One sample T IntervalExample: speeding

• 90% confidence interval

• n=23

• Check conditions

• Model: tn-1

• Confidence interval: 31.0±1.52 = (29.48, 32.52)

• STAT TESTS 8 TIntervalInput: Data or Stats

One-sample t-test

• Null hypothesis

• test statistic

• Model: tn-1

• P-value depends the alternative hypothesis

One sample T TestExample: speeding

• t-statistic = 1.13

• P-value = 0.136 (one-sided)

• STAT TESTS 2: T-TEST (for one sample)

Chapter 24 Comparing Means

Math2200

Example: AA battery

• Brand name vs. generic batteries• The same CD player, the same CD, volume at 5• 6 pairs of AA alkaline batteries, randomized run

orderBrand name Generic

194.0 190.7

205.5 203.5

199.2 203.5

172.4 206.5

184.0 222.5

169.5 209.4

Plot the data

• Boxplot– Generic batteries laste

d longer and were more consistent

– Two outliers?– Is this difference really

large enough?

Comparing two means

• Parameter of interest

• Standard error

Comparing Two Means (cont.)

• Because we are working with means and estimating the standard error of their difference using the data, we shouldn’t be surprised that the sampling model is a Student’s t.– The confidence interval we build is called a

two-sample t-interval (for the difference in means).

– The corresponding hypothesis test is called a two-sample t-test.

Sampling Distribution for the Difference Between Two Means

• When the conditions are met, the standardized sample difference between the means of two independent groups

can be modeled by a Student’s t-model with a number of degrees of freedom found with a special formula.

• We estimate the standard error with

1 2 1 2

1 2

y yt

SE y y

2 21 2

1 21 2

s sSE y y

n n

A two-sample t-interval

• Margin of error– – What degrees of freedom?

• Confidence interval

What is df?

Between and

Assumptions and Conditions

• Independence– Randomization– 10% condition

• Normal population assumption– Nearly normal condition– n<15, do not use these methods if seeing severe ske

wness– n<40, mildly skewness is OK. But should remark outli

ers– n>40, the CLT works well. The skewness does not m

atter much.• Independent group assumption

– Think about how the data are collected

Example: AA battery

• Parameter of interest

• Check conditionsHistogram of x

x

Fre

qu

en

cy

160 170 180 190 200 210

0.0

0.5

1.0

1.5

2.0

Histogram of y

y

Fre

qu

en

cy

190 200 210 220 230

01

23

4

Example: AA battery

• The sampling distribution is t with df=8.98– Critical value for 95% CI

– 95% CI: (206.0-187.4)±16.5 = (2.1, 35.1)

Testing the difference between two means• Price offered for a used camera buying from a friend vs.

buying from a stranger. Does friendship has a measurable effect on pricing?

x y

15

02

00

25

03

00

Buying from a friend Buying from a stranger

275 260

300 250

260 175

300 130

255 200

275 225

290 240

300

Two-sample t-test

• Null hypothesis

• T-test statistic

• Standard error

• df (by the complicated formula)

• P-value (one-sided or two-sided)

Example: friend vs. stranger

• Specify hypotheses

• Check conditions (boxplots)

• When conditions are satisfied, do a two-sample t-test– Observed difference 281.88-211.43 = 70.45– se = 18.70– Df = 7.622948 – P-value = 0.00600258 (two-sided)

Back Into the Pool

• Remember that when we know a proportion, we know its standard deviation. – Thus, when testing the null hypothesis that

two proportions were equal, we could assume their variances were equal as well.

– This led us to pool our data for the hypothesis test.

Back Into the Pool (cont.)

• For means, there is also a pooled t-test.– Like the two-proportions z-test, this test

assumes that the variances in the two groups are equal.

– But, be careful, there is no link between a mean and its standard deviation…

Back Into the Pool (cont.)

• If we are willing to assume or we are told that the variances of two means are equal, we can pool the data from two groups to estimate the common variance and make the degrees of freedom formula much simpler.

• We are still estimating the pooled standard deviation from the data, so we use Student’s t-model, and the test is called a pooled t-test.

The Pooled t-Test

• Estimate of the common variance

• se of the sample mean difference

• t-statistic

The Pooled t-Test

• Df = n1 + n2 – 2

• Confidence interval

1 2 1 2df pooledy y t SE y y

When should we pool?

• Most of the time, the difference is slight• There is a test that can test this condition, but it

is very sensitive to failure of assumptions and does not work well for small samples.

• In a comparative randomized experiment, experiment units are usually selected from the same population. If you think the treatment only changes the mean but not the variance, we can assume equal variances.

T-83 Plus

– STAT TESTS + 0: 2-SampTInt• Data: 2 Lists or STATS: Mean, sd, size of each s

ample• Whether to pool the variance

– STAT TESTS + 4: 2-SampTTest• One-sided or two-sided• Two-tail, lower-tail, upper-tail• Whether to pool the variance

What Can Go Wrong?

• Watch out for paired data.– The Independent Groups Assumption

deserves special attention. – If the samples are not independent, you can’t

use two-sample methods.

• Look at the plots.– Check for outliers and non-normal

distributions by making and examining boxplots.

What have we learned?

• We’ve learned to use statistical inference to compare the means of two independent groups.– We use t-models for the methods in this chapter.– It is still important to check conditions to see if our

assumptions are reasonable.– The standard error for the difference in sample means

depends on believing that our data come from independent groups, but pooling is not the best choice here.

• The reasoning of statistical inference remains the same; only the mechanics change.