A quality control worker in a Beanie Babies factory sampled

A quality control worker in a Beanie Babies factory sampled 150 toys and found that 25 were defective.

What is the standard error of the proportion of defective toys?

What is the probability that 20% (.2) or less of the population of toys is defective?

What is the probability that 10% (.1) or less of the population of toys is defective?

What is the probability that more than 25% (.25) of the population of toys is defective?

Based on this information, what is the 95% Confidence Interval for the true proportion of defective toys?

Solution

What is the standard error of the proportion of defective toys?

As

p^ = 25/150 = 0.166666667

Then

SE = sqrt(p^ (1-p^) / n)

= sqrt(0.166666667*(1-0.166666667)/150)

= 0.030429031 [ANSWER]

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What is the probability that 20% (.2) or less of the population of toys is defective?

We first get the z score for the critical value. As z = (x - u) / s, then as          
          
x = critical value =    0.2      
u = mean =    0.166666667      
          
s = standard deviation =    0.030429031      
          
Thus,          
          
z = (x - u) / s =    1.095445114      
          
Thus, using a table/technology, the left tailed area of this is          
          
P(z <   1.095445114   ) =    0.863339161 [ANSWER]

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What is the probability that 10% (.1) or less of the population of toys is defective?

We first get the z score for the critical value. As z = (x - u) / s, then as          
          
x = critical value =    0.1      
u = mean =    0.166666667      
          
s = standard deviation =    0.030429031      
          
Thus,          
          
z = (x - u) / s =    -2.190890228      
          
Thus, using a table/technology, the left tailed area of this is          
          
P(z <   -2.190890228   ) =    0.014229869 [ANSWER]

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What is the probability that more than 25% (.25) of the population of toys is defective?

We first get the z score for the critical value. As z = (x - u) / s, then as          
          
x = critical value =    0.25      
u = mean =    0.166666667      
          
s = standard deviation =    0.030429031      
          
Thus,          
          
z = (x - u) / s =    2.738612785      
          
Thus, using a table/technology, the right tailed area of this is          
          
P(z >   2.738612785   ) =    0.00308495 [ANSWER]

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Based on this information, what is the 95% Confidence Interval for the true proportion of defective toys?

Note that              
              
p^ = point estimate of the population proportion = x / n =    0.166666667          
              
Also, we get the standard error of p, sp:              
              
sp = sqrt[p^ (1 - p^) / n] =    0.030429031          
              
Now, for the critical z,              
alpha/2 =   0.025          
Thus, z(alpha/2) =    1.959963985          
Thus,              
Margin of error = z(alpha/2)*sp =    0.059639805          
lower bound = p^ - z(alpha/2) * sp =   0.107026862          
upper bound = p^ + z(alpha/2) * sp =    0.226306471          
              
Thus, the confidence interval is              
              
(   0.107026862   ,   0.226306471   ) [ANSWER]