Review of Statistics. Estimation of the Population Mean Hypothesis Testing Confidence Intervals ...

Review of Statistics

Estimation of the Population Mean Hypothesis Testing Confidence Intervals Comparing Means from Different Populations Scatterplots and Sample Correlation

Estimation of the Population Mean

One natural way to estimate the population mean, , is simply to compute the sample average from a sample of n i.i.d. observations. This can also be motivated by law of large numbers.

But, is not the only way to estimate . For example, Y1 can be another estimator of .

Copyright © 2003 by Pearson Education, Inc.

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Key Concept 3.1

In general, we want an estimator that gets as close as possible to the unknown true value, at least in some average sense.

In other words, we want the sampling distribution of an estimator to be as tightly centered around the unknown value as possible.

This leads to three specific desirable characteristics of an estimator.

Three desirable characteristics of an estimator.

Let denote some estimator of , Unbiasedness: Consistency: Efficiency.

Let be another estimator of , and suppose both and are unbiased. Then is said to be more efficient than if

Properties of It can be shown that E( )= and

(from law of large numbers), is both unbiased and consistent.

But, is efficient?

Examples of alternative estimators.

Example 1: The first observation Y1?

Since E(Y1)= , Y1 is an unbiased estimator of . But,

if n≥2, is more efficient than Y1.

Example 2:

where n is assumed to be an even number.The mean of is and its variance is

Thus is unbiased and, because Var( ) → 0 as n→∞, is consistent.

However, is more efficient than .

In fact, is the most efficient estimator of among all unbiased estimators that are weighted averages of Y1, … , Yn. (Weighted average implies that the estimators are all unbiased.)

Hypothesis Testing

The hypothesis testing problem (for the mean): make a provisional decision, based on the evidence at hand, whether a null hypothesis is true, or instead that some alternative hypothesis is true. That is, test

Copyright © 2003 by Pearson Education, Inc.

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Key Concept 3.5

Terminology

Significance level; p-value; critical value Confidence interval; acceptances region,

rejection region Size; power

p-value = probability of drawing a statistic (e.g. ) at least as adverse to the null as the value actually computed with your data, assuming that the null hypothesis is true.

The significance level of a test is a pre-specified probability of incorrectly rejecting the null, when the null is true.

Calculating the p-value based on :

where is the value of actually observed. To compute the p-value, you need to know the

distribution of . If n is large, we can use the large-n normal

approximation.

where denotes the standard deviation of the distribution of .

Calculating the p-value with Y known

Type I and Type II Error

Type I Error (Red) ， Type II Error (Blue)

虛 無 分 配對 立 假 設 下 的虛 無 分 配

臨 界 值

α/2

α/2β

對立假設

Confidence Intervals

A 95% confidence interval for is an interval that contains the true value of Y in 95% of repeated samples.

Digression: What is random here? the confidence interval— it will differ from one sample to the next; the population parameter, , is not random.

In practice, is unknown—it must be estimated.Estimator of the variance of Y:

Fact: If (Y1, … , Yn ) are i.i.d. and E(Y4)< ∞ , then Why does the law of large numbers apply?

Because is a sample average. Technical note: we assume E(Y4)< ∞ because

here the average is not of Yi , but of its square.

For the first term, Define , then , and

W1, … , Wn are i.i.d.

. Thus W1, … , Wn are i.i.d. and Var(Wi) < ∞ , so

Therefore,

For the second term, because

Therefore,

Computing the p-value with estimated:

The p-value and the significance level

With a prespecified significance level (e.g. 5%): reject if |t| > 1.96. equivalently: reject if p ≤ 0.05. The p-value is sometimes called the marginal

significance level.

What happened to the t-table and the degrees of freedom? Digression: the Student t distribution If Yi, i = 1,…, n is i.i.d. N(Y, ), then the t-statistic

has the Student t-distribution with n – 1 degrees of freedom. The critical values of the Student t-distribution is tabulated in the back of all statistics books. Remember the recipe? Compute the t-statistic Compute the degrees of freedom, which is n – 1 Look up the 5% critical value If the t-statistic exceeds (in absolute value) this

critical value, reject the null hypothesis.

Comments on this recipe and the Student t-distribution

The theory of the t-distribution was one of the early triumphs of mathematical statistics. It is astounding, really: if Y is i.i.d. normal, then you can know the exact, finite-sample distribution of the t-statistic – it is the Student t. So, you can construct confidence intervals (using the Student t critical value) that have exactly the right coverage rate, no matter what the sample size. But….

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Comments on Student t distribution, ctd.

If the sample size is moderate (several dozen) or large (hundreds or more), the difference between the t-distribution and N(0,1) critical values are negligible. Here are some 5% critical values for 2-sided tests:

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10 2.2320 2.0930 2.0460 2 1.96

5% t -distributioncritical value

degrees of freedom(n-1)

Comments on Student t distribution, ctd.

So, the Student-t distribution is only relevant when the sample size is very small; but in that case, for it to be correct, you must be sure that the population distribution of Y is normal. In economic data, the normality assumption is rarely credible. Here are the distributions of some economic data. Do you think earnings are normally distributed? Suppose you have a sample of n = 10 observations

from one of these distributions – would you feel comfortable using the Student t distribution?

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Comments on Student t distribution, ctd.

Consider the t-statistic testing the hypothesis that two means (groups s, l) are equal:

Even if the population distribution of Y in the two groups is normal, this statistic doesn’t have a Student t distribution!

There is a statistic testing this hypothesis that has a normal distribution, the “pooled variance” t-statistic – see SW (Section 3.6) – however the pooled variance t-statistic is only valid if the variances of the normal distributions are the same in the two groups. Would you expect this to be true, say, for men’s v. women’s wages?

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2 2 ( )s l

s l

s l s l

s ss l

n n

Y Y Y Yt

SE Y Y

The Student-t distribution – summary

The assumption that Y is distributed N(Y, ) is rarely plausible in practice (income? number of children?)

For n > 30, the t-distribution and N(0,1) are very close (as n grows large, the tn–1 distribution converges to N(0,1))

The t-distribution is an artifact from days when sample sizes were small and “computers” were people

For historical reasons, statistical software typically uses the t-distribution to compute p-values – but this is irrelevant when the sample size is moderate or large.

For these reasons, in this class we will focus on the large-n approximation given by the CLT

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Summary on The Student t-distribution

If Y is distributed , then the t-statistic has the Student t-distribution (tabulated in back of all stats books)

Some comments: For n > 30, the t-distribution and N(0,1) are very close. The assumption that Y is distributed is

rarely plausible in practice (income? number of children?)

The t-distribution is an historical artifact from days when sample sizes were very small.

In this class, we won’t use the t distribution - we rely solely on the large-n approximation given by the CLT.

Confidence Intervals

A 95% confidence interval for is an interval that contains the true value of Y in 95% of repeated samples.

Digression: What is random here? the confidence interval— it will differ from one sample to the next; the population parameter, , is not random.

A 95% confidence interval can always be constructed as the set of values of not rejected by a hypothesis test with a 5% significance level.

This confidence interval relies on the large-n results that is

approximately normally distributed and

Summary:

From the assumptions of:(1) simple random sampling of a population, that is,

(2)we developed, for large samples (large n):

Theory of estimation (sampling distribution of ) Theory of hypothesis testing (large-n distribution

of t-statistic and computation of the p-value). Theory of confidence intervals (constructed by

inverting test statistic).Are assumptions (1) & (2) plausible in practice? Yes

Tests for Difference between Two Means

Let be the mean hourly earning in the population of women recently graduated from college and let be population mean for recently graduated men. Consider the null hypothesis that earnings for these two populations differ by certain amount d, then

Replace population variances by sample variances, we have the standard error

and the t-statistic is

If both nm and nw are large, the t-statistic has a standard normal distribution.

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Summarize the relationship between variables

Scatterplots:

The population covariance and correlation can be estimated by the sample covariance and sample correlation.

The sample covariance is

The sample correlation is

It can be shown that under the assumptions that (Xi , Yi) are i.i.d. and that Xi and Yi have finite fourth moments,

It is easy to see that the second term converges in probability to zero because so

by Slutsky’s theorem.

By the definition of covariance, we haveTo apply the law of large numbers on the first term,

we need to have

which is satisfied since

The second inequality follows by applying the Cauchy-Schwartz inequality, and the last inequality follows because of the finite fourth moments for (Xi , Yi).

The Cauchy-Schwartz inequality is

Applying the law of large numbers, we have

Also, , therefore

Scatterplots and Sample Correlation