Reviewing Confidence Intervals

Anatomy of a confidence level• A confidence level always consists of two

pieces:• A statistic being measured• A margin of error

X mThe margin of error can be determined by many different methods depending on what kind of distribution we are using:

normal, t-test, paired tests etc

Go to applet that demonstrates the concept of a confidence level

Simple example• Suppose that we know the standard

deviation for the active ingredient in a drug is 0.025 mg and the variation in amount is normally distributed. If we measure a sample of the drug and find the amount of active ingredient present is 0.15 mg, what would be the acceptable range of active ingredient at the 90% confidence level?

Solution…

• Use the correct z-value for 90%

95% of area left of this point5% of area left of this

point

The correct z values are -1.645 and +1.645 and are usually denoted z* to indicate that these are special ones chosen with a

particluar confidence level “C” in mind. In this example C = 90%

X m

0.15 (1.645)0.025

0.15 0.041

mg mg

mg

Another way to express this is:

The amount of active ingredient is (0.109,0.191) mg at the 90% level

Using the z-score formula we get:

*X

z

* * , * 1.645z X z z 0.15 1.645 0.025 0.15 1.645 0.025X

90% of the readings will be expected to fall in the range (0.109,0.191) mg

Using Confidence Intervals when Determining the True value of a Population Mean

• We rarely ever know the population mean – instead we can construct SRS’s and measure sample means.

• A confidence interval gives us a measure of how precisely we know the underlying population mean

• We assume 3 things:• We can construct “n” SRS’s• The underlying population of sample means is

Normal• We know the standard deviation

This gives …

Confidence interval for a population mean:

* *X z X zn n

We measure this

We infer this

Number of samplesor tests

Example: Fish or Cut Bait?

A biologist is trying to determine how many rainbow trout are in an interior BC lake. To do this he uses a large net that filters 6000 m3 of lake water in each trial. He drops the net in a specific area and records the mean number of fish caught in 10 trials. This represents one SRS. From this he is able to determine a mean and standard deviation for the number of fish in 100 SRS’s. Each SRS has the same = 9.3 fish with a sample mean of 17.5 fish. How precisely does he know the true mean of fish/6000 m3? Use C = 90%If the volume of the lake is60 million m3, how many trout are in the lake?

Solution:

• Since C = 0.90, z* = 1.645

* *

9.3 9.317.5 1.645( ) 17.5 1.645( )10 10

z X zn n

X

There is a 90% chance that the true mean number of fish/6000 m3 lies in the range (16.0,19.0) Total number of fish: He is 90% confident that there are between 160 000 and 190 000 fish in the lake.

Why should you be skeptical of this result?

Margin of Error

• When testing confidence limits you are saying that your statistical measure of the mean is:

• ie: X = 3.2 cm +/- 1.1 cm with a 90% confidence

estimate +/- the margin of error

Math view…

• Mathematically the margin of error is:

• You can reduce the margin of error by• increasing the number of samples you test• making more precise measurements (makes

smaller)

*zn

Matching Sample Size to Margin of Error

• An IT department in a large company is testing the failure rate of a new high-end graphics card in 200 of its work stations. 5 cards were chosen at random with the following lifetime per failure (measured in 1000’s of hours) and = 0.5:

1 2 3 4 5

1.4 1.7 1.5 1.9 1.8

Provide a 90% confidence level for the mean lifetime of these boards.

1.4 1.7 1.5 1.9 1.81.66

5X

0.5* 1.66 1.645( ) 1.66 0.37

5X z

n

IT is 90% confident that the mean lifetime of these boards is between 1290 and 2030 hours.

HoweverHowever – these are expensive boards and accounting wants to have the margin of error reduced to 0.10 with a 90% confidence level. What should IT do?

2* ( * )m z n zmn

IT needs to test 68 machines!

Using other statistical tests…

• The margin of error can be estimated in many different ways…

• Consider 7.37

• Here we are using a confidence iterval to test the likelihood of the null hypothesis

The main idea…• Margin of error shows you the range in a

confidence interval

• The value of ME depends on the confidence level you set and the type of statistical analysis that is appropriate