the normal curve. probability distribution imagine that you rolled a pair of dice. what is the...
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![Page 1: The Normal Curve. Probability Distribution Imagine that you rolled a pair of dice. What is the probability of 5-1? To answer such questions, we need to](https://reader035.vdocuments.site/reader035/viewer/2022062421/56649cf85503460f949c92fb/html5/thumbnails/1.jpg)
The Normal Curve
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Probability Distribution
• Imagine that you rolled a pair of dice. What is the probability of 5-1?
• To answer such questions, we need to compute the whole population for possible results of rolling two dices.
• A standard dice has six faces. The probability of a number on a dice is 1/6.
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Probability Distribution
• We have two unrelated populations. That is, the results of the dices are not dependent.
• So, the probability of 5-1 is equal to 1/6 * 1/6 = 1/36. That means, in the perfect universe of math, if we roll two dice for 36 times, than one of the results will be 5-1.
• We can see that in the table below
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Probability Distribution1-1 1-2 1-3 1-4 1-5 1-62-1 2-2 2-3 2-4 2-5 2-63-1 3-2 3-3 3-4 3-5 3-64-1 4-2 4-3 4-4 4-5 4-65-1 5-2 5-3 5-4 5-5 5-66-1 6-2 6-3 6-4 6-5 6-6
So, the probability of 5 – 1 is 1/36. But remember, it is the probability of the event that the first dice (and only the first one) will be 5 and the second one will be 1
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Probability Distribution
• What if we would like to find the probability of the number 6 in that table? – That is the events when the sum of the two dices
will be six.• Which combinations of two dices produce the
score 6?– Let’s see in the same table
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Probability Distribution
• So, the probability of the score 6 is 5/36. • That is, in the perfect conditions 5 of 36 trails will result in the
score 6.
1-1 1-2 1-3 1-4 1-5 1-62-1 2-2 2-3 2-4 2-5 2-63-1 3-2 3-3 3-4 3-5 3-64-1 4-2 4-3 4-4 4-5 4-65-1 5-2 5-3 5-4 5-5 5-66-1 6-2 6-3 6-4 6-5 6-6
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Probability Distribution
• Now, to see the distributions of the possible score, let’s compute the possibilities of each scores ranging between 2 – 12. Let’s relook at the table
1-1 1-2 1-3 1-4 1-5 1-62-1 2-2 2-3 2-4 2-5 2-63-1 3-2 3-3 3-4 3-5 3-64-1 4-2 4-3 4-4 4-5 4-65-1 5-2 5-3 5-4 5-5 5-66-1 6-2 6-3 6-4 6-5 6-6
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Probability DistributionScore f Cum. f Prop. cum prop.
12 1 36 0,028 111 2 35 0,056 0,97210 3 33 0,083 0,9179 4 30 0,111 0,8338 5 26 0,139 0,7227 6 21 0,167 0,5836 5 15 0,139 0,4175 4 10 0,111 0,2784 3 6 0,083 0,1673 2 3 0,056 0,0832 1 1 0,028 0,028
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Probability Distribution
• Now, let’s look at the shape of distribution.
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dices
12,0010,008,006,004,002,000,00
Frequency
6
5
4
3
2
1
0
Histogram
Mean =7,00Std. Dev. =2,449
N =36
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Basic Characteristics of Normal Curve
• There are different kinds of normal curves. • Their means and standard deviations differ
from each other. But, the matter of concern is their shape.
• Three possible conditions for normal curves are
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Basic Characteristics of Normal Curve
• As you can see, the main similarity of the different normal curves is their symmetrical shape. – That is, the left half of the normal distribution is a mirror
image of the right half• They are unimodal distributions, with the mode at
the center• Mode, median and mean have the same value. • Their tails never touches to the x axis. So, we
describe the area under the curve as proportion. That is, the total area under the curve is equal to 1
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Basic Characteristics of Normal Curve
• The equation of normal curve is:
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Basic Characteristics of Normal Curve
• As you can see, the value of Y is determined by N, sd, mean. So, different distributions with different N, sd and mean have different normal distributions. Basically,
• N= the area under the curve• Mean= location of the center of the curve• SD= rapidity with which the curve approaches
to x axis.
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Basic Characteristics of Normal Curve
• As you can remember, to standardize different distributions, we use z scores. The mean and sd of z scores is 0 and 1, respectively. So, if we reorganize the formula for z, it becomes
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Basic Characteristics of Normal Curve
• As you can see, last formula indicates that z score determines the area in the normal curve.
• So, using the standardized z scores, we can compute areas in the normal curve
• In book, you can see these areas in Table A, pp. 552
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Area Under the Normal Curve
• In Table A, – z scores, – the area between z scores and mean, – and the area above z scores are presented.
• Note 1: the areas are bigger when z score is close to zero (the mean)
• Note 2: the sum of the two areas (the area between z scores and mean, AND the area above z scores) is .50. Because, this area represents the half of the distribution
• Note 3: All z scores in the distribution are positive, since the possitive area are the mirror image of negative area
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Area Under the Normal Curve
• Using this table, we can calculate – The area above a certain score• How many students (what proportion of scores) got
(was) higher than 70 in the final prep. exam
– The area under a certain score• How many students got lower than 35 points in the
final prep. exam
– The area two known scores• How many students got a score btw. 35 and 70 points in
the final prep. exam
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Area Under the Normal Curve
• The area above a certain score• The mean of the prep. Final exm = 60• The SD is 10• 2500 students took the final exam