assume data is normally distributed round answers to three d

(assume data is normally distributed, round answers to three decimal places)

Solution

a)

We first get the z score for the two values. As z = (x - u) sqrt(n) / s, then as          
x1 = lower bound =    9      
x2 = upper bound =    12      
u = mean =    13      
n = sample size =    45      
s = standard deviation =    15      
          
Thus, the two z scores are          
          
z1 = lower z score = (x1 - u) * sqrt(n) / s =    -1.788854382      
z2 = upper z score = (x2 - u) * sqrt(n) / s =    -0.447213595      
          
Using table/technology, the left tailed areas between these z scores is          
          
P(z < z1) =    0.036819135      
P(z < z2) =    0.327360423      
          
Thus, the area between them, by subtracting these areas, is          
          
P(z1 < z < z2) =    0.290541288   [ANSWER]

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b)

We first get the z score for the critical value. As z = (x - u) sqrt(n) / s, then as          
          
x = critical value =    13      
u = mean =    13      
n = sample size =    45      
s = standard deviation =    15      
          
Thus,          
          
z = (x - u) * sqrt(n) / s =    0      
          
Thus, using a table/technology, the right tailed area of this is          
          
P(z >   0   ) =    0.5 [ANSWER]

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c)

First, we get the z score from the given left tailed area. As          
          
Left tailed area =    0.025      
          
Then, using table or technology,          
          
z =    -1.959963985      
          
As x = u + z * s / sqrt(n)          
          
where          
          
u = mean =    13      
z = the critical z score =    -1.959963985      
s = standard deviation =    15      
n = sample size =    45      
Then          
          
x = critical value =    8.617387297   [ANSWER]