Answer:
a) The 99% confidence interval would be given by (24.409;24.979)  
b) n=464
Step-by-step explanation:
A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".  
The margin of error is the range of values below and above the sample statistic in a confidence interval.  
Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".  
Part a
The confidence interval for the mean is given by the following formula:  
 (1)
 (1)  
In order to calculate the critical value  we need to find first the degrees of freedom, given by:
 we need to find first the degrees of freedom, given by:  
 
  
Since the Confidence is 0.95 or 95%, the value of  and
 and  , and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-T.INV(0.025,46)".And we see that
, and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-T.INV(0.025,46)".And we see that  
  
Now we have everything in order to replace into formula (1):  
 
  
 
  
So on this case the 95% confidence interval would be given by (503.01;546.99)
Part b
The margin of error is given by this formula:  
 (1)
 (1)  
And on this case we have that ME =7 msec, we are interested in order to find the value of n, if we solve n from equation (1) we got:  
 (2)
 (2)  
The critical value for 95% of confidence interval is provided,  from part a, replacing into formula (2) we got:
 from part a, replacing into formula (2) we got:  
 
  
So the answer for this case would be n=464 rounded up to the nearest integer