1 Lecture 3 Ordinary Least Squares Assumptions, Confidence Intervals, and Statistical Significance.

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Lecture 3

Ordinary Least Squares Assumptions, Confidence Intervals, and Statistical Significance

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Sampling Terminology Parameter

fixed, unknown number that describes the population Statistic

known value calculated from a sample a statistic is often used to estimate a parameter

Variability different samples from the same population may yield

different values of the sample statistic Sampling Distribution

tells what values a statistic takes and how often it takes those values in repeated sampling

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Parameter vs. StatisticA properly chosen sample of 1600 people across the United States was asked if they regularly watch a certain television program, and 24% said yes. The parameter of interest here is the true proportion of all people in the U.S. who watch the program, while the statistic is the value 24% obtained from the sample of 1600 people.

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Parameter vs. Statistic The mean of a population is denoted by µ – this

is a parameter. The mean of a sample is denoted by – this is

a statistic. is used to estimate µ.x

x The true proportion of a population with a certain

trait is denoted by p – this is a parameter. The proportion of a sample with a certain trait is

denoted by (“p-hat”) – this is a statistic. is used to estimate p.

p̂ p̂

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The Law of Large NumbersConsider sampling at random from a population with true mean µ. As the number of (independent) observations sampled increases, the mean of the sample gets closer and closer to the true mean of the population. ( gets closer to µ ) x

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The Law of Large NumbersGambling

The “house” in a gambling operation is not gambling at all the games are defined so that the gambler has a

negative expected gain per play (the true mean gain after all possible plays is negative)

each play is independent of previous plays, so the law of large numbers guarantees that the average winnings of a large number of customers will be close the the (negative) true average

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Figure 10.1: Odor Threshhold

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Sampling Distribution The sampling distribution of a statistic

is the distribution of values taken by the statistic in all possible samples of the same size (n) from the same population to describe a distribution we need to specify

the shape, center, and spreadwe will discuss the distribution of the sample

mean (x-bar).

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Case StudyDoes This Wine Smell Bad?

Dimethyl sulfide (DMS) is sometimes present in wine, causing “off-odors”. Winemakers want to know the odor threshold – the lowest concentration of DMS that the human nose can detect. Different people have different thresholds, and of interest is the mean threshold in the population of all adults.

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Case Study

Suppose the mean threshold of all adults is =25 micrograms of DMS per liter of wine, with a standard deviation of =7 micrograms per liter and the threshold values follow a bell-shaped (normal) curve. Assume we KNOW THE VARIANCE!!!

Does This Wine Smell Bad?

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Where should 95% of all individual threshold values fall? mean plus or minus about two standard

deviations

25 2(7) = 11

25 + 2(7) = 39

95% should fall between 11 & 39

What about the mean (average) of a sample of n adults? What values would be expected?

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Sampling Distribution What about the mean (average) of a sample of

n adults? What values would be expected? Answer this by thinking: “What would happen if we

took many samples of n subjects from this population?” (let’s say that n=10 subjects make up a sample) take a large number of samples of n=10 subjects from

the population calculate the sample mean (x-bar) for each sample make a histogram of the values of x-bar examine the graphical display for shape, center, spread

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Case Study

Mean threshold of all adults is =25 micrograms per liter, with a standard deviation of =7 micrograms per liter and the threshold values follow a bell-shaped (normal) curve.

Many (1000) repetitions of sampling n=10 adults from the population were simulated and the resulting histogram of the 1000x-bar values is on the next slide.


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Case StudyDoes This Wine Smell Bad?

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Mean and Standard Deviation of Sample Means

If numerous samples of size n are taken from

a population with mean and standard

deviation , then the mean of the sampling

distribution of is (the population mean)

and the standard deviation is:

( is the population s.d.) n

X

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Mean and Standard Deviation of Sample Means

Since the mean of is , we say that is

an unbiased estimator of X X

Individual observations have standard deviation , but sample means from samples of size n have standard deviation

. Averages are less variable than individual observations.

X

n

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Sampling Distribution ofSample Means

If individual observations have the N(µ, )

distribution, then the sample mean of n

independent observations has the N(µ, / )

distribution. (Note, σ is KNOWN)

X

n

“If measurements in the population follow a Normal distribution, then so does the sample mean.”

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Case Study

Mean threshold of all adults is =25 with a standard deviation of =7, and the threshold values follow a bell-shaped (normal) curve.


