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Unit 5: Distributional ShapeLesson 5:Probability and the Normal Curve : 

Unit 5: Distributional ShapeLesson 5:Probability and the Normal Curve EDER 5210 – Educational Statistics Dr. Robin K. Henson University of North Texas © 2002 University of North Texas Dr. Robin K. Henson © 2002 Next Slide

Normal curve = Exact percentages of area under the curve : 

Normal curve = Exact percentages of area under the curve Exact percentages = Exact probabilities Dr. Robin K. Henson © 2002 University of North Texas Next Slide

Randomly draw one score : 

Randomly draw one score Prob. of Z <=0 .50 Dr. Robin K. Henson © 2002 University of North Texas Next Slide

In a normal curve, we can use the Z score to identify the probability of obtaining a score < or > the score in question. : 

In a normal curve, we can use the Z score to identify the probability of obtaining a score < or > the score in question. Area under the curve represents probability of observations. Dr. Robin K. Henson © 2002 University of North Texas Next Slide

IQ Test: Your Z = 1.00 : 

IQ Test: Your Z = 1.00 .50 + .34 ~ ~ .84 1.00 - .84 ~ ~ .16 Dr. Robin K. Henson © 2002 University of North Texas Next Slide

But what if the Z score isn’t a nice round #? : 

But what if the Z score isn’t a nice round #? * For any point in the distribution there is a probability above and below the Z score. * We can use a table to calculate the probabilities. Dr. Robin K. Henson © 2002 University of North Texas Next Slide

GRE verbal = 550 : 

GRE verbal = 550 Probability of scoring better than 550? Dr. Robin K. Henson © 2002 University of North Texas Next Slide

Use an Area under Standard Normal Curve Table for Values of Z – Table C.1: Hinkle, Wiersma, & Jurs (1998) : 

Use an Area under Standard Normal Curve Table for Values of Z – Table C.1: Hinkle, Wiersma, & Jurs (1998) *Table gives:1) Area between (Z=0) and Z.2) Area beyond Z (into the tail). Dr. Robin K. Henson © 2002 University of North Texas Next Slide

Probability of scoring better than Z = .50? : 

Probability of scoring better than Z = .50? Dr. Robin K. Henson © 2002 University of North Texas Next Slide

Slide 10: 

Dr. Robin K. Henson © 2002 University of North Texas Next Slide

Probability of scoring better than Z = -1.38? : 

Probability of scoring better than Z = -1.38? .9162 Dr. Robin K. Henson © 2002 University of North Texas Next Slide

Tabled Z values can be positive or negative. : 

Tabled Z values can be positive or negative. Can calculate exact probabilities given a Z score. Assumes a normal distribution for your scores. Dr. Robin K. Henson © 2002 University of North Texas Next Slide

Slide 13: 

Dr. Robin K. Henson © 2002 University of North Texas Go to next slide for answer. Q: You have just received your midterm exam back and earned a Z score of -.23. Assuming the students’ scores were normally distributed, you did better than what proportion of other students? a. .409 b. .091 c. .591 d. .909 e. None of the above. Click mouse to continue

Answer : 

Answer a. Yes, the area beyond a Z = .23 to the tail is .409 per the Standard Normal Curve table. Note that you have to consider the absolute value of the tabled areas because the tabled areas apply to both sides of the normal distribution of Z scores (i.e., the distribution is perfectly symmetrical). b. No, this is the area between Z and the mean for Z = .23. We care about the area in the tail to find out the proportion you did better than. c. No, this is the proportion higher than Z = .23, and is irrelevant here. d. No, this is irrelevant. e. No, (a) is correct. Click mouse to go to next slide.

Slide 15: 

Dr. Robin K. Henson © 2002 University of North Texas Go to next slide for answer. Q: Because she likes to show off her immense knowledge of statistics, your obnoxious classmate tells you that you did better than 79.1% of the other students’ scores (which happen to be normally distributed) on your midterm exam. If your obnoxious classmate is right, this means your Z score must have been: a. .791 b. .810 c. .209 d. .190 e. None of the above. Click mouse to continue

Answer : 

Answer a. No, the Z score is not just the percentage. b. Yes, if you did better than 79.1% of the scores, then you did better than .791 proportion of the area under a normal curve. You know that if you are above the mean then you must have done better than .500. So, now all you need to find out is the area between the mean and where your Z score is, or .791 - .500 = .291. The area between the mean and your Z is therefore .291. Now you can use the Standard Normal Curve table to find out what Z score would you have if you have a proportion of .291 between you and the mean. Just scan the table column for “Area between Mean and Z” until you find .291. The Z score associated with this is Z = .81. It probably would help tremendously if you also sketched out the distribution and marked the areas you are considering – much like the distributions in this presentation. c. No, irrelevant. d. No, irrelevant. e. No, (b) is correct.

Unit 5: Distributional ShapeLesson 5:Probability and the Normal Curve : 

Unit 5: Distributional ShapeLesson 5:Probability and the Normal Curve EDER 5210 – Educational Statistics Dr. Robin K. Henson University of North Texas © 2002 University of North Texas Dr. Robin K. Henson © 2002 End

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