A random sample of 64 SAT scores of students applying for merit
scholarships showed an average of 1400 with a standard deviation of
240. Provide a 95% confidence interval for the SAT scores of all the
students who applied for the merit scholarships.

Hi lanee19!!


I will start with some concepts that lead us to the solution of the problem:

What we want is that the confidence level be 0.95. 
Supose that we have a standard normal variable, in that case we need
to find Q that:
P(Z > Q) = 0.025. 

Note that due the symmetry of the normal distribution around 0, if Q
is such that P(Z > Q) = 0.025 , then we will have that P(Z < -Q) =
0.025.

Working with that probabilities we can conclude that:
P(-Q < Z < Q) = 1 - P(Z > Q) - P(Z < -Q) =
              = 1 - 0.025 -0.025 =
              = 0.95
This means that the interval (-Q,Q) defines a 95% confidence interval.

Use the following table to find the value of Z that correspond to a
probability of 0.025. You will find Z = 1.96
"z-distribution Table":
http://www.math2.org/math/stat/distributions/z-dist.htm

Then for a standard normal variable the interval (-1.96,+1.96) defines a 95%
confidence interval.

If you have a large number n of sample size (say n > 30) and knowing
the Mean and the STD (standard deviation) we can calculate the
extremes of the 95% confidence interval by the formula:

Mean +/- (1.96 * STD)/sqrt(n) 


For this problem we have that:

n = 64 (n > 30)
Mean = 1400
STD = 240

The extremes of the confidence interval are:

left extreme = 1400 - (1.96 * 240)/sqrt(64) =
             = 1400 - 58.8 =
             = 1341.2

right extreme = 1400 + (1.96 * 240)/sqrt(64) =
              = 1400 + 58.8 =
              = 1458.8

Then the 95% confidence interval for the SAT scores of all the
students who applied for the merit scholarships is:
(1341.2, 1458.8)

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For reference visit the following page:
"ConŻdence Intervals":
http://math.cofc.edu/faculty/diamond/104confint.pdf

-----------------------------------------------------------

I hope that this helps you. If you find something unclear, wrong
and/or incomplete, please let me know by using the clarification
request, I will gladly respond to your requests.


Best regards.
livioflores-ga

Thank you, everything was perfectly correct!