Estimating – with confidence!

Case study…

The amount of potassium in the blood varies slightly from day to day and this fact (along with routine measurement errors) means that successive readings of a patient’s potassium level will show a small variation with a standard deviation of = 0.2 mmol/L with a range of 3.5 – 5.0 mmol/L being cosidered normal. A significantly different value could indicate possible renal failure. A patient has 1 K-test performed and presents a

reading of 3.4 mmol/L. How reliable is this 1 measure? Can we quantify our confidence in this reading?

A confidence Interval If we assume that the errors in measuring K

are normally distributed then we can use z-scores to help quantify our confidence in this reading:

The reading of X=3.4 mmol/L is our estimate of the true value of the parameter “K blood-level concentration”

A 90% confidence interval would represent a range of readings that we would expect to get 90% of the time.

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%

Using the z-score formula we get:

*X

z

* * , * 1.645z X z z

3.4 1.645 0.2 3.4 1.645 0.2X

90% of the readings will be expected to fall in the range (3.1,3.7) mmol/L

Suppose 5 tests were performed over a number of days and eachtime gave a result of 3.4 mmol/L. How would that change the range of numbers in the confidence interval?

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!

Important Caveats…

Read page 426 carefully!Data must be a SRSOutliers can wreak havoc!We “fudged” our knowledge of in general

we don’t know thisPoorly collected data or bad experiment design

cannot be overcome by fancy formulas!

Examples…

6.13 6.18 6.19 6.30

In conclusion…

This whole discussion rests on your understanding of z-scores. If you are OK with this then just review the new terms and try the previous examples

If you are still “rusty” or un-sure about z-scores, come and see me!