Let x represent the scores of an achievement test given to 1

Let x represent the scores of an achievement test given to 100 students. assume that x can be modeled as a guassian random variable with the mean of 60 and standard deviation of 8. calculate

a) Probability that your score is within 8 points of the mean

b)probability that your socre is either below or above the mean by at least 10 points

c) what should your score be to be placed in the top 10% of the class?

Solution

a)

We first get the z score for the two values. As z = (x - u) / s, then as          
x1 = lower bound = 60 - 8 =    52      
x2 = upper bound = 60 + 8 =    68      
u = mean =    60      
          
s = standard deviation =    8      
          
Thus, the two z scores are          
          
z1 = lower z score = (x1 - u)/s =    -1      
z2 = upper z score = (x2 - u) / s =    1      
          
Using table/technology, the left tailed areas between these z scores is          
          
P(z < z1) =    0.158655254      
P(z < z2) =    0.841344746      
          
Thus, the area between them, by subtracting these areas, is          
          
P(z1 < z < z2) =    0.682689492   [ANSWER]

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b)

We first get the z score for the two values. As z = (x - u) / s, then as          
x1 = lower bound = 60 - 10 =    50      
x2 = upper bound = 60 + 10 =    70      
u = mean =    60      
          
s = standard deviation =    8      
          
Thus, the two z scores are          
          
z1 = lower z score = (x1 - u)/s =    -1.25      
z2 = upper z score = (x2 - u) / s =    1.25      
          
Using table/technology, the left tailed areas between these z scores is          
          
P(z < z1) =    0.105649774      
P(z < z2) =    0.894350226      
          
Thus, the area between them, by subtracting these areas, is          
          
P(z1 < z < z2) =    0.788700453      

Thus, those outside this interval is the complement =    0.211299547   [ANSWER]

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c)

First, we get the z score from the given left tailed area. As          
          
Left tailed area =    0.9      
          
Then, using table or technology,          
          
z =    1.281551566      
          
As x = u + z * s,          
          
where          
          
u = mean =    60      
z = the critical z score =    1.281551566      
s = standard deviation =    8      
          
Then          
          
x = critical value =    70.25241252   [ANSWER]