##### R Project 4: Normal Distribution
##### Name:
##### Version Number:


#############################
###### PART 1 ###############
###### GRAPH  ###############
#############################

## DENSITY FUNCTION
	## C1: Create x-values - code
  xvalues_part1 <- seq(-3.7,3.7,by = 0.01)


	## C2: Create y-values - code
  yvalues_part1 <- dnorm(xvalues_part1, mean = 0, sd = 1)


	## C3: Create Plot - code
	## Remember to save your plot and also submit it to Gradescope.
  plot(xvalues_part1, yvalues_part1, type = "l", main = "Standard Normal Probability Density Function (PDF)", xlab = "Variable", ylab = "Density", col = "paleturquoise3")



#############################
###### PART 1 ###############
###### QUESTIONS  ###########
#############################

## Question 4: Largest approximate y value
	# Answer: 0.398942
  largest_y_value <- dnorm(0, mean = 0, sd = 1)





## Question 5: Why stop at the x-values from C1 instead of something like -1 and +1?
	# Answer: There are a couple of reasons to stop at the x-values from C1 instead of something like -1 and +1. One of the reasons is that it covers more than 99.7% of the data. Another reason is that you can observe the behavior of the tails.





## Question 6: When calculating a probability, how would this be represented on the graph?
	# Answer: If I were to calculate a probability based on this distribution, I would represent a probability based on the standard normal distribution on the standard normal probability density function graph.Particularly, I would visualize the area under the curve within a specific range of x-values.





## Question 7: Standard Normal Questions
	## a) What is the mean and variance of the standard normal distribution?
			## Mean = 0
			## Variance = 1


	## b) What random variable abbreviation do we usually use to represent the standard normal distribution?
			## Answer: Z


	## c) Based on graph in Part 1, what do the values on the horizontal axis represent?
			## Answer: Based on graph in Part 1, the values on the horizontal axis represent z-scores.






#############################
###### PART 2 ###############
###### GRAPH  ###############
#############################

## CUMULATIVE DISTRIBUTION FUNCTION
## X ~ N(mean = ????; variance = ????) (see PDF for mean and variance values)


	## C8: Create x-values - code
  xvalues_part2 <- seq(410, 550, by = 7)


	## C9: Create y-values - code
  yvalues_part2 <- pnorm(xvalues_part2, mean = 480, sd = 23, lower.tail =  TRUE)


	## C10: Create Plot - code
  plot(xvalues_part2, yvalues_part2, type = "l", main = "Normal CDF", xlab = "x-values", ylab = "P(X <= x): Cumulative Probability", col = "tomato2")


	## C11: Cumulative Probabilities - code
  cumulative_probabilities <- c(0.17, 0.36, 0.50, 0.64, 0.83)


	## C12: Find x-values associated with cumulative probabilities - code
  quantile_for_k <- qnorm(cumulative_probabilities)


	## C13: Overlay points on plot - code
  points(quantile_for_k, cumulative_probabilities, pch = 24, bg = "lavenderblush3", col = "lavenderblush3")


	## C14: Add text at each point - code
	## Remember to save your plot and also submit it to Gradescope.  <- this is the only plot from Part 2 you need to submit.
  text(quantile_for_k, cumulative_probabilities, labels = paste("(", round(quantile_for_k,2), ", ", cumulative_probabilities,")", sep = ""), pos = 4)





#############################
###### PART 2 ###############
###### QUESTIONS  ###########
#############################

## Question 15: What do the y-values approach as x goes to +/- infinity?
	# As x goes towards -infinity: 0


	# As x goes towards +infinity: 1





## Question 16: Pick one of the points on your graph from Part 2.  Write a probability statement involving the $x$- and $y$- coordinate values that describes how they relate to each other. Do not use the value corresponding to y = 0.5.

	# Point you will use: (410,0.001169302)
  pnorm(410, mean = 480, sd = 23, lower.tail = TRUE)
	# Probability Statement: y = P(X <= 410)





## Question 17: Create and solve your own probability problem.
## Must be of the form where you solve for a particular value, given the probability.
## Do not use any of the points from Q16.
## Do not use the mean value.
## Include your code and your final answer.
## Do NOT use a table or your calculator.

	# Question: P(X <= a) = 0.25. Solve for a.


	# Code:


	# Answer: