For 95 out of 100 (large) samples, the interval

will contain the true population mean.

nx x96.1

But we don’t know ?!

Inference for the Mean of a Population

To estimate , we use a confidence interval around x.

The confidence interval is built with , which we replace with s (the sample std. dev.) if is not known.

nx x96.1

t-distributions

ns

The “standard error” of x.

nsx

t

The “standard error” of x.

For an SRS sample, the one-sample t-statistic has the t-distribution with n-1 degrees of freedom.

(see Table D)

t-distributions

t-distributions with k (=n-1) degrees of freedom – are labeled t(k), – are symmetric around 0, – and are bell-shaped – … but have more variability than Normal

distributions, due to the substitution of s in the place of .

Example: Estimating the level of vitamin C

Data:

26 31 23 22 11 22 14 31 Find a 95% confidence interval for . A: ( , ) Write it as “estimate plus margin of error”

STATA Exercise 1

STATA Exercise 2

STATA Exercise 2

STATA Exercises 3 and 4

Paired, unpaired tests

“Paired” tests compare each individual between two variables and ask whether the mean difference (“gain” in this example) is zero.

Ho: mean(pretest - posttest) = mean(diff) = 0

STATA Exercise 5

STATA Exercise 6

Robustness of t procedures

t-tests are only appropriate for testing a hypothesis on a single mean in these cases:– If n<15: only if the data is Normally distributed

(with no outliers or strong skewness)– If n≥15: only if there are no outliers or strong

skewness– If n≥40: even if clearly skewed (because of the

Central Limit Theorem)

Comparing Two Means

Comparing Two Means

Suppose we make a change to the registration procedure. Does this reduce the number of mistakes?

Basically, we’re looking at two populations: – the before-change population (population 1)– the after-change population (population 2)

Is the mean number of mistakes (per student) different? Is 1 – 2 = 0 or 0?

Comparing Two Means

Notice that we are not matching pairs. We compare two groups.

Comparing Two Means

Population Variable MeanStandard Deviation

1 x1 1 1

2 x2 2 2

Comparing Two Means

PopulationSample

SizeSample Mean

Sample Standard Deviation

1 n1 x1 s1

2 n2 x2 s2

Comparing Two Means

The population, really, is every single student using each registration procedure, an infinite number of times.– Suppose we get a “good” result today: how do we

know it will be repeated tomorrow? We can’t repeat the procedure an infinite

number of times, we only have a “sample”: numbers from one year.

We estimate (1 – 2) with (x1 – x2) .

Comparing Two Means

Remember is a Random Variable. To estimate we need both and the margin of error around , which is

So we need to know ,or rather, the appropriate standard error for this estimation.

Because we are estimating a difference, we need the standard error of a difference.

nt x*x

nx

xx

=0

Comparing Two Means

If the standard error for is

Then the standard error for (x1 – x2) is

1

1

n

1x

2

22

1

21

nn

2

22

1

21

2121

nn

xxt

STATA uses the Satterthwaite approximation as a default. This t* does not have a t-distribution because we are replacing two standard deviations by their sample equivalents.

Two-sample significance test

STATA uses the Satterthwaite approximation as a default. This t* does not have a t-distribution because we are replacing two standard deviations by their sample equivalents.

STATA Exercise 7

Paired, unpaired tests

“Paired” tests compare each individual between two variables and ask whether the mean difference (“gain” in this example) is zero.

Ho: mean(pretest - posttest) = mean(diff) = 0 “Unpaired” tests take the mean of each variable and test

whether the difference of the means is zero.Ho: mean(pretest) - mean(posttest) = diff = 0

STATA Exercise 5

STATA Exercise 8ttest ego, by(group) unequal

Robustness and Small Samples

Two-sample methods are more robust than one-sample methods.– More so if the two samples have similar shapes

and sample sizes. STATA assumes that the variances are the same (what

the book calls “pooled t procedures”), unless you tell it the opposite, using the unequal option.

Small samples, as always, make the test less robust.

Pooled two-sample t procedures

Pooled two-sample t procedures

Suppose the two Normal population distributions have the same standard deviation.

Then the t-statistic that compares the means of samples from those two populations has exactly a t-distribution.

Pooled two-sample t procedures

The common, but unknown standard deviation of both populations is . The sample standard deviations s1 and s2 estimate .

The best way to combine these estimates is to take a “weighted average” of the two, using the dfs as the weights:

2

11

21

222

2112

nn

snsnsp

(assuming is the same for both populations)

21

11

nnsp

Here, t* is the value for the t(n1 + n2 – 2) density curve with area C between – t* and t*.

To test the hypothesis Ho: 1 = 2, compute the pooled two-sample t statistic

And use P-values from the t(n1 + n2 – 2) distribution.

21

21

11nn

s

xxt

p

THE POOLED TWO-SAMPLE T PROCEDURES

ttest ego, by(group)