Suppose all the us womens height is a normal distribution N

Suppose all the us women\'s height is a normal distribution N (64,2.7) A SRS of size 9 is taken from from them what is the probability of the average height of this sample is less that 63?

The time X that a light bulb can last is a distribution with mean 1 year and standard deviation 0.5 year. Suppose we have 100 such light bulbs what is the probability that the average life length of the light bulb is greater than 1.02 year?

Solution

a)

We first get the z score for the critical value. As z = (x - u) sqrt(n) / s, then as          
          
x = critical value =    63      
u = mean =    64      
n = sample size =    9      
s = standard deviation =    2.7      
          
Thus,          
          
z = (x - u) * sqrt(n) / s =    -1.111111111      
          
Thus, using a table/technology, the left tailed area of this is          
          
P(z <   -1.111111111   ) =    0.133260263 [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 =    1.02      
u = mean =    1      
n = sample size =    100      
s = standard deviation =    0.5      
          
Thus,          
          
z = (x - u) * sqrt(n) / s =    0.4      
          
Thus, using a table/technology, the right tailed area of this is          
          
P(z >   0.4   ) =    0.344578258 [ANSWER]