Chapter 3b (Normal Curves) When is a data point ( raw score) considered unusual?

Chapter 3b (Normal Curves) In Chapter 3- We have 3 “tasks” 1)Find the “Z Score” of a raw score “x” (where x is “normally distributed”) 2) Find the ‘relative position” of a raw score “x” in terms of it’s “percentile” score 3)(Given a “word problem” ), make a conclusion about the relative position of “x” – or -- find the raw score “x” needed to satisfy some condition

Chapter 3b– Some Definitions Z Score : A measure of the relative position of a data point when the population is “approximately normally” distributed z is the number of standard deviations and the direction a “raw score” is away from the population’s mean. Eg. Given X ~ N( 27,5) Then if x=36, z= 1.8 Z Table – a list giving the proportion of scores below a specific z score. In this text, it is called: Table A, Standard Normal Probabilities.

Here is a histogram of vocabulary scores of 947 seventh graders. What is the z score for a raw score of 10 ? Answer : z = What is the z score for a raw score of 5 ? Answer ; z=

Your Turn : given What is a child’s z score, if her vocabulary score was 4 ? What is a “standardized” score for a vocabulary score of 12? Would you consider a vocab score of 9 unusual?

X~N(3.8, 4.3) Find the z score for a value of 9. Is 9 an “unusual score” ? Z= = ( 9-3.8)/4.3 = 1.21 for some people “yes” others “no” For some people, anything over 1  is “unusual”. For others, they only think in “Percentile score” ---so lets find a percentile !

Task #2: Finding the ‘relative position” of a raw score “x” in terms of it’s “percentile” score P(X < x score ) the “proportion” of data points less then or equal to a raw score “x”. If given in %, it is often referred to as the “percentile” score P(Z <z score ) the “proportion” of standardized points less than or equal to the or equal to the “standardized” score “z”. Then we will use Table A to find the “proportion” of scores “less than”

IQ ~N(100,15) P(x<122) = ? What proportion of scores are below an IQ of 122. P(x<122)= = P(z< 1.47) Using Table A, the proportion of data points below z=1.47 is “ An IQ of 122 puts an individual in the percentile “

IQ ~N(100,15) P(x<82) = ? What proportion of scores are below an IQ of 82. Your turn

IQ ~N(100,15) P(x>82) = ? What proportion of scores are above an IQ of 82. CAUTION: Table A “only” gives probabilities/proportions ‘below” a specified z score A”twist” on the wording

IQ ~N(100,15) P(x>82) = ? What proportion of scores are above an IQ f 82. “FROM definition of density curves” “ P(x>82) = 1-P(x<82) P(x>82) = P(z>-1.20) = 1-P(z<-1.20) = = ALTERNATIVE : use of Table A –due to symmetry P(z >#) = P(z -1.20) = P(z<+1.20) =.8849 A”twist” on the wording ANSWER:

Your Turn Given that female heights (in inches) are distributed approximately : x ~N(64,2.7). 1. What proportion of females are below 60”? 2. What is the probability of randomly selecting a female who is taller than 70” ?

P( 80 < x<110) This is equivalent to finding P(x<120) – P(x<80) P(z<0.67) –P(z<-1.33) = =.6568 *Note: You are subtracting “Areas” =--never subtract z scores

Your Turn. Given that female heights (in inches) are distributed approximately : X ~N(64,2.7). What proportion of females are between 61” and 67” i.e. P( 61 < x< 67) = ?

Finding a z score corresponding to a “percentile” (using Table A “backwards” ) What z score is “just above” a 70 percentile? i.e. P(Z<z score ) =.70 what is z score ? Find Table A entry (“inside”) that is closest to.7000 …… ……….. Find the corresponding “z score” 0.52

Chapter 3c – Friday

Finding a “raw score” corresponding to a “percentile” (using Table A “backwards” ) First remember : or Finding a x score is really dependent on finding a ‘z score” first e.g. Given IQ’s ~N(100,15) what IQ just is in the 90 percentile? Find “z score” corresponding to a Table A entry closest to.9000 … ……. Z=1.28 X= (15) = = 119

Your turn Given female heights (in inches) are distributed approximately : x ~N(64,2.7). What height just puts a female into the bottom 25% (ie. what is Q1!)

Task #3 – Word Problems Incomes in a local city are approximately normally distributed with a mean of $36,000 and a standard deviation of $8000 What percentage of incomes are below $30,000? 1) sketch the density curve, labeling the mean, etc appropriately. 2) identify the area under the curve to the left of x = 30,000 (by shading it in) 3) Calculate the z score for x=30,000 z= _________ 4) using Table A find the P(Z < z calculated in step 3 ) “4b” look at your sketch does your answer make sense?

Task #3 – Word Problems (continued) Incomes in a local city are approximately normally distributed with a mean of $36,000 and a standard deviation of $8000 What income is ”just” in the bottom 20% ? 1) sketch the density curve, labeling the mean, etc appropriately. 2) identify the area that is “approximately” in the bottom 20% 3) Find a z score that’s “closest” to )Calculate x score from z score. “4b” look at your sketch does your answer make sense?

Your turn WS Ch 3