review normal distributions –draw a picture. –convert to standard normal (if necessary) –use...

Review• Normal Distributions

– Draw a picture.– Convert to standard normal (if necessary)– Use the binomial tables to look up the value.

– In the case of a reverse look up we may have to now solve for x.

Estimator and Point Estimate

An estimator is a “sample statistic” (such as the sample mean, or sample standard deviation) used to approximate a population parameter.

A Point Estimate is a single value or point used to approximate a population parameter. A point estimator may be biased or unbiased.

Central Limit Theorem

Take ANY random variable X and compute and for this variable. If samples of size n are randomly selected from the population, then:

1) For large n, the distribution of the sample means, will be approximately a normal distribution,

2) The mean of the sample means will be the population mean and

3) The standard deviation of the sample means will be

n

x

The Sampling Distribution of X and the Central Limit Theorem

Assume a population with = 80, = 6. If a sample of 36 is taken from this population, what is the probability that the sample mean is larger than 82?

Sketch the curve of x and identify area of interest

The Sampling Distribution of X and the Central Limit Theorem

Convert 82 to z value

First, calculate the standard deviation of the sampling distribution

Then calculate the z value

Use the tables to find probability of interest

82 802

1x

x

xz

61.0

36x n

82 ( 2) .5 .4772 .0228P x P z

Practice Problems

• #6.34 page 310

• #6.41 page 311

Example

Suppose you took a sample of 100 accounts from Visa and found the mean balance was $213.

What can you say about the average balance?

Example


What can you say about the average balance of all VISA accounts?

ANS: That it is approximately $213. (This is a point estimate.

Example


If in addition you know the standard deviation for all accounts is $112, what can you say about the average balance of all VISA accounts?


If in addition you know the standard deviation for all accounts is $112, what can you say about the average balance of all VISA accounts?

You can create an interval (called a confidence interval) that you can be 95% sure the mean is in.

Confidence Interval

A confidence interval (or interval estimate) is a range of values that estimates the true value of the population parameter.

This is associated with a degree of confidence, which is a measure the probability that a randomly selected confidence interval encloses the population parameter.

Confidence LevelThe confidence level is equal to 1- , and is split between the two tails of the distribution

Usually the confidence level is:

90% (meaning )

95% (meaning ) OR

99% (meaning ) 10.05.01.

Confidence Intervals

The Confidence Interval is expressed as:

E is called the margin of error.

For samples of size > 30,

nzxzx

Ex

x

22

n

szx 2

Example

Suppose you took a sample of 100 accounts from Visa and found the mean balance was $213. If in addition you know the standard deviation for all accounts is $112, what can you say about the average balance of all VISA accounts? Build a 95% confidence interval.

Example


95.21213

Example


100

112645.1213

Example


42.18213100

112645.1213

Example


100

112575.2213

Example


84.28213100

112575.2213

Age of STFX Students

Find the 90% confidence interval for the mean.

0.10 50n ,6.2 ,4.21 sx

Age of STFX Students

Find the 90% confidence interval for the mean.

22.0 20.850

6.24.21 50

6.24.21

0.10 50n ,6.2 ,4.21

05.005.0

22

zz

nzx

nzx

sx

Sample Size

The sample size needed to estimate so as to be (1-)*100 % confident that the sample mean does not differ from more than E is:

…round up

2

2

E

zn

Overview

• Confidence Intervals

Practice Problems

• #7.11 page 329

• #7.19 page 331

• #7.21 page 331

26

Homework

• Review Chapter 7.1, 7.2

• Read Chapters 7.3-7.5

review normal distributions –draw a picture. –convert to standard normal (if necessary) –use...

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