It is said that happy and healthy workers are more efficient

It is said that happy and healthy workers are more efficient and productive. A company that manufactures exercising machines claims that only 30 percentage a companies that provides such facilities on site. A sample of 240 such companies Los Angeles showed that 80 of them provide such facilities on site.

Solution

a)

Note that              
              
p^ = point estimate of the population proportion = x / n =    0.333333333          
              
Also, we get the standard error of p, sp:              
              
sp = sqrt[p^ (1 - p^) / n] =    0.030429031          
              
Now, for the critical z,              
alpha/2 =   0.015          
Thus, z(alpha/2) =    2.170090378          
Thus,              
Margin of error = z(alpha/2)*sp =    0.066033747          
lower bound = p^ - z(alpha/2) * sp =   0.267299586          
upper bound = p^ + z(alpha/2) * sp =    0.399367081          
              
Thus, the confidence interval is              
              
(   0.267299586   ,   0.399367081   ) [ANSWER]

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b)

As we can see in the confidence interval, 0.30 is inside it. So, there is no significant evidence that the proportion of companies that provide such facilities on site is distinct from the overall proportion. [CONCLUSION]