1. What is a continuous distribution - properties and finding probabilities.
2. What is a normal distribution - properties and construction.
3. Identify if a random variable follows a normal distribution or not.
4. What is a standardized normal distribution and the relationship between the standardize normal random variable and any given normal random variable.
5. How to use the normal table.
6. Apply normal random variables in real world situations.

Terms: Normal random variable, Standardized normal random variable, Z, Bell-shape distribution.

1. Class Discussion Problems

1. Let X be the SAT scores. Based on the previous record, the distribution shape of the SAT scores is symmetric about the mean, and most of scores are around the mean and a few scores are away from the mean.

The probability is the area underneath the curve within the interval of interest.

For example, The proportion of SAT scores between 450 and 700 is denoted as P(450 < X < 700).

NOTE: For any continuous r.v.,

P(a < X < b) = P(a <= X < b) = P(a < X <= b) = P(a <= X < = b)

*** The Equality sign does not contribute to the probability in continuous case. ***

This says: P(X = a ) = 0.

1. What is the proportion of SAT scores to be higher than 650?

(Hint:) You can do this using Empirical Rule.

2. What is the probability of having an SAT score between 350 and 550?

(Hint) You can also do this by empirical rule.

How do you solve the following ones:

3. What is the probability of having an SAT score between 450 and 600?
4. What is the probability of having an SAT score higher than 750?
5. Find median, Q1 SAT scores.
6. CMU will admit the top 60% of students. What is the minimum SAT scores for the admission to CMU?
7. A student scored 720. She claimed it is in the top 5%. Is her claim correct?

3. Z~N(0,1). Find P(0 < Z > 1.28), P(Z > 1.28) , P(Z < -1.28)

4. X~N(-20,10) Then, Is it true for each of the following statements:

1. P(X > -20) = .5,
2. (b) P(X > 0) = P(X < 0),
3. (c) P(X > -40) = P( X < 0),
4. (d) P( X > -20) < P(X < 0)

5. According to experience, the computer price follows approximately a normal distribution with average price $1200 and a s.d. $300.

1. What will the probability that a PC will cost $1550 or higher?
2. What is the 25^th percentile PC price?
3. Your PC costs you $900. What is the proportion of PCs having prices lower than yours?
4. A computer price analyst indicated that the average PC price will be about 40^th percentile of the current prices by the end of the year. What will be the average PC price by the end of the year based on this analyst?
5. If a PC price is outside two s.d. of the average price, it is considered rare. Is a PC of $800 considered as a rare price?

1. Give three r. v. that reasonably follow normal distribution, and three variables which do not follow normal distributrion.

2. Z ~ N(0,1).

Find P(0 < Z < 1.96), P(Z < 1.96), P( Z < -1.96), P(Z > 1.96), P(-1.96 < Z < 0)

Find zo so that P(0 < Z < zo) = .24

Find zo so that P(Z > zo) = .975

Find Q3

3. X ~ (50, 20)

Find the corresponding z-value for X = -20, 0, 20, 50, 80, 110

Find P(20 < X < 60), P(10 < X < 40), P( X > 85), P( X < 60)

Find xo so that P(X < xo) = .025

Find xo so that P(20 < X < xo) = .75

4. According to experience, the time for using computer per day for students follows approximately a normal distribution with average 100 minutes and a s.d. 40 minutes.

1. What will the probability that a student will be on computer for more than 3 hours per day?
2. What is the 25^th percentile of the amount of time on computer for students?
3. You are on computer for 30 minutes per day. What is the proportion of students who spend less time on computer than you?
4. If the amount of time spent on computer is either in the top 1% or low 1% of user are considered rare, what will be the amount of time on computer to be considered rare on the higher end and lower end, respectively?