BUSOBA 2320: Normal Distribution MATH

OPTIONAL PRACTICE

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For the normal probability distribution, the area to the left of the mean is

.5

Area (bell curve) is always

positive. The Z-score could be negative, but not the area. 

Because the Normal distribution is symmetric, the mean = median is

the point of symmetry. Half of the area under the curve is above the mean and half is below the mean.

Characteristics of Normal Probability Distribution:

The mean, median, and the mode are equal

The mean of the distribution can be negative, zero, or positive

The distribution is symmetrical

The standard deviation can be

any positive value. σ > 0. Doesn’t have to be 1. 

In a standard normal distribution, the range of values of Z is from

-∞ to +∞ 

Standard Normal Distribution Definition:

All Normal variables can have any value between minus infinity and positive infinity. This is the theoretical definition.

If given a question, like this “Z ~ N(0. 1). P(-1.80 < Z < -1.00) = “, the line underneath the bell curve will be

• z.

  • Also, will use interior areas tables, find probability between 0 and -1.80 AND 0 and -1.00, then will subtract the two to find answer. 

Z ~ N(0,1). P(Z < a) = 0.025. Find the value of “a” For this question,

the tail of the left side of the bell curve will be shaded and have a probability of .025. This leads the z score (a) to be negative. 

With probability 0.95, what values of X are in the centered, symmetric range? For this question, find the

z score, then use that and given mean and s.d. to find x.

For questions about inter-quartile range, we are dealing with

.25 as a probability. 

When asked to find probability with limited information, like the following question: X is a Normal random variable with = 5. What is the probability that a value of the variable will be within 5 units of its mean?. 

• Make a bell curve with mean as the ACTUAL symbol μ and x’s as μ - # and μ + #. Can use x as is to find z scores, which will be used to find probability (your answer). 

  • If the question above decreases or increases s.d., x STAYS THE SAME.

What is the probability that a value of the variable will be more than 5 units away (no direction specified) from its mean? For this question,

both tails of bell curve will be shaded. You must find the probabilities within and then MULTIPLY by 2. 

More than 5 units away is the same as more than

1 standard deviation away, since σ = 5 (units)

“described by a Normal distribution with a mean of 10% and a standard deviation of 5%.” Despite being a percent,

10% can still be put on the line (x).

“Losing money” MEANS

bell curve’s tail on the left will be shaded. It’s negative.

Also, means x will be 0.

“Breakeven” is

0.

If you have a question with multiple parts (a, b, c, etc.)

continue to use information from previous parts. 

Higher chance of losing money –

more risk – with more variability in returns.

If trying to find z score,

add two probabilities together, then divide by 2. If z score can be found in the middle of it, choose the lesser of the two z scores (decimal places section). 

For normal distribution questions, keep going back to

z score equation. Will help you solve many answers. Also, use common sense.

The manager wants to stock enough mailing tubes each week so that the probability of running out of tubes is no higher than 0.05. What is the lowest such stock level? MEANS we

are solving for x. 

For any Normal random variable, what is the probability that the variable has a value that is within 2.5 standard deviations away from the mean? For this question, we can

put a line (z) underneath the bell curve and solve.  Also, “within” indicates we are using interior area table. 

For inter quartile questions

, the bell curve will be drawn with an interior area. That area is .50. Half of that is .25.