Unit Normal DistributionThis is the simplest of the family of Normal Distributions, also calledthe z distribution. It is a...
The Z ScoreThe number of standard deviations from the mean is called the z-scoreand can be found by the formula:          ...
ExampleFind the z-score corresponding to a raw score of 132 from a normaldistribution with mean 100 and standard deviation...
A z-score of 1.7 was found from an observation coming from anormal distribution with mean 14 and standard deviation3. Find...
Area Under the Unit Normal CurveThe area under the unit normal curve may represent several things likethe probability of a...
ExamplesExample no. 1        Find the are between z = 0 and z = +1Solution:        From the table, we locate z = 1.00 and ...
Example no. 2         Find the area between z = -1 and z = 0Solution         As you can see, there is no negative value of...
Example no. 3          Find the area below z = -1Solution:          Since the whole area under the curve is 1, then the wh...
Example no. 4         Find the area between z = -0.70 and z = 1.25Solution:         The area between z = -0.70 and z = 0 i...
Example no. 5        Find the area between z = 0.68 and z = 1.56.Solution:        The area between z = 0 and z = 0.68 is 0...
Activity : Plot the followingI.   Find the Z SCORE   II. Find the area under the unit                             normal c...
The Normal Distribution Curve
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The Normal Distribution Curve

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The Normal Distribution Curve

  1. 1. Unit Normal DistributionThis is the simplest of the family of Normal Distributions, also calledthe z distribution. It is a distribution of a normal random variable with amean equal to zero (μ = 0) and a standard deviation equal to one (μ = 1).It is represented by a normal curve.Characteristics:-It is symmetrical about the vertical line drawn through z = 0-the curve is symptotic to the x-axis. This means that both positive and negative ends approach the horizontal axis but do not touch it.-Mean, Median, and Mode coincide with each other.
  2. 2. The Z ScoreThe number of standard deviations from the mean is called the z-scoreand can be found by the formula: z= x-x SD Where: z= the z score x= raw score SD= standard deviation
  3. 3. ExampleFind the z-score corresponding to a raw score of 132 from a normaldistribution with mean 100 and standard deviation 15.SolutionWe compute 132 - 100 z = ________ = 2.133 15 -2 -1 0 +1 +2.133 z
  4. 4. A z-score of 1.7 was found from an observation coming from anormal distribution with mean 14 and standard deviation3. Find the raw score.SolutionWe have x - 14 1.7 = _______ 3To solve this we just multiply both sides by the denominator3, (1.7)(3) = x - 14 5.1 = x - 14 x = 19.1
  5. 5. Area Under the Unit Normal CurveThe area under the unit normal curve may represent several things likethe probability of an event, the percentile rank of a score, or thepercentage distribution of a whole population. For example, the areunder the curve from z = z1 to z = z2, which is the shaded region in figure7.6, may represent the probability that z assumes a value between z 1 andz 2. Fig. 7.6 The Probability That z1 and z2. z1 z2
  6. 6. ExamplesExample no. 1 Find the are between z = 0 and z = +1Solution: From the table, we locate z = 1.00 and get the corresponding areawhich is equal to o.3413 2nd 0.3413 1 3 s r t d 1.0 0.3413 0 +1
  7. 7. Example no. 2 Find the area between z = -1 and z = 0Solution As you can see, there is no negative value of z, so we need thepositive value. Hence, the area is also 0.03413 0.3414 -1 0
  8. 8. Example no. 3 Find the area below z = -1Solution: Since the whole area under the curve is 1, then the whole areais divided into two equal parts at z = 0. This means that the area to theleft of z = 0 is 0.5. To get the area below z = -1 means getting the area tothe left of z = -1. The area below z = -1 is then equal to 0.5000 – 0.3414= 0.1587. 0.1587 -1 0
  9. 9. Example no. 4 Find the area between z = -0.70 and z = 1.25Solution: The area between z = -0.70 and z = 0 is 0.2580, while thatbetween z = 0 and z = 1.25 is 0.3944. Therefore, the area between z = -0.70 and z = 1.25 is 0.2580 + 0.3944 = 0.6524. We add the two areas sincethe z values are on both side of the distribution. 0.6524 -0.7 0 1.25
  10. 10. Example no. 5 Find the area between z = 0.68 and z = 1.56.Solution: The area between z = 0 and z = 0.68 is 0.2518, while the areabetween z = 0 and z = 1.56 is 0.4406. Since the two z values are on thesame side of the distribution, we get the difference between the twoareas. Hence, the area between z = 0.68 and z = 1.56 is 0.4406 – 0.2518= 0.1888. 0.1888 0 0.68 1.56
  11. 11. Activity : Plot the followingI. Find the Z SCORE II. Find the area under the unit normal curve for the following1. Raw Score = 128 values of z. Mean = 95 SD = 3 1. Below z = 1.05 2. Above z = 1.522. Raw Score = 98 3. Above z = -0.44 Mean = 112 4. Below z = 0.23 SD = 1.5 5. Between z = -0.75 and z = 2.02 6. Between z = -0.51 and z = -2.173. Raw Score = 102 7. Between z = -1.55 and z = 0.55 Mean = 87 SD = 1.8
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