Class 4

Normal DistributionsNormal Distributions

Sampling DistributionsSampling Distributions

Central Limit TheoremCentral Limit Theorem

Normal Random Variable

• Bell shaped curve

-3 -2 -1 0 1 2 3

xexf

x2

2

1

2

1)(

Computing Normal Probabilities

• We have computed probabilities for Z, a standard normal. What is E(Z)? Var(Z)?

• It turns out that if X is any normal random variable with mean and standard deviation , then (X - )/ is a standard normal random variate. As a result, we write .

X

Z

Computing Normal Probs. (cont.)

• Suppose that X has a normal distribution with = 5 and = 3. Can you graph the distribution of X?

• What is the P{X 11}?

}3

511

3

5{}11{

XPXP

Computing Normal Probs (cont.)

• What is P{-1 X 9}?

}3

59

3

5

3

51{

XP

Using EXCEL

• Select the Function Wizard (fx) statistical/normdist.

• The syntax of this function looks like normdist(x, , , true or false).

• If the fourth argument is true, this will return P{X x} where X is a normal(, ).

Example

• How can you interpret the computation of Z?• The lifetime of a tire has a normal distribution

with a mean of 40,000 miles and a standard deviation of 3,000 miles. It is desired to set a warranty on these tires such that 10% of the tires fall under warranty. What is the required value (in miles)?

Example (cont.)

• How many standard deviations do we have to go out on a (any) normal distribution to cut off 10%?

• Therefore, if w is the warranty limit, we have:

Using EXCEL

• The function norminv(prob, , ) will return the value on a normal(, ) distribution that cuts off prob to the left.

• Try norminv(.1, 40000, 3000).

Summary (So far)

• Describe Data Graphically and Numerically

• Populations vs. Samples

• Further description of populations--Random Variables

• Discrete

• Continuous

Now we will return to sampling and apply what Now we will return to sampling and apply what we have learned.we have learned.

Sampling• Reasons for sampling as opposed to taking a census

• Cost

• Speed

• Analysis

• Feasibility

• Types• Nonrandom

• Random» Simple Random Sample: A sample where all samples of size n have

the same chance of being chosen.» Systematic» Stratified» Cluster

Judgment or Convenience

Sampling Distributions

• Basic idea: Imagine all simple random samples of size n that can be drawn from a population. Each sample has its own characteristics (such as a sample mean). We might wonder about the likelihood of seeing a particular characteristic in our sample. This is the idea behind a sampling distribution.

Example

• Infinite population:

• For future reference:

X p(x)1 0.22 0.24 0.6

= 1(.2) + 2(.2) + 4(.6) = 3

2 = (1-3)2(.2) + (2-3)2(.2) + (4-3)2(.6)

= 1.6

Distribution of X

0.2 0.2

0

0.6

0

0.1

0.2

0.3

0.4

0.5

0.6

1 2 3 4

Pro

babi

liti

es

Distribution of Sample Mean (n = 2)

0.04

0.08

0.04

0.24 0.24

0

0.36

0

0.1

0.2

0.3

0.4

0.5

0.6

1 1.5 2 2.5 3 3.5 4

Pro

b

Distribution of Sample Mean (n = 3)

0.008

0.024 0.024

0.08

0.144

0.072

0.216 0.216

0

0.216

0

0.05

0.1

0.15

0.2

0.25

1.000 1.333 1.667 2.000 2.333 2.667 3.000 3.333 3.667 4.000

Pro

b

Distribution of Mean (n = 4)

0.0000

0.0200

0.0400

0.0600

0.0800

0.1000

0.1200

0.1400

0.1600

0.1800

0.2000

1.000 1.250 1.500 1.750 2.000 2.250 2.500 2.750 3.000 3.250 3.500 3.750 4.000

Pro

b

Distribution of Sample Mean (n = 7)

0.0000

0.0200

0.0400

0.0600

0.0800

0.1000

0.1200

0.1400

1.000

1.143

1.286

1.429

1.571

1.714

1.857

2.000

2.143

2.286

2.429

2.571

2.714

2.857

3.000

3.143

3.286

3.429

3.571

3.714

3.857

4.000

Pro

b

Homework

• For the following population:

• Compute and .

• Generate the sampling distribution of for a sample of size n = 3.

X

X p(x)0 0.51 0.32 0.2

The Central Limit Theorem

• Let be computed by taking a simple random sample of size n from a population with mean and standard deviation . Then for large n, the distribution of will be approximately normal. Large n means:

• n 1 when sampling from a normal distribution,

• n 30 when sampling from any distribution.

X

X

As always when sampling from an infinite population or a population

of size N where N>>n,

.)(,)(2

nXVarandXE

A Quick Note

n

nXVar

X

X

22)(

Using the CLT

• Incomes in a community are normally distributed with a mean of $25,000 and standard deviation of $8,000. If we take a sample of size 4, what is the probability that the average income in the sample is greater than $29,000?

Income Example

40004

8000

000,25

4

000,8

000,25

n

n

X

X

000,4

000,25000,29}000,29{

X

XXPXP

1587.

}1{

ZP

• What is the probability that a single income selected will fall above $29,000?

Income Example

}000,8

000,25000,29{}000,29{

X

PXP

3085.0

}5.0{

ZP

• What proportion of the population will fall above $29,000?

Using the CLT• A company produces lids for tin cans. The

lids are supposed to be 4 inches in diameter. The standard deviation of tin can lids is .012 inches. Because a worker suggested that the machine is in need of adjustment, the foreman has taken a sample of 100 lids and found that inches. Should they shut down production to make the adjustment?

003.4x