Given a population where the probability of sucesses where t

Given a population where the probability of sucesses where the pie = O.40, calculate the probabilities below if a sample of 300 is taken.

A)Calculate that the probability the portion of succeses in the sample will be less than 0.42.

B)What is the probability the proportion of sucesses in the sample will be greater than 0.43.

round four decimal places as needed.

Solution

Here, the standard deviation of proportions is

s(p^) = sqrt(p(1-p)/n) = sqrt(0.40*(1-0.40)/300) = 0.028284271

a)

We first get the z score for the critical value. As z = (x - u) / s, then as          
          
x = critical value =    0.42      
u = mean =    0.4      
          
s = standard deviation =    0.028284271      
          
Thus,          
          
z = (x - u) / s =    0.707106787      
          
Thus, using a table/technology, the left tailed area of this is          
          
P(z <   0.707106787   ) =    0.760249941 [ANSWER]

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b)

We first get the z score for the critical value. As z = (x - u) / s, then as          
          
x = critical value =    0.43      
u = mean =    0.4      
          
s = standard deviation =    0.028284271      
          
Thus,          
          
z = (x - u) / s =    1.060660181      
          
Thus, using a table/technology, the right tailed area of this is          
          
P(z >   1.060660181   ) =    0.144422181 [ANSWER]