STATISTICS - CLUTCH CH.7: THE STANDARD NORMAL DISTRIBUTION (Z-SCORES)

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2 Z-SCORES You have to standardize normal distributions in order to find You standardize by changing all the values into z-scores The z-score represents how many a value is away from the The sign of the z-score tells you where the value lies If the value is above the mean, the z-score is going to be If below the mean, Even though this may seem simple, you should always draw to visualize the problem from here on! EXAMPLE 1: Everyone comes in pre-med and then gets to Organic Chemistry and says, Screw this. The grades in the class are normally distributed with a mean of 30 and a standard deviation of 10. Determine the z-score corresponding to scores of 60, 10 and 30. z = x-μ x σ x x = observation μ x = mean σ x = SD EXAMPLE 2: You and your friend are in different sections of the Intro to Statistics class. She earned a 70 in a class that had an average of 50 with a standard of 10. You, on the other hand, earned a 65 in a class that had an average of 60 with a standard deviation of 2. Who technically did better? Page 2

3 PRACTICE 1: Assuming a data set is normally distributed with a mean of 100 and a standard deviation of 20, what is the z- score that represents a value of 120? PRACTICE 2: Referring to the data set in Practice 1, what is the z-score that represents a value of 110? PRACTICE 3: Referring to the data set in Practice 1, what is the z-score that represents a value of 119? PRACTICE 4: Assuming a data set is normally distributed with a mean of 100 and a standard deviation of 15, what is the z- score that represents a value of 119? PRACTICE 5: Referring to the data set in Practice 4, how and why is this z-score different from the z-score from Practice 3? PRACTICE 6: Referring to the data set in Practice 4, what is the probability of finding values between 115 and 130? Page 3

4 Z-TABLE The empirical rule gives you an estimate of probabilities within set What about intervals which aren t so simple? (i.e. between z = 1.28 and z = 1.96) You can use the z-table to get probabilities for any Below is a sample of the table: 0 z z The table has probabilities that represent the area under the curve from Example: the highlighted number is the probability of finding a z-score between 0 and 0.27 Another way of saying this is:.1064 = P(0 z 0.27) EXAMPLE 1: What is the probability of finding a z-score between 0 and 1.96? EXAMPLE 2: What is the probability of finding a z-score between and 1.28? EXAMPLE 3: Find P(z > 2.58). EXAMPLE 4: Find P(z > -3.65). Page 4

5 TWO Z-SCORES We saw how to use the z-table can help us find intervals using one z-score, but what about two? Three types of situations will happen when working with an interval between : on the same side (right) on opposite sides on the same side (left) z1 z2 z1 z2 z1 z2 P(0 z z 2) P(0 z z 1) the two probabilities P(0 z z 1) + P(0 z z 2) the two probabilities P(0 z z 1) P(0 z z 2) the two probabilities Once the two probabilities are looked up in the table, it s simply a matter of EXAMPLE 1: What is the probability of finding a z-score between and -1.65? EXAMPLE 2: What is the probability of finding a z-score between 1.96 and 2.33? EXAMPLE 3: Find P(-2.33 z 2.17). EXAMPLE 4: Find (-3.65 < z < -4.00) Page 5

6 PRACTICE 1: What is the probability of finding a z-score between and 1.65? PRACTICE 2: What is the probability of finding a z-score between and -3.99? PRACTICE 3: Find P(-2.33 z 2.17) PRACTICE 4: Find P(z > 3.76). PRACTICE 5: Find P(z < -.72). Page 6