Assume that X is normal with mean 10 and standard deviation

Assume that X is normal with mean 10 and standard deviation 3. Find the value k such that

(a) P(X > k) = 0.5

(b) P(X > k) = 0.95

(c) P(k < X < 10) = 0.2

(d) P(k < X 10 < k) = 0.95

(e) P(k < X 10 < k) = 0.99

Solution

a)

First, we get the z score from the given left tailed area. As          
          
Left tailed area =    0.5      
          
Then, using table or technology,          
          
z =    0      
          
As k = u + z * s,          
          
where          
          
u = mean =    10      
z = the critical z score =    0      
s = standard deviation =    3      
          
Then          
          
k = critical value =    10   [ANSWER]

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b)

First, we get the z score from the given left tailed area. As          
          
Left tailed area =    0.05      
          
Then, using table or technology,          
          
z =    -1.644853627      
          
As k = u + z * s,          
          
where          
          
u = mean =    10      
z = the critical z score =    -1.644853627      
s = standard deviation =    3      
          
Then          
          
k = critical value =    5.065439119   [ANSWER]

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c)

As 0.5 of the values are less than 10, then the left tailed value of k here is 0.5 - 0.2 = 0.3.

First, we get the z score from the given left tailed area. As          
          
Left tailed area =    0.3      
          
Then, using table or technology,          
          
z =    -0.524400513      
          
As k = u + z * s,          
          
where          
          
u = mean =    10      
z = the critical z score =    -0.524400513      
s = standard deviation =    3      
          
Then          
          
k = critical value =    8.426798462   [ANSWER]

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d)

As the variable is now X-10, the mean is now shifted by 10 units. The new mean = 0.

This is the middle 95% of values.

Now, by symmetry, the right tailed area of k is 0.975.

First, we get the z score from the given left tailed area. As          
          
Left tailed area =    0.975      
          
Then, using table or technology,          
          
z =    1.959963985      
          
As k = u + z * s,          
          
where          
          
u = mean =    0      
z = the critical z score =    1.959963985      
s = standard deviation =    3      
          
Then          
          
k = critical value =    5.879891954   [ANSWER]

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e)

As the variable is now X-10, the mean is now shifted by 10 units. The new mean = 0.

This is the middle 99% of values.

Now, by symmetry, the right tailed area of k is 0.995.

First, we get the z score from the given left tailed area. As          
          
Left tailed area =    0.995      
          
Then, using table or technology,          
          
z =    2.575829304      
          
As k = u + z * s,          
          
where          
          
u = mean =    0      
z = the critical z score =    2.575829304      
s = standard deviation =    3      
          
Then          
          
k = critical value =    7.727487911