Answer:
(a): The 95% confidence interval is (46.4, 53.6)
(b): The 95% confidence interval is (47.9, 52.1)
(c): Larger sample gives a smaller margin of error.
Step-by-step explanation:
Given
 -- sample size
 -- sample size
 -- sample mean
 -- sample mean
 --- sample standard deviation
 --- sample standard deviation
Solving (a): The confidence interval of the population mean
Calculate the standard error




The 95% confidence interval for the z value is:

Calculate margin of error (E)



The confidence bound is: 



 --- approximated
 --- approximated



 --- approximated
 --- approximated
<em>So, the 95% confidence interval is (46.4, 53.6)</em>
Solving (b): The confidence interval of the population mean if mean = 90
First, calculate the standard error of the mean




The 95% confidence interval for the z value is:

Calculate margin of error (E)



The confidence bound is: 



 --- approximated
 --- approximated



 --- approximated
 --- approximated
<em>So, the 95% confidence interval is (47.9, 52.1)</em>
Solving (c): Effect of larger sample size on margin of error
In (a), we have:
 
      
In (b), we have:
 
    
<em>Notice that the margin of error decreases when the sample size increases.</em>