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Inferential Statistics Normall Curve • Specific p f bell-shaped p curve that is unimodal, symmetric, and defined mathematically. – Ubi Ubiquitous i – Helps clarify the probability of particular events – Let’s see an example with a coin toss Basis off Inferential f l Statistics 1. The approximate shape of the normal curve is everywhere. 2 The bell shape of the normal curve may 2. be translated into percentages ( (standardization). ) 3. A distribution of means produces a bellshaped curve even if the original distribution of individual scores is not bell-shaped, as long as the means are from sufficiently large samples (central li i theorem) limit h ) Sample l Size and d Distributions b Standardization d d • Comparing z scores: – Statistics Exam • Mean = 78, SD = 6, Your Score = 88 • z = 1.67 – Cognition Exam • Mean = 76, SD = 5, Your Score = 85 • z = 1.8 Transforming z scores into Percentiles Guinness and d Normall Curves • 1900s: W.S. Gosset hired for qualityy control q – Brewing and bottling both require q ap precise amount of yeast – Can’t test every bottle and every barrel • Need a sample! Centrall Limit Theorem h • A distribution of sample means approaches pp a normal curve as the sample size increases. – Even when the original distribution of scores is not normally distributed! Sampling Distribution of Means • A distribution composed of many means that are calculated ffrom all possible samples of a given size, all taken ffrom the same p population. p – Less variability than the actual scores. – Why does this distribution have less variability? y Sampling Distribution of Means • We cannot use the standard deviation for this distribution. distribution Standard d d Error • Standard deviation of a distribution of sample p means. • New Symbols: σM σM = μM σ N Quick k Review 1. As sample size increases, the mean of the sampling p g distribution of means approaches the mean of the population p p of individual scores Quick k Review 2. The standard error is smaller than the standard deviation and as sample p size increases, standard error decreases. Quick k Review 3. The shape of the distribution of means will approximate pp normal if the distribution of the population of individual scores is normal or if the size of each sample that comprises it is sufficientlyy large, g , at least 30. – Central Limit Theorem Back k to z Scores • Remember that we are often working with samples, p , not entire p populations. p – We need a new way to create z scores ( M − μM ) z= σM – z statistic: How many standard errors a sample mean is from the population mean A Practicall Example l • We conduct an IQ test in a class of 40 and find that the class average g is 106. – Population: Mean = 100, SD = 15 – How does our class average measure up when compared p with this p population? p A Practicall Example l ( M − μM ) z= σM μ M = μ = 100 σM = σ 15 15 = = = 2.372 N 40 6.325 A Practicall Example l ( M − μ M ) (106 − 100 ) z= = = 2.53 σM 2.372 To convert this z statistic to a percentage, percentage consult a z table (at the back of the book)