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.
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
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
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
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
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
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
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
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
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
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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