1 For a random variable that is normally distributed with 1

1. For a random variable that is normally distributed, with = 115.57 and = 24.9251, the probability that a simple random sample of 35 items will produce a mean that is less than 113 is equal to

2. 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 less than 119 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 =    113      
u = mean =    115.57      
n = sample size =    35      
s = standard deviation =    24.9251      
          
Thus,          
          
z = (x - u) * sqrt(n) / s =    -0.610000563      
          
Thus, using a table/technology, the left tailed area of this is          
          
P(z >   -0.610000563   ) =    0.270930717 [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 =    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 left tailed area of this is          
          
P(z >   0.690000271   ) =    0.754902992 [ANSWER]