What two properties must be satisfied by a continuous probability distribution / probability curve?

1. What two properties must be satisfied by a continuous probability distribution / probability curve?
2. Explain what the mean µ tells us about a normal curve, and explain what the standard deviation σ tells us about a normal curve.
3. Explain how to compute the z value. What does the z value tell us about the value of the random variable?
4. Let x be a normally distributed random variable with µ=30 and σ=5. Find the z value for each of the following observed values of x:
   a. X = 25
   b. X = 15
   c. X = 30
   d. X = 40
   e. X = 50
5. If the random variable z has a standard normal distribution, sketch and find each of the following probabilities.
   a. P( 0 < z 2)
   c. P( z < 1.7)
   d. P( z < -1.6)
6. Weekly demand at a grocery store for a brand of breakfast cereal is normally distributed with a mean of 800 boxes and a standard deviation of 75 boxes. What is the probability that the weekly demand is:
   a. 960 boxes or less?
   b. More than 1005 boxes?
   c. Between 750 and 850 boxes?
7. What does the Central Limit Theorem tell us about the sampling distribution and sample mean?
8. In each of the following cases, determine whether the sample size n is large enough to say that the sampling distribution of p (p hat) is a normal distribution.
   a. P=0.4; n=100
   b. P=0.1; n=10
   c. P=0.1, n=50