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Math 1107 Introduction to Statistics

- Lecture 11
- The Normal Distribution

Math 1107 The Normal Distribution

Drawing Conclusions from Representative

Data Making Decisions Looking for Relationships

Analyzing Specific Data Looking for

Outliers Looking for Relationships

- Descriptive Statistics
- Visualization, Summarization, Outliers
- Categorical Data Analysis

- Inferential Statistics
- Sampling Central Limit Theorem
- Confidence Intervals, Hypothesis Testing,

Regression, ANOVA, etc.

Math 1107 The Normal Distribution

- There are many types of distributions
- Binomial 2 outcomes (success or failureH or

T) - Poisson Infinite possibilities, with discrete

occurrences - Normal Bell Shaped continuous distribution

Math 1107 The Normal Distribution

- A family of continuous random variables whose

outcomes range from minus infinity to plus

infinity. - Bell shaped and symmetric about the mean µ.
- Mean µ, Median µ, Mode µ.
- The standard deviation is s .
- The area under the normal curve below µ is .5.
- The area above µ is also .5.
- Probability that a Normal Random Variable

Outcome - Lies within /- 1 std dev of the mean is .6826
- Lies within /- 2 std dev of the mean is .9544
- Lies within /- 3 std dev of the mean is .9974

Math 1107 The Normal Distribution

Math 1107 The Normal Distribution

0

1

2

3

-1

-2

-3

Math 1107 The Normal Distribution

- The Standard Normal Distribution looks like a

Normal Distribution, but has important

statistical properties - mean 0
- std dev 1

- Remember from earlier in the semester that
- The Std Normal Distribution enables the

calculation of Z-scores - Z-Scores can be compared against ANY populations

using any scale

Math 1107 The Normal Distribution

- Remember from earlier in the semester that
- The Std Normal Distribution enables the

calculation of Z-scores - Z-Scores can be compared against ANY populations

using any scale - Z-scores are stated in units of standard

deviations - So, typical Z-scores will range from 0 (the

mean) to 3 and can be negative or positive. - Andmost importantlywe can use Z-scores to

determine the associated probability of an

outcome.

Math 1107 The Normal Distribution

How do we use a z-score to find a

probability? Z(x-mu)/std dev

Where, X is a value of interest from the

distribution Mu the average of the

distribution Std dev the std dev of the

distribution.

Math 1107 The Normal Distribution

Prior to solving any Normal Distribution

problem using Z-scores, ALWAYS draw a sketch of

what you are doing. This will provide you with a

guide for what is a reasonable answer.

Math 1107 The Normal Distribution

Example Watts Corporation makes lightbulbs with

an average life of 1000 hours and a std dev of

200 hours. Assuming the life of the bulbs is

normally distributed, what is the probability of

buying a bulb at random that lasts for up to 1400

hours?

X1400 Mu 1000 Std dev 200 So,

Z(1400-1000)/200 2. A z-score of 2 equals

.4772. We add .5 to this and get a probability

of .9772.

Math 1107 The Normal Distribution

Example Unlucky Larry bought a Watts

Corporation bulb and it only lasted 800 hours.

What is the probability that a bulb selected at

random would last between 800 and 1000 hours?

X800 Mu 1000 Std dev 200 So,

Z(800-1000)/200 -1. A z-score of -1 equals

.3413. So, there is a 34.13 chance of selecting

a bulb at random that generates between 800 and

1000 hours of light.

Math 1107 The Normal Distribution

Example What is the probability of selecting a

bulb at random that generates less than 800 hours?

The total area under the curve less than the

average is .50 or 50. So, if we know the area

between 800 and 1000 is .3413, then the area less

than 800 is .5-.3413 or .1587.

What is the probability of selecting a bulb at

random that generates more than 800 hours?

The total area under the curve more than the

average is .50 or 50. So, if we know the area

between 800 and 1000 is .3413, then the area less

than 800 is .5.3413 or .8413.

Math 1107 The Normal Distribution

Example Coca Cola Bottlers produce millions of

cans of coke a year. The average can holds 12

ounces with a std dev of .2 ounces. What is the

probability of getting a coke with between 11.8

and 12 ounces?

X11.8 ounces Mu 12 Std dev .2 So,

Z(11.8-12)/.2 -1. A z-score of -1 equals

.3413.

Math 1107 The Normal Distribution

Example Coca Cola Bottlers produce millions of

cans of coke a year. The average can holds 12

ounces with a std dev of .8 ounces. What is the

probability of getting a coke with between 11.8

and 12 ounces?

X11.8 ounces Mu 12 Std dev .8 So,

Z(11.8-12)/.8 -.25. A z-score of -.25 equals

.0987, or 9.87

Math 1107 The Normal Distribution

Example from Page 243 Airlines have designed

their seats to accommodate the hip width of 98

of all males. Men have hip widths that are

normally distributed with a mean of 14.4 inches

and a standard deviation of 1.0. What is the

minimum hip width that airlines cannot

accommodate? This is the 98th percentile.

Math 1107 The Normal Distribution

In this example, we are working backward. We

know the Probability (98) and we want to know

the value that generates this probability. Given

the Z formula, we now solve for x.

Z(x-mu)/std dev

2.05(x-14.4)/1 2.05 x-14.4 2.0514.4

x-14.414.4 16.45 x