Please include all R code and R output.

Problem 1
SAT scores (out of 1600) of high school students in Illinois are distributed normally with mean of 1100 and a standard deviation of 200.
1. Suppose we award a certicate of excellence to all students who score at least 1350 on the SAT. What percentage of students will be awaraded from Illinois? (Hint: use pnorm)
2. Let's randomly select 100 students from Illinois. Dene the sampling distribution of mean SAT scores for 100 students. Specify the parameters.
Problem 2
1. Randomly generate 1,000 samples from Normal distribution with mean 5 and standard deviation 1 using rnorm after set.seed(1) and save these samples in norm.sample.
Draw a histogram of norm.sample with "breaks=50" and "probability=TRUE". What is the sample mean and sample variance of norm.sample?
2. Create a vector called z.scores by computing the Z scores for each of the elements in norm.sample. Again draw a histogram of z.scores with "breaks=50" and "probabil-ity=TRUE". What is the sample mean and sample variance of z.scores?
3. Let's compare the histograms of norm.sample and z.scores. How did the sample mean and sample variance change from norm.sample to z.scores?
4. Now increase the sample size to 100,000 and repeat 1 and 2. Name the vector of those samples as norm.sample2 and z.scores2 respectively.
5. Draw a probability distribution (curve) for standard normal distribution using function the dnorm. Place the plot of the probability distribution (curve) for standard normal and the histogram of z.scores2 side by side on the same row. What can you conclude?
(Hint : When you draw a curve for standard normal distribution let the domain of x to be x = seq(-3, 3, length=1000) )
Problem 3
Height of 10 years olds, regardless of gender, closely follow a normal distribution with mean 55 inches and standard deviation 6 inches.
1. What is the probability that a randomly chosen 10 year old is shorter than 48 inches?
2. What is the probability that a randomly chosen 10 year old is between 60 and 65 inches?
3. The height requirement for Batman the Ride at Six Flags Magic Mountain is 54 inches.
What percent of 10 year olds cannot go on this ride?
Problem 4
In Graduate Record Examination (GRE) suppose we want to look at percentage of answer-ing one particular question correctly. The percentage of answering that specic question correctly is 35%. Let's randomly generate 100 people among the GRE test takers. Denote X the number of people who answered this problem correctly among 100 people.
1. What is the probability distribution for X? Specify the parameters of that distribution.
2. What is the probability that less than 40 people answer this question correctly? (Hint: Use pbinom or dbinom)
3. Using Central Limit Theorem, if you were to write X as a normal distribution, what would be the mean and variance of X? (Hint : See the lecture note 3.4 page 26)
4. Using the normal approxmiation what is the probability that P(30≤ X≤ 40)?
Problem 5
Suppose a newspaper article states that the distribution of auto insurance premiums for residents of California is approximately normal with a mean of $1,650. The article also states that 25% of California residents pay more than $1,800.
1. What is the Z-score that corresponds to the top 25% of the standard normal distribu-tion? (Hint: Use qnorm)
2. What is the mean insurance cost? What is the cuto for the 75th percentile?
3. Identify the standard deviation of insurance premiums in LA.