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BA 253: Business Statistics
Today
More Chapter 6
Normal Distribution
9/25/16
Next Week
More Normal
Sampling Distributions
Normal Distribution (Chapter 6)
Inches
Ex: Annual snowfall at DMR is normally distributed with mean 232 inches and standard
deviation 58 inches. Let X = snowfall for this upcoming season. Estimate:
a) P(X = 150) = 0%
b) P(X ≤ 150) = 8%
Annual Snowfall in Inches at DMR (μ = 232, σ = 58)
c) P(150 ≤ X ≤ 300) = 80%
d) P(X ≥ 300) = 12%
400
e) Middle 70%,
350
f) Top 15%.
300
o Standard Normal
250
Distribution: μ = 0, σ = 1.
200
x
150
o z

100
o Solve all…….
50
0
0
5
10
15
Season (1983/1984 to 2002/2003)
o http://davidmlane.com/hyperstat/z_table.html
Calculations:
1.
2.
3.
4.
Draw picture.
Calculate z-score
Look up z-score in correct table (may have to subtract from 100% or 50%)
Verify answer seems OK
z-score Calculations


P(1.00 ≤ z ≤ 2.00)
Which z-score has 65% to the right of it?
20
Example 2:
Historically, demand for a product is normally distributed with mean 550 and standard
deviation 82. Calculate:
a) P(D ≤ 550) and P(D ≥ 550)
b) Demand under 450 is considered “low.” What is the probability of low demand?
c) Demand between 450 and 600 is “typical.” What is this probability?
d) Demand between 600 and 700 is “high.” What is this probability?
e) The supplier only has 700 units on hand. If demand is greater than 700, the supplier
will have a shortage, which of course, is bad for business. What is the probability that
demand is greater than 700?
f) In order to avoid shortages (as in part e)), how much inventory should the supplier
keep on hand so that shortages only happen during the top 1% highest demand? In other
words, to be 99% certain that all demand will be met?
Examples of How the Normal Distribution is Used
1. What is the probability that Apple goes out of business?
a. µ = 35%, σ = 75%
b. P(X ≤ -100%)
2. To get into graduate school, you need a GRE score in the top 30%. The GRE
average is 1200 ± 200. What score do you need?
3. What percent of the Normal distribution is between z = ±1?
a. Does this agree with empirical rule?
b. How about between z = ±2.5?
4. In quality control (of a production process), the process is considered “in control”
if the outputs are within 3 standard deviations of the norm. How often is the
process out of control?
a. Further, another measure of quality is called “Six sigma.” How often is
six sigma out of control?
5. The best apples off a tree are big (obviously) but not too big – the huge ones don’t
taste as good. In fact, any apple between -0.5 ≤ z ≤ 2 is probably the best. What
percent of apples are the best?