(Population distribution)

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Central Limit Theorem

“No matter what distribution the population values follow, the sample mean will follow a Normal distribution if the sample size is large.”

If a random sample of size n is selected from ANY population with mean and standard

deviation , then when n is “large” the sampling distribution of the sample mean is approximately Normal:

is approximately N(µ, / )

X

nX

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Central Limit Theorem:Sample Size

How large must n be for the CLT to hold?depends on how far the population

distribution is from Normal the further from Normal, the larger the sample

size needed a sample size of 25 or 30 is typically large

enough for any population distribution encountered in practice

recall: if the population is Normal, any sample size will work (n≥1)

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Central Limit Theorem:Sample Size and Distribution of x-bar

n=1

n=25n=10

n=2

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Provides methods for drawing conclusions about a population from sample dataConfidence Intervals

What is the population mean?

Tests of Significance Is the population mean larger than 66.5?

This would be ONE-SIDED

Statistical Inference

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1. SRS from the population of interest

2. Variable has a Normal distribution N(, ) in the population

3. Although the value of is unknown, the value of the population standard deviation is known

Inference about a MeanSimple Conditions-will be relaxed

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A level C confidence interval has two parts1. An interval calculated from the data,

usually of the form:estimate ± margin of error

2. The confidence level C, which is the probability that the interval will capture the true parameter value in repeated samples; that is, C is the success rate for the method.

Confidence Interval

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Case StudyNAEP Quantitative Scores

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4. The 68-95-99.7 rule indicates that and are within two standard deviations (4.2) of each other in about 95% of all samples.

x

267.8=4.2272=4.2 x276.2=4.2+272=4.2+x

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So, if we estimate that lies within 4.2 of , we’ll be right about 95% of the time.

x

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Take an SRS of size n from a Normal population with unknown mean and known std dev. . A level C confidence interval for is:

Confidence IntervalMean of a Normal Population

n

σzx

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Confidence IntervalMean of a Normal Population

LOOKING FOR z*

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Using the 68-95-99.7 rule gave an approximate 95% confidence interval. A more precise 95% confidence interval can be found using the appropriate value of z* (1.960) with the previous formula. Show how to find in Table B.2 in next lecture

267.884=4.116272=1)(1.960)(2. x276.116=4.116272=1)(1.960)(2. x

We are 95% confident that the average NAEP quantitative score for all adult males is between 267.884 and 276.116.

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But the sample distribution is narrower than the population distribution, by a

factor of √n.

Thus, the estimates

gained from our samples

are always relatively

close to the population

parameter µ.

nSample means,n subjects

n

Population, xindividual subjects

x

x

If the population is normally distributed N(µ,σ),

so will the sampling distribution N(µ,σ/√n).

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Red dot: mean valueof individual sample

Ninety-five percent of all sample

means will be within roughly 2

standard deviations (2*/√n) of

the population parameter

Because distances are

symmetrical, this implies that

the population parameter

must be within roughly 2

standard deviations from the

sample average , in 95% of

all samples.

n

This reasoning is the essence of statistical inference.

x

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Summary: Confidence Interval for the Population Mean

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Hypothesis Testing

Start by explaining when σ is known Move to unknown σ should be

straightforward

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The statement being tested in a statistical test is called the null hypothesis.

The test is designed to assess the strength of evidence against the null hypothesis.

Usually the null hypothesis is a statement of “no effect” or “no difference”, or it is a statement of equality.

When performing a hypothesis test, we assume that the null hypothesis is true until we have sufficient evidence against it.

Stating HypothesesNull Hypothesis, H0

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The statement we are trying to find evidence for is called the alternative hypothesis.

Usually the alternative hypothesis is a statement of “there is an effect” or “there is a difference”, or it is a statement of inequality.

The alternative hypothesis should express the hopes or suspicions we bring to the data. It is cheating to first look at the data and then frame Ha to fit what the data show.

Stating HypothesesAlternative Hypothesis, Ha

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One-sided and two-sided tests

A two-tail or two-sided test of the population mean has these null and alternative

hypotheses:

H0: µ = [a specific number] Ha: µ [a specific number]

A one-tail or one-sided test of a population mean has these null and alternative

hypotheses:

H0: µ = [a specific number] Ha: µ < [a specific number] OR

H0: µ = [a specific number] Ha: µ > [a specific number]

The FDA tests whether a generic drug has an absorption extent similar to the known

absorption extent of the brand-name drug it is copying. Higher or lower absorption

would both be problematic, thus we test:

H0: µgeneric = µbrand Ha: µgeneric µbrand two-sided

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The P-valueThe packaging process has a known standard deviation = 5 g.

