all cases SD and 3SD2.15% of all cases >3SD0.13% of all cases>130 Mean and -1SD34.13% of all cases SD and -2SD13.59% of all cases SD and -3SD2.15% of all cases70-80 < -3SD0.13% of all cases<70 5

 If you want to compare individuals in different distributions  Z scores are comparable because they are standardized in units of standard deviations. 6

 Standard score X: the individual score : the mean : standard deviation Sample or population? 7

 Mean=0, standard deviation=1 8

Mean and SD for Z distribution? Mean=25, SD=2, what is the z score for 23, 27, 30? 9

 Z scores across different distributions are comparable  Z scores represent the distances from the mean in a same measurement  Raw score 12.8 (mean=12, SD=2)  z=+0.4  Raw score 64 (mean=58, SD=15)  z=+0.4 Equal distances from the mean 10

 Eric competes in two track events: standing long jump and javelin. His long jump is 49 inches, and his javelin throw was 92 ft. He then measures all the other competitors in both events and calculates the mean and standard deviation:  Javelin: M = 86ft, s = 10ft  Long Jump: M = 44, s = 4  Which event did Eric do best in? 11

 Standardize(x, mean, standard deviation)  (x-average(array))/STDEV(array) 12

 Raw scores below the mean has negative z scores  Raw scores above the mean has positive z scores  Representing the number of standard deviations from the mean  The more extreme the z score, the further it is from the mean, 13

 84% of all the scores fall below a z score of +1 (why?)  16% of all the scores fall above a z score of +1 (why?)  This percentage represents the probability of a certain score occurring, or an event happening  If less than 5%, then this event is unlikely to happen 14

 In a normal distribution with a mean of 100 and a standard deviation of 10, what is the probability that any one score will be 110 or above? What about 6σ 15

 NORM.DIST(z,mean,standard_dev,cumulative)  z: The z score value for which you want the distribution.  mean: The arithmetic mean of the distribution.  cumulative: A logical value that determines the form of the function. If cumulative is TRUE, NORM.DIST returns the cumulative distribution function; if FALSE, it returns the probability mass function (which gives the probability that a discrete random variable is exactly equal to some value). 16

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 The probability associated with z=1.38  41.62% of all the cases in the distribution fall between mean and 1.38 standard deviation,  About 92% falls below a 1.38 standard deviation  How and why? 18

 What is the probability to fall between z score of 1.5 and 2.5  Z=1.5, 43.32%  Z=2.5, 49.38%  So around 6% of the all the cases of the distribution fall between 1.5 and 2.5 standard deviation. 19

 What is the percentage for data to fall between 110 and 125 with the distribution of mean=100 and SD=10 20

 The probability of a particular score occurring between a z score of +1 and a z score of

 Compute the z scores where mean=50 and the standard deviation =5  55  50  60  57.5  46 22

 The math section of the SAT has a μ = 500 and σ = 100. If you selected a person at random:  a) What is the probability he would have a score greater than 650?  b) What is the probability he would have a score between 400 and 500?  c) What is the probability he would have a score between 630 and 700? 23

 Expected response rate: obtain based on historical data  Number of responses needed: use formula to calculate 24

 n=number of responses needed (sample size)  Z=the number of standard deviations that describe the precision of the results  e=accuracy or the error of the results  =variance of the data  for large population size 25

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 Z=1.96 (usually rounded as 2)  =2.78  e=0.2  n=278 (responses needed)  assume response rate is 0.4  Sample size=278/0.4=695 27

 Z=1.96 (usually rounded as 2)  5-point scale (suppose most of the responses are distributed from 1-5)  error tolerance=0.4  assume response rate is 0.6  What is sample size? 28

 How to collect data so that conclusions based on our observations can be generalized to a larger group of observations.  Population: A group that includes all the cases (individuals, objects, or groups) in which the researcher is interested.  Sample: A subset of cases selected from a population  Parameter: A measure (e.g., mean or standard deviation) used to describe the population distribution.  Statistic: A measure (e.g., mean or standard deviation) used to describe the sample distribution 29

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 A method of sampling that enables the researchers to specify for each case in the population the probability of its inclusion in the sample.  The purpose of probability sampling is to select a sample that is as representative as possible of the population.  It enables the researcher to estimate the extent to which the findings based on one sample are likely to differ from what would be found by studying the entire population. 31

 A sample designed in such a way as to ensure that 1) every member of the population has an equal chance of being chosen, 2)every combination of N members has an equal chance of being chosen.  Example: Suppose we are conducting a cost- containment study of 10 hospitals in our region, and we want to draw a sample of two hospitals to study intensively. 32

 A method of sampling in which every Kth member in the total is chosen for inclusion in the sample.  K is a ratio obtained by dividing the population size by the desired sample size.  Example: we had a population of 15,000 commuting students and our sample was limited to 500, so K=30. So we first choose any one student at random from the first 30 students, then we select every 30 th student after that until reach

 A method of sampling obtained by 1) dividing the population into subgroups based on one or more variables central to our analysis, and 2) then drawing a simple random sample from each of the subgroups.  Proportionate stratified sample: the size of the sample selected from each subgroup is proportional to the size of that subgroup in the entire population. 34

 The size of the sample selected from each subgroup is deliberately made disproportional to the size of that subgroup in the population  A sample (N=180), with 90 whites (50%), 45 blacks (25%) and 45 Latinos (25%). 35

 Helps estimate the likelihood of our sample statistics and enables us to generalize from the sample to the population.  But population in most of times unknown  The sampling distribution is a theoretical probability distribution (which is never really observed) of all possible sample values for the statistics in which we are interested. 36

37 If we select 3 of them, what will be the difference for mean and standard deviation?

 A theoretical probability distribution of sample means that would be obtained by drawing from the population all possible samples of the same size 38 Mean Income of 50 Samples of Size 3 from 20 individuals

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 A process whereby we select a random sample from a population and use a sample statistic to estimate a population parameter.  Point estimate: A sample statistic used to estimate the exact value of a population parameter. Point estimate usually results in some sort of sampling error, therefore has less accuracy.  Confidence interval (CI): A range of values defined by the confidence level within which the population parameter is estimated to all. Sometimes confidence intervals are referred as a margin of error.  Confidence level: the likelihood, expressed as a percentage or probability, that a specified interval will contain the population parameter.  Margin of error: the radius of a confidence interval. 43

 Confidence intervals are defined in terms of confidence levels.  A 95% confidence level, there is a 0.95 probability – or 95 chances out of 100- that a specified interval will contain the population mean.  Most common confidence levels are: 90%, 95%, 99%  Margin of error is the radius of a confidence level. So if we select a 95% confidence level, we would have a 5% chance of our interval being incorrect. 44

MeanStandard Deviation Sample Distribution Population Distribution 45

 Follow these steps  Calculate the standard error (standard deviation) of the mean  Decide on the level of confidence, and find the corresponding Z value  Calculate the confidence interval  Interpret the results 46

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49 If we do 10 different samples, with 95% confidence level and come out with the confidence interval, only 1 out of the 10 confidence intervals does not intersect with the vertical line which is the true population mean

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 If other factors do not change  If the sample size goes up, the width gets smaller  If the sample size goes down, the width gets bigger  If the value of the sample standard deviation goes up, the width gets bigger  If the value of the sample standard deviation goes down, the width gets smaller  If the level of confidence goes up (95% to 99%), the width gets bigger  If the level of confidence goes down (99% to 95%), the width gets smaller. 52