Answer:
The 95% confidence interval for the mean would be given by (56.604;68.596)
The 99% confidence interval for the mean would be given by (53.196;72.004)
Step-by-step explanation:
1) Previous concepts
When we compute a confidence interval for the mean, we are interested on the parameter population mean, and we use the info from the sample to estimate this parameter.
2) Basic operations
The sample mean can be calculated with the following formula
Using excel we can use this function to calculate the mean:
=AVERAGE(54.2,59.8,61.8,63.3,65.1,71.4)
The value obtained is 
In order to find the sample deviation we can use this formula
And using excel we can use this function to calculate the sample standard deviation:
=STDEV.S(54.2,59.8,61.8,63.3,65.1,71.4)
The value obtained is 
The sample size for this case is n=6, n<30 so then is better use the t distribution to calculate the margin of error. First we need to calculate the degrees of freedom, on this case 
The formula for the confidence interval would be given by:
(1)
3) Part a
If we want a 95% or 0.95 of confidence, then the value for the signficance is 
, and 
, and 
so we can find the critical t value with the following formula in excel:
=T.INV(0.975,5)
And we got 
And we can replace into equation (1) and we got:
And using excel with the following formulas we got:
=62.6-2.571*(5.713/SQRT(6)) = 56.604
=62.6+2.571*(5.713/SQRT(6)) = 68.596
So the 95% confidence interval for the mean would be given by (56.604;68.596)
4) Part b
If we want a 99% or 0.99 of confidence, then the value for the signficance is 
, and 
, and 
so we can find the critical t value with the following formula in excel:
=T.INV(0.995,5)
And we got 
And we can replace into equation (1) and we got:
And using excel with the following formulas we got:
=62.6-4.032*(5.713/SQRT(6)) = 53.196
=62.6+4.032*(5.713/SQRT(6)) = 72.004
So the 99% confidence interval for the mean would be given by (53.196;72.004)
