Week 4 – Continuous Distributions

386331. What is the area under a standard normal curve? Why is it that number?

2. A statistics student was about to hand in some homework when his friend pointed at his answers and said, “You may want to recheck those.” The friend saw this:

The probability of the event is
a. 1/3
b. 1.5
c. -0.9
d. 0.77

Which answer(s) was the friend concerned about, and why?

3. Explain what the mean µ tells us about a normal curve, and explain what the standard deviation σ tells us about a normal curve. What is the relationship between the two? (Focus on the height of the peak/ how wide or narrow the curve is – for example, as the standard deviation increases, the height of the peak___________________)

4. Explain how to compute the z value. Start with the formula. Tell us what each variable means. Then, explain what the z value tells us about the value of the random variable. Use the numbers in 5a (below) to aid in your explanation.

5. Let x be a normally distributed random variable with µ=50 and σ=1.36. Find the z value for each of the following observed values of x: Round to TWO decimal places if needed. Show your work.

a. X = 51.14
b. X = 50
c. X = 53.01
d. X = 52.1
e. X = 48.8

6. In #5,
a. which response (a, b, c, d, e) is the most uncommon due to how far it is from the mean?
b. which response(s) are within one standard deviation of the mean? How do you know?

7. If the random variable z has a standard normal distribution, sketch (insert your photos of your sketches) and find each of the following probabilities using a z table. You may need to reference an online z table.
a. P( -2 < z <0.46)
b. P( z 1.68)

Examples of what sketches might look like (but you need to add a scale of standard deviation at the bottom from -3 to +3):

7. Weekly demand at a grocery store for a brand of breakfast cereal is normally distributed with a mean of 275 boxes and a standard deviation of 25 boxes. What is the probability that the weekly demand is:
a. 250 boxes or less?
b. More than 310 boxes?
c. Between 240 and 320 boxes?
** Show your problem set up and work for #7