Entry represents area under the cumulative standardized norm

Entry represents area under the cumulative standardized normal distribution from -infinity to Z How much is P(-0.1 LE Z,5), where X N(2.4)? (rounded to four decimal Places) How much is the positive number a, such that for Z N(0,1), P(|Z|>a)=03628?(rounded to two decimal places) How much is the value of b, such that for X N(3LEXLEb)=0.1952?(rounded to one decimal place)

Solution

1.

z1 = lower z score =    -0.1      
z2 = upper z score =     2      
          
Using table/technology, the left tailed areas between these z scores is          
          
P(z < z1) =    0.460172163      
P(z < z2) =    0.977249868      
          
Thus, the area between them, by subtracting these areas, is          
          
P(z1 < z < z2) =    0.517077705   [ANSWER]

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2.

We first get the z score for the critical value. As z = (x - u) / s, then as          
          
x = critical value =    5      
u = mean =    2      
          
s = standard deviation =    4      
          
Thus,          
          
z = (x - u) / s =    0.75      
          
Thus, using a table/technology, the right tailed area of this is          
          
P(z >   0.75   ) =    0.226627352 [ANSWER]

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3.

Getting the outward areas of the two extreme tails,          
          
alpha/2 = (1-middle area)/2 =    0.1814      
          
Thus, the z values bounding these tails, using table/technology,          
          
z1 = lower z value =    -0.910042683      
z2 = upper z value =    0.910042683      

Hence, a = 0.910042683 [ANSWER]

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4.

Left tailed area of x = 3:

We first get the z score for the critical value. As z = (x - u) / s, then as          
          
x = critical value =    3      
u = mean =    3.2      
          
s = standard deviation =    4      
          
Thus,          
          
z = (x - u) / s =    -0.05      
          
Thus, using a table/technology, the left tailed area of this is          
          
P(z <   -0.05   ) =    0.480061194
  
Thus, the left tailed area of b should be 0.480061194 + 0.1952 = 0.675261194

First, we get the z score from the given left tailed area. As          
          
Left tailed area =    0.675261194      
          
Then, using table or technology,          
          
z =    0.454488021      
          
As b = u + z * s,          
          
where          
          
u = mean =    3.2      
z = the critical z score =    0.454488021      
s = standard deviation =    4      
          
Then          
          
b = critical value =    5.017952083   [ANSWER]