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STATISTICS - CLUTCH
CH.7: THE STANDARD NORMAL DISTRIBUTION (Z-SCORES)
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.
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?
z = x-μx
σx
x = observation
μx = mean
σx = SD
STATISTICS - CLUTCH
CH.7: THE STANDARD NORMAL DISTRIBUTION (Z-SCORES)
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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?
STATISTICS - CLUTCH
CH.7: THE STANDARD NORMAL DISTRIBUTION (Z-SCORES)
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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 .00 .01 .02 .03 .04 .05 .06 .07 .08 .09 0.0 .0000 .0040 .0080 .0120 .0160 .0199 .0239 .0279 .0319 .0359 0.1 .0398 .0438 .0478 .0517 .0557 .0596 .0636 .0675 .0714 .0753 0.2 .0793 .0832 .0871 .0910 .0948 .0987 .1026 .1064 .1103 .1141 0.3 .1179 .1217 .1255 .1293 .1331 .1368 .1406 .1443 .1480 .1517 0.4 .1554 .1591 .1628 .1664 .1700 .1736 .1772 .1808 .1844 .1879
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 3: Find P(z > 2.58).
EXAMPLE 2: What is the probability of finding a z-score
between -1.28 and 1.28?
EXAMPLE 4: Find P(z > -3.65).
.1064
STATISTICS - CLUTCH
CH.7: THE STANDARD NORMAL DISTRIBUTION (Z-SCORES)
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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
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 -2.58 and -1.65?
EXAMPLE 3: Find P(-2.33 ≤ z ≤ 2.17).
EXAMPLE 2: What is the probability of finding a z-score
between 1.96 and 2.33?
EXAMPLE 4: Find (-3.65 < z < -4.00)
P(0 ≤ z ≤ z2) – P(0 ≤ z ≤ z1)
________ the two probabilities
P(0 ≤ z ≤ z1) + P(0 ≤ z ≤ z2)
________ the two probabilities
P(0 ≤ z ≤ z1) – P(0 ≤ z ≤ z2)
________ the two probabilities
STATISTICS - CLUTCH
CH.7: THE STANDARD NORMAL DISTRIBUTION (Z-SCORES)
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PRACTICE 1: What is the probability of finding a z-score between -2.57 and 1.65?
PRACTICE 2: What is the probability of finding a z-score between -1.42 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).
STATISTICS - CLUTCH
CH.7: THE STANDARD NORMAL DISTRIBUTION (Z-SCORES)
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