The probability that an amateur archer hits a target 50 feet

The probability that an amateur archer hits a target 50 feet away is 20%. Suppose we take a sample of 100 archers.

a. What is the standard error of the proportion?

b. What is the probability that the proportion of archers who hit the target will be less than 10%?

Solution

a)

standard error of proportion (sp) = sqrt[p(1-p)/n]

= sqrt(0.20*(1-0.20)/100)

= 0.04 [answer]

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b)

We first get the z score for the critical value.

As z = (p - po) / sp, then as          
          
p = critical value =    0.1      
po = mean =    0.2      
          
sp = standard deviation =    0.04      
          
Thus,          
          
z =    -2.5      
          
Thus, using a table/technology, the left tailed area of this is          
          
P(z <   -2.5   ) =    0.006209665 [ANSWER]