Stat Unit 5--Winter 2008 - Review z scores express a...

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Review z- scores express a specific value from a distribution relative to the mean and standard deviation z-score at mean will be zero A score 1 standard deviation above mean will be 1 A score 1 standard deviation below mean will be -1 If distribution is normal, we can use the table of z to determine the fraction of cases occurring Above z, below z or between 2 different z-scores δ μ X or s X X z - - =
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If e =80, s = 20, what fraction of scores are between 60 and 75? • z 1 = (60-80)/20 = -1 • z 2 = (75-80)/20 = -.25 p(z< -.25) = .401; from area C of table of z p(z< -1) = .274; from area C of table of z P(-.25< z> -.1)= .401-.274 = .127 12.6% of cases fall in this interval
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Introduction to probability and INFERENCE Statistical theory holds that “randomness” is predictable to a degree On the flip of a single coin, there are only two possible outcomes Head on fair coin will occur ½ the time Tail on fair coin will occur ½ the time On the two flips, there are 4 possible outcomes HH, HT, TT, TH Each of these will occur 25% of the time The chances of a head and tail are 50% (HT, TH)
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Terms The number of times we flip the coin (or take other action) will be N The probability of a specific outcome (a head) is p The probability of not getting that outcome is (1-p) or q
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With 3 flips, the number of potential outcomes becomes 8 (2 3 ) HHH, HHT, HTH, HTT, TTT, THT, TTH, THH Chance of 3 heads is 1/8 or 12.5% HHH , HHT, HTH, HTT, TTT, THT, TTH, THH Chance of 2 heads is 3/8 or 37.7% HHH, HHT , HTH , HTT, TTT, THT, TTH, THH Chance of 1 head is 3/8 or 37.5% HHH, HHT, HTH, HTT , TTT, THT , TTH , THH Chance of 0 heads is 1/8 or 12.5% HHH, HHT, HTH, HTT, TTT , THT, TTH, THH
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With 10 flips, the number of outcomes grows dramatically to 2 10 or 1024 For instance: HTTHTHHTTH or TTHTHHTTTH The probability of each outcome can be graphed Looks like this
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N=10, p=.5 Number of Heads
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Very low probability of 0 heads Low probability of 9 heads 4, 5 and 6 heads are more common outcomes Between 4 and 6, we have well over half of the outcomes Turns out that it’s about 65% of the outcomes that are either 4, 5, or 6 Can calculate the chances of any kind of probability outcome with a complex formula for binomial distributions We will not
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Binomial calculator gives us opportunity to do this for many sizes of N, other values of p http://onlinestatbook.com/java/binomialProb.html What do these distributions start to resemble? Normal distribution
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N=50, p=.5 Number of Heads
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Chances of extreme outcome ((like no heads or all heads) get very small as the number of trials or cases gets very large This is the logic underlying inferential statistics We do not know what will happen on any single trial However, we can measure the chance of any outcome occurring Other random processes, besides coin flips, also have knowable patterns of outcomes Sampling theory
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Sampling concepts Universe— the theoretical definition of the group to be studied Americans EMU students Likely Michigan voters Individuals with Type II diabetes
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