Comparing Means

Chapter 24

déjà vu’

Comparing two means is very similar to comparing two proportions.

this is our t-value; it works a lot like a z-score. You can use the InvT function in the calculator to get your p-value.

With degrees of freedom calculated from a special formula

Special Degrees of Freedom Formula

Don’t worry, we will usually use the calculator for this. If you do this by hand, always round down.

Our Conditions

1. Independence1. Is it randomized and less than 10% of the population?

2. Normal Population1. Nearly Normal Condition

1. If n<15, histogram should show almost now skewness, and be unimodal

2. If n = 40 or close to it, should be mostly normal, moderate skewness can be tolerated.

3. Independent Groups1. Are the two groups being tested independent of each

other? You just have to think through this one.

A confidence interval.

Confidence Interval:

With Standard Error equal to

Step by Step; two sample t-interval

Brand Name Generic

194 190.7

205.5 203.5

199.2 203.5

172.4 206.5

184.0 222.5

169.5 209.4

The data table below shows the lives of two batteries tested under identical experiment conditions (in minutes). Let’s make a confidence interval for the differences of means.

Plan

PlanIdentify the parameter you wish to estimate, and the populations you want to make statements about.

Make a picture

ResponseI have measurements of the lifetimes of 12 batteries, 6 from a brand name manufacturer and 6 from a generic manufacturer. I want to create a 95% confidence interval to contain the true difference of

Picture: Check the box plots on page 551.

What model will we use?

Model• Check the appropriate

assumptions and conditions.

ResponseIndependent Groups: Batteries manufactured by two different companies should be independent.

Randomization: The batteries were involved in a experiment, meaning randomization should have been employed. Its also fair to assume that 6 batteries is less than 10% of the population of either brand.

Nearly Normal: Check the histogram on page 552. Do you know how to make one in your calculator?

The conditions are met for a Student’s t-model. We’ll use a two-sample t-interval.

Show

MechanicsShow all of your calculations. We need to calculate several values here. 1. The standard error2. Approximate degrees of

freedom.3. Critical t-value (from table)4. Margin of Error.

ResponseUse the calculator; 2 Sample T-interval. a. Input the data into List 1 and

List 2 in the calculator. b. Use the 2-sample T interval

function in the calculator (Stat>Test>0 2SampT-Int)

c. Leave both Frequency options at 1, and select NO for pooled.

d. Calculate

Show cont.

Important Pieces1. The interval. Your

calculator will automatically use List1 – List 2. If that yields negative results, you must interpret them correctly.

ResponseThe calculator gave (-35.1, -2.069). Since we used Generic – Brand, our answer will be (2.069, 35.1).

Tell

Tell• Interpret your results both

statistically and in the context of the problem.

Response. • The 95% confidence interval

is (2.1, 35.1) minutes.• I am 95% confident that hte

interval from 2.1 minutes to 35.1 minutes captures the mean amount of time generic batteries outlast name brand batteries.

A test for difference between two means.

When doing a test between two means we use…

Example

• Refer to the example about cameras on pages 554-558 in your book for a hypothesis test.