review a random variable where x can take on a range of values, not just particular ones. examples:...

Review

A random variable where X can take on a range of values, not just particular ones.

Examples:

Heights

Distance a golfer hits the ball with their driver

Time to run 100 meters

Electricity usage of a home.

Review

There are two types of continuous distributions we discuss now: uniform and normal distributions.

A density curve is the graph of a continuous probability distribution.

1) This curve always positive (or 0)

2) The area under the curve is 1.

Review For a density curve depicting the

probability distribution of a continuous random variable, – the total area under the curve is 1, – there is a direct correspondence between area

and probability.– Only the probability of an event occurring in

some interval can be evaluated. – The probability that a continuous random

variable takes on any particular value is zero.

Example

Find the probability x is at most 5 and at least 2.

x

P(x)

0 2 4 6 8 10

0.1

0

)52(P x

Example

Find the probability x is at most 5 and at least 2.

x

P(x)

0 2 4 6 8 10

0.1

0

)52(P x

Example

Find the probability x is at most 5 and at least 2.

x

P(x)

0 2 4 6 8 10

0.1

0

)52(P x

Example

Find the probability x is at most 5 and at least 2.

x

P(x)

0 2 4 6 8 10

0.1

0

3.0)1.0)(3()52(P x

Normal Distributions

This is the most common observed distribution of continuous random variables. A normal distribution corresponds to bell-shaped curves.

2

/ 22 2)(

xe

y

Normal Distributions

Shape of this curve is determined by µ and σ – µ it’s centered, σ is how far it’s spread out.

Standard Normal Distribution

The Standard Normal Distribution is a normal probability distribution that has a mean of 0 and a standard deviation of 1.

In this way the formula giving the heights of the normal curve is simplified greatly.

Z-score

We represent a standard normal variable with a z instead of an x.

Convert any normal distribution to a standard normal distribution by using the

z-score.

x

z

Standard Normal Probabilities

P(0 z 1) = 0.3413

This can found in a table in the back of the text (Table IV). The table only gives the areas under the curve to the right between 0 and z. To find other intervals requires some tricks

Finding Probabilities when given z-scores.

For a given z-score, the probability can be found in a table in the back of the text (Table IV), also see inside front cover.

Note: The table only gives the areas under the curve to the right between 0 and z. To find other intervals requires some tricks.

Examples

Use the tables in the back of the book to find the following.

a) P(0 z 2.43) = 0.4925

b) P(-2.43 z 0) = 0.4925

c) P(1.20 z 2.30) =0.4893 - 0.3849=0.1044

d) P(-1.50 z 2.4) = 0.4918 + 0.4332 = 0.925

e) P( z 1.8) = 0.4641 + 0.5 = 0.9641

Problems

Problems 5.3, 5.4, 5.12

Problems 5.22, 5.26, 5.28, 5.30, 5.36, 5.40, 5.48

Keys to success

Learn the standard normal table and how to use it.

We will be using these tables through out the course.

5.4 How do you know when a data set is normal?

3 methods

5.4 How do you know when a data set is normal?

Method 1:

• A data set is approximately normal if it is symmetric and bell-shaped.

5.4 How do you know when a data set is normal?

Method 2:

• A set of data is approximately normal if the data set satisfies the empirical rule:

– Within 1 sd: 68% of the data.– Within 2 sd: 95% of the data.– Within 3 sd: 99.7% of the data.

5.4 How do you know when a data set is normal?

Method 3:

• Find IQR and standard deviation. If the data is approximately normal, then

3.1s

IQR

Example

You are given a data set and determine that: 1. IQR=0.44 and

2. s=0.33

Would you suspect this data is normally distributed?

Example

You are given a data set and determine that: 1. IQR=0.44 and

2. s=0.33

Would you suspect this data is normally distributed?

33.133.0

44.0

s

IQR

Example

You are given a data set and determine that: 1. IQR=0.44 and

2. s=0.33

Would you suspect this data is normally distributed? YES

33.133.0

44.0

s

IQR

Problems

Problems 5.3, 5.4, 5.12

Problems 5.22, 5.26, 5.28, 5.30, 5.36, 5.40, 5.48

Problem 5.54

25

Homework

• Review Chapter 5.1-5.4

• Read Chapters 6.1-6.3 for next week

• Midterm on Thursday, – 7:00-8:30 in PS 1072– Covers chapters 1-4

• Quiz during class on Tuesday

• Next class - optional tutorial