H0: µ = 227 g versus Ha: µ ≠ 227 g

The average weight from your four random boxes is 222 g.

What is the probability of drawing a random sample such as yours if H0 is true?

Tests of statistical significance quantify the chance of obtaining a

particular random sample result if the null hypothesis were true.

This quantity is the P-value.

This is a way of assessing the “believability” of the null hypothesis given

the evidence provided by a random sample.

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Interpreting a P-value

Could random variation alone account for the difference between the null

hypothesis and observations from a random sample?

A small P-value implies that random variation because of the sampling process

alone is not likely to account for the observed difference.

With a small P-value, we reject H0. The true property of the population is

significantly different from what was stated in H0.

Thus small P-values are strong evidence AGAINST H0.

But how small is small…?

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P = 0.1711

P = 0.2758

P = 0.0892

P = 0.0735

P = 0.01

P = 0.05

When the shaded area becomes very small, the probability of drawing such a

sample at random gets very slim. Oftentimes, a P-value of 0.05 or less is

considered significant: The phenomenon observed is unlikely to be entirely due

to chance event from the random sampling.

Significant P-value

???

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The significance level

The significance level, α, is the largest P-value tolerated for rejecting a true null hypothesis (how

much evidence against H0 we require). This value is decided arbitrarily before conducting the test.

If the P-value is equal to or less than α (p ≤ α), then we reject H0.

If the P-value is greater than α (p > α), then we fail to reject H0.

Does the packaging machine need revision?

Two-sided test. The P-value is 4.56%.

* If α had been set to 5%, then the P-value would be significant.

* If α had been set to 1%, then the P-value would not be significant.

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Implications

We don’t need to take lots of

random samples to “rebuild” the

sampling distribution and find

at its center.

n

n

Sample

Population

All we need is one SRS of

size n, and relying on the

properties of the sample

means distribution to infer

the population mean .

THE WHOLE POINT OF THIS!!!!

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If σ is Estimated

Usually we do not know σ. So when it is estimated, we have to use the t-distribution which is based on sample size.

When estimating σ using σ, as the sample size increases the t-distribution approaches the normal curve.

^

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Conditions for Inferenceabout a Mean Data are from a SRS of size n. Population has a Normal distribution

with mean and standard deviation . Both and are usually unknown.

we use inference to estimate . Problem: unknown means we cannot

use the z procedures previously learned.

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When we do not know the population standard deviation (which is usually the case), we must estimate it with the sample standard deviation s.

When the standard deviation of a statistic is estimated from data, the result is called the standard error of the statistic.

The standard error of the sample mean is

Standard Error

x

sn

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When we estimate with s, our one-sample z statistic becomes a one-sample t statistic.

By changing the denominator to be the standard error, our statistic no longer follows a Normal distribution. The t test statistic follows a t distribution with k = n – 1 degrees of freedom.

One-Sample t Statistic

ns

μxt

nσ

μxz 00

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The t Distributions The t density curve is similar in shape to the

standard Normal curve. They are both symmetric about 0 and bell-shaped.

The spread of the t distributions is a bit greater than that of the standard Normal curve (i.e., the t curve is slightly “fatter”).

As the degrees of freedom k increase, the t(k) density curve approaches the N(0, 1) curve more closely. This is because s estimates more accurately as the sample size increases.

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The t Distributions

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Critical Values from T-Distribution

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How do we find specific t or z* values?We can use a table of z/t values (Table B.2). For a particular confidence level C, the appropriate t or z*

value is just below it, by knowing the sample size. Lookup α=1-C. If you want 98%, lookup .02, two-tailed

We can use software. In Excel when n is large, or σ is known:

=NORMINV(probability,mean,standard_dev)

gives z for a given cumulative probability.

