1. CHAPTS 3 & 5 Z Score, and Basic Concepts of Probability

2. Z-Scores
Locations of Scores & Standardized Distributions

3. Z-Scores every score in a distribution. Ex. IQ score of 120
The purpose of Z-scores or Standards Scores is to identify and describe the exact location of
every
score in a
distribution.
Ex. IQ score
of 120

4. IQ-Score Z-Score

5. Z-Scores Characteristics of Z-Scores
1. The mean of the Z-scores is equal to zero.
2. Every distribution of Z-scores has standard deviation of 1.
3. The Shape of the distribution of Z-scores is identical to the shape of the distribution of raw scores.

8. Z-Scores
Transformation of X values or individual scores into Z-scores serves 2 purposes
1. It tells the exact location of the score within the distribution
2. Scores can be compared to other distributions that also have been transformed into Z-scores.

10. WHAT z SCORES REALLY MEAN
Because
Different z scores represent different locations on the x-axis, and
Location on the x-axis is associated with a particular percentage of the distribution
z scores can be used to predict
The percentage of scores both above and below a particular score, and
The probability that a particular score will occur in a distribution

11. Z-Scores
X= σ(Z)+µ
µ= X- σZ
σ= (X-µ)/Z
If X=60
µ=50
σ= Z=?

12. Problems
1. Identify the z-score value corresponding to each of the following locations in a distribution?
A. Bellow the mean by ¾ standard deviation.
B. Above the mean by 1.5 standard deviation.

13. Problems
2. For a population with µ=20 and σ=4, find the z-score for each of the following scores?
A. X=18
B. X=28

14. Problems
3. For a population with µ=60 and σ=2, find the X value corresponding to each of the following z-scores?
A. Z=0.50
B. Z=-0.25

15. Problems
3. For a population with µ=60 and σ=2, find the X value corresponding to each of the following z-scores?
A. Z=0.50
B. Z=-0.25

16. Problems
4. In a distribution with µ=50 a score of X=48 corresponding to Z= What is the standard deviation for this distribution?

17. Problems
5. In a distribution with σ=12, a score of X=56 corresponding to Z= What is the mean for this distribution?

18. Problems
6. A normal-shaped distribution with µ=40 and σ=8 is transformed into z-scores. The resulting distribution of
z-scores has a mean of_______ and a standard deviation of_________

19. Problems
7. A distribution of English exam scores has µ=70 and σ=4. A distribution of History exams has a µ=65 and σ=1.5. For which exam would a score of X=78 have a higher standing.

20. Quiz # 2
Please read the sample review questions for the z-scores, and take the Quiz # 2 next week in class.

21. Probability
Probability

22. Probability
Number of times an event occurs in an infinite series of trials
P= (#of times an event occurs/Total number of trials) P=(f/N)100
For probability to be accurate it is necessary to use random sampling.

23. Random Sampling Random Sampling has 2 requirements:
1. Each individual in the population must have an equal chance of being selected.
2. If more than one individual or a group of individuals is to be selected for the sample there must be constant probability for each and every selection.

24. Probability

28. Probability facts Mega Ball/Power Ball= 1/292m Lottery = 1/67m
Hit by Lightening = 1/10m
Victim of Crime in the
U.S = 1/8000
Chance of dying in a flight 1/16m

29. Violent Crime rate Comparison per 1,000 residents
Your chances of becoming a victim
in Florida 1 in 205
in Miami is 1 in 85

30. Property Crime Rate Comparison per 1,000 residents
Your chances of becoming a victim
in Florida 1 in 31
in Miami 1 in 19

32. CHOOSING A Critical Value
Always select an alpha (C) value which is closest to your alpha level (upper or lower). If the upper and the lower values are even, then select the lower value because, it gives you a higher critical value for the Z. Ex: next

33. CHOOSING A Critical Value
Ex. If alpha (C) is 0.01 for a two tailed test alpha will be You need to select between the upper number and the lower number (the numbers are equally close to 0.005). The lower number gives you the Z critical value of Z=2.58 and the upper number gives you the Z critical value of Z=2.57. Your best choice is to select the Z=2.58. The lower the alpha the larger the critical region. The larger the critical regions the more power to reject the Null.

