1 For a random variable that is normally distributed with 1

1. For a random variable that is normally distributed, with = 116.15 and = 25.4617, the probability that a simple random sample of 38 items will produce a mean that is greater than 119 is equal to

2. For a random variable that is normally distributed, with = 115.62 and = 33.72, the probability that a simple random sample of 36 items will produce a mean that is greater than 110 is equal to

Solution

1.

We first get the z score for the critical value. As z = (x - u) sqrt(n) / s, then as          
          
x = critical value =    119      
u = mean =    116.15      
n = sample size =    38      
s = standard deviation =    25.4617      
          
Thus,          
          
z = (x - u) * sqrt(n) / s =    0.690000271      
          
Thus, using a table/technology, the right tailed area of this is          
          
P(z >   0.690000271   ) =    0.245097008 [ANSWER]

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2.

We first get the z score for the critical value. As z = (x - u) sqrt(n) / s, then as          
          
x = critical value =    110      
u = mean =    115.62      
n = sample size =    36      
s = standard deviation =    33.72      
          
Thus,          
          
z = (x - u) * sqrt(n) / s =    -1      
          
Thus, using a table/technology, the right tailed area of this is          
          
P(z >   -1   ) =    0.841344746 [answer]