Since we want the middle C probability, the probability we require is (1 - C)/2

Example: For a 98% confidence level (NOTE: This is now for 1% on each side)

= NORMINV (.01,0,1) = −2.32635 (= neg. z*)

Ex. For a 98% confidence level, z*=t=2.326

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ExcelTDIST(x, degrees_freedom, tails)

TDIST = p(X > x ), where X is a random variable that follows the t distribution (x positive). Use this function in place of a table of critical values for the t distribution or to obtain the P-value for a calculated, positive t-value.

X is the standardized numeric value at which to evaluate the distribution (“t”). Degrees_freedom is an integer indicating the number of degrees of freedom. Tails specifies the number of distribution tails to return. If tails = 1, TDIST returns the one-

tailed P-value. If tails = 2, TDIST returns the two-tailed P-value.

TINV(probability, degrees_freedom)

Returns the t-value of the Student's t-distribution as a function of the probability and the degrees of freedom (for example, t*).

Probability is the probability associated with the two-tailed Student’s t distribution. Degrees_freedom is the number of degrees of freedom characterizing the distribution.

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Sampling Distribution of ̂0 and ̂1 Based on Simulation Assume the relationship between grades and

hours studied for an entire population of students in an econometrics class looks like this

The upward sloping line suggests that more

studying results in higher grades. The

equation for the line is E(Grades) = 50 + 2 ×

Hours, suggesting that if a person spent 10 hours studying their grade would be 70

points.

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Sampling Distribution of ̂0 and ̂1 Based on Simulation

The more typical situation involves a sample—not a population Our goal is to learn about a population’s slope and intercept via sample

data A plot of the trend line from a random sample of 20 observations

from the econometric student population looks like this

The random sample has an intercept of 55

and a slope of 1.5, while the population

sample’s intercept was 50 with a slope of 2.

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Additional random samples with 20 observations each result in different slopes and intercepts

If a computer calculated all the possible slope estimates (with the same size random sample n) we could graph the distribution of possible valuesThen use it to conduct confidence intervals and

hypothesis tests


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The following graph represents 10,000 samples of 20 observations from the population


Observations

• Roughly centered around 2 (the population’s slope)

• Has a standard deviation of 0.55

• Appears to be normally distributed

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Based on this simulation we can make the following statementsEstimated slope and intercepts are random

variables Value is dependent upon random sample gathered

Mean of the different estimates is equal to the population value

Distribution of the estimators is approximately normal… this is a huge implication.


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A linear estimator satisfies the condition that it is a linear combination of the dependent variable

The estimator for the population slope is

The Linearity of the OLS Estimators

Known as the ordinary least squares (OLS)

estimator.

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The Variance of the OLS Estimator

The variance of the OLS slope estimator describes the dispersion in the distribution of OLS estimates around its mean

Var( ̂1) is smaller if the variance of Y is smaller

The smaller the variance of Y—the less likely we are to observe extreme samples

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Hypothesis Testing

Hypothesis tests are conducted analogously to those concerning population means or proportions

Suppose someone alleges the population slope equals a and the alternative hypothesis is that 1 does not equal a

Formally, the null and alternative hypothesis are

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Hypothesis Testing

The farther ̂1 is from a the more plausible the alternative hypothesis

Formalized via the T-statistic

T represents the number of standard deviations the sample slope is from the slope hypothesized under the null The larger this number, the more plausible the alternative

hypothesis Would expect to observe a sample slope that deviates from the

true slope by more than 1.96 standard deviations, at most 5% of the time

T

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Hypothesis Testing

Would decide in favor of the alternative when

Where is the desired significance level

If the sample is used to estimate the standard deviation of the slope, s ̂, use the t distribution rather than the normal distribution to determine critical values

To test one-sided alternatives proceed in a similar fashion to that used when testing hypotheses for means and proportions

|T| t

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Hypothesis Testing

Alternative approachCalculate p-values rather than comparing a z-

or t-statistic to the relevant critical valuesStart with the z- or t-statistics and calculate

the probability of observing the value of ̂1 or larger

With the z-statistic use the relationship to estimate the value of

|T| t

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The Multiple Regression Model The multiple regression model has the following

assumptions The dependent variable is a linear function of the explanatory

variables The errors have a mean of zero The errors have a constant variance The errors are uncorrelated across observations The error term is not correlated with any of the explanatory

variables The errors are drawn from a normal distribution No explanatory variable is an exact linear function of other

explanatory variables (important with dummy variables)

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Interpretation of the Regression Coefficients The value of the dependent variable will

change by j units with a one unit change in the explanatory variableHolding everