Pp # 3 CHAPTS 3 & 5 normal distribution (Z Score), and Basic Concepts of Probability

Z-Scores Locations of Scores & Standardized Distributions

Z-Scores every score in a distribution. 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. An IQ score of 120

Z-Score IQ-Score

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.

Z-Scores X= (σZ)+µ µ= X- σZ σ= (X-µ)/Z If X=60 µ=50 σ=5 Z=?

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.

WHAT z SCORES REALLY MEAN 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

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.

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

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

Problems 4. In a distribution with µ=50 a score of X=48 corresponding to Z= -0.50. What is the standard deviation for this distribution?

Problems 5. In a distribution with σ=12, a score of X=56 corresponding to Z= -0.50. What is the mean for this distribution?

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_________

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.

SPSS

SPSS next https://www.youtube.com/watch?v=Soi1iXxpGmA

Click on the Orange Star to see the Z-Scores

DE-7 means very small =0.000

Please read the sample review questions for the z-scores, and take the Quiz # 3 on Blackboard.

Probability

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

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 (next slide).

A random sample requirements: A random sample requires that A random sample requirements: The probabilities cannot change during a series of selections There must be sampling with replacement (put it back in the jar). Every individual has an equal chance of being selected

Plural of “Die” Is “Dice Probability36X6=216

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

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

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

See p.714 TEXT

Z-Scores Calculators http://www.measuringu.com/zcalcp.php https://www.easycalculation.com/statistics/p-value-for-z-score.php <iframe src='https://www.easycalculation.com/statistics/embedded_p-value-for-z-score.php' width='500' height='' frameborder='0'></iframe>

*CHOOSING A Critical Value Always select an alpha value (smaller portion) 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

CHOOSING A Critical Value Ex. If alpha is 0.01 for a two tailed test then (smaller portion) will be 0.005 in each tail or 0.01/2= 0.005.You need to select between the lower number 0.0049 and the upper number 0.0051 (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 lower # which gives you a higher Z=2.58. The larger the critical values the more power to reject the Null.

Finding The 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)= 1.b. For a normal distribution, what is the probability of selecting a z-score less than 1.50? P(Z<1.50)=

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)= 0.1587 (or 15.87%) 1.b. For a normal distribution, what is the probability of selecting a z-score less than 1.50? P(Z<1.50)= 0.9332 or 93.32%

Finding Proportions/Probabilities/% for specific Z-Score Values: Examples 2.a 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 1.50? P(Z>1.50)=

Finding Proportions/Probabilities/% for specific Z-Score Values: Examples 2.a What Proportion of the normal distribution correspond to Z-Score values less than 1.00? P(Z<1.00) = 0.8413 (or 84.13%). 2.b For a normal distribution, what is the probability of selecting a z-score greater than 1.50? P(Z>1.50)=0.0668 or 6.68%

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)

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) =0.3085 or 30.85% 4. What proportion of the normal distribution is contained in the tail beyond Z= -0.50? P(Z<-.050) =0.3085 or 30.85%

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? 6. What proportion of a normal distribution falls between z = -1.15 and z = +1.15?

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: 0.0919 6. What proportion of a normal distribution falls between z = -1.15 and z = +1.15? Answer: 0.7498

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?

Probability Questions 9A. 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 %.

Probability Questions 9B.For a normal distribution with μ= 500 and σ= 100, what score separates the top 70% of the distribution from the rest? Hint: First look for the critical value of Z across from the larger portion or body B=0.70 Z value will be in the negative zone. Then use the X formula to find the score X= σ(Z)+µ

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=70 miles per hour with a standard deviation of σ=5. The distribution was approximately normal. Given the information, what percentage of the cars are traveling between 60 and 80 miles per hour? P(60<X<80)=? Hint: First find the Z-scores for each X (speeds, 60 and 80). Then, find the proportions (d or mean to Z) for the Z-scores and add them together. Convert to %

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 Proportions and add them together. Subtract from 1 and convert to %. See next slide

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 = (1100 - 1000)/175 = 0.57 Z2 = (1250 - 1000)/175 = 1.43 Now find the Proportion in the body or larger portion (b) for the Z=0.57,which is 0.7157 and the proportion in the tail (smaller portion) for z=1.43 which is 0.0764. Add them together= 0.7921, subtract from 1. 1-0.7921=0.2079 Convert to 20.79%

calculate the probability The Binomial Test and one sample Sign Test The Binomial Test follows the same four-step procedures for hypothesis Testing . i.e. 1. H0=No difference, H1= Difference 2. Critical regions 3. Calculation 4. Make a Decision The Binomial Test Calculator: http://www.vassarstats.net/textbook/ch5apx.html http://www.socscistatistics.com/tests/binomial/Default2.aspx https://graphpad.com/quickcalcs/binomial1.cfm ** http://stattrek.com/online-calculator/binomial.aspx

Use SPSS to calculate the probability The Binomial Test and one sample Sign Test 1. Click on (Variable View)name your variable i.e., GENDER 2. Click at the bottom of the (VALUE label) give 0 to male, and 1 to female. Click on (DATA View)-- Type in your DATA. 3. Click on Analyze  Non-Parametric Tests One Sample Scan Data 4. [automatically compare observed data to hypothesized]  Run 5. Double Click on the table to see the graph and details

Please read the sample review questions for the Probability, and take the Quiz # 4.