Part 1) Answer the following questions:
1. In 2005, 1,475,623 students heading to college took the SAT. The distribution of scores in the math section of the SAT follows a normal distribution with mean =520 and standard deviation =115.
1) Calculate the -score for an SAT score of 720.
2) What percentile is 720?
3)What math SAT score is 1.5 standard deviations above the mean
4) What can you say about this SAT score?
2. 1)For 2012, the SAT math test had a mean of 514 and standard deviation 117. If one person took the SAT math test and scored 700 , what percentile is this score?
2) The ACT math test is an alternate to the SAT and is approximately normally distributed with mean 21 and standard deviation 5.3. If a second person took the ACT math test and scored 30, what percentile is this score.
3) Which person performed better?
part 2) Normal Law for Uniform Random Variable.
(i) For a uniform random variable X on (0,1), what are the mean and stdev of X?
(ii) For a uniform random sample X1, …, Xn , what are the mean and stdev of (X1.+…+ Xn)/n?
(iii) In Excel, rand( ) and randarray( ) can be used to generate uniform random variables for one or high dimensions.
a) In Excel, in A1, type in “n=1”
In A2 type in “=rand ( )” and then copy to A3:A201. (This generates 200 samples from uniform(0,1) for n=1)
b) In B1, type in “n=10”
In B2 type in “=average(randarray(10))” and copy to B3:B201. (this generates 200 sample means for n=10)
c) In C1, type in “n=30”
In C2 type in “=average(randarray(30))” and copy to C3:C201. (this generates 200 sample means for n=30)
(iv) a) highlight A2:A201 –>insert –> chart –> all charts –> histogram
b) highlight B2:B201–>insert –> chart –> all charts –> histogram
c) highlight C2:C201–>insert –> chart –> all charts –> histogram
Compare the shapes of these four histograms. Do you see the normal law?
(v) Calculate the sample means and sample standard deviations for each sample.
For example, for a) in A202, you can type in “=average(A2:A201)” and in A202, you can type in “=stdev(A2:A201)”
Compare the simulated means and stdevs with the theoretical values in (ii).