34. Finding Proportions/Probabilities/% for specific Z-Score Values: Examples
1.a. What Proportion of the normal distribution correspond to Z-Score values greater than 1.00? P(Z>1.00)=
2.a. For a normal distribution, what is the probability of selecting a z-score less than 1.50? P(Z<1.50)=

35. Finding Proportions/Probabilities/% for specific Z-Score Values: Examples
1.a. What Proportion of the normal distribution correspond to Z-Score values greater than 1.00? P(Z>1.00)= (or 15.87%)
2.a. For a normal distribution, what is the probability of selecting a z-score less than 1.50? P(Z<1.50)= or 93.32%

36. Finding Proportions/Probabilities/% for specific Z-Score Values: Examples
1.b What Proportion of the normal distribution correspond to Z-Score values less than 1.00? P(Z<1.00) =
2b. For a normal distribution, what is the probability of selecting a z-score greater than 150? P(Z>1.50)=

37. Finding Proportions/Probabilities/% for specific Z-Score Values: Examples
1.b What Proportion of the normal distribution correspond to Z-Score values less than 1.00? P(Z<1.00) = (or 84.13%).
2.b For a normal distribution, what is the probability of selecting a z-score greater than 150? P(Z>1.50)= or 6.68%

38. Finding Proportions/Probabilities/% for specific Z-Score Values: Examples
3. What proportion of the normal distribution is contained in the tail beyond Z= -0.50? P(Z<-.50)
4. What proportion of the normal distribution is contained in the tail beyond Z=0.50? P(Z>.50) =

39. Finding Proportions/Probabilities/% for specific Z-Score Values: Examples
3. What proportion of the normal distribution is contained in the tail beyond Z= -0.50? P(Z<-.050) = or 30.85%
4. What proportion of the normal distribution is contained in the tail beyond Z=0.50? P(Z>.050) = or 30.85%

40. Finding Proportions/Probabilities% for specific Z-Score Values: Examples
5. What proportion of a normal distribution is located between z = 1.00 and z = 1.50? Answer:
6. What proportion of a normal distribution falls between z = and z = +1.15? Answer:

41. Finding the Z-Score Location that Correspond to Specific Proportion: Examples
7.For a normal distribution what Z-score separates the top 10% from the remainder of the distribution?
8. For a normal distribution, what z-score values form the boundaries that separate the middle 60% of distribution from the rest of the scores?

42. Probability Questions
9. What is the probability of randomly selecting an individual with an IQ score less than 120?
Hint: First convert the X to Z using Z formula µ=100, σ=15, X=120
Then, find the proportion for the body and convert to %.

43. Probability Questions
10. The highway department conducted a study measuring driving speeds on a local section of interstate highway, They found an average speed of µ=58 miles per hour with a standard deviation of σ=10. The distribution was approximately normal. Given the information, what proportion of the cars are traveling between 55 and 65 miles per hour? P(55<X<65)=?
Hint: First find the Z-scores for each X (speeds, 55 and 65).
Then, find the proportions (d) for the Z-scores and add them together. Convert to %

44. Probability Questions
11. Suppose GRE scores are normally distributed with mean 1000 and standard deviation 175. What percentage of test takers will score between 1100 and 1250?
Hint: First find the Z-scores for each X(1100 and 1250). Now find the Proportion and add them together. Subtract from 1 and convert to %.
See next slide

45. Probability Questions
Find the z score for each score: First convert the Xs to Z using Z formula µ=1000, σ=175. Z = (Score - Mean)/Standard deviation X1=1100 , X2=1250 Z1 = ( )/175 = 0.57 Z2 = ( )/175 = 1.43 Now find the Proportion in the body (b) for the Z=0.57,which is and the proportion in the tail for z=1.43 which is Add them together= subtract from 1.
= Convert to 20.79%

46. Please read PP #4 for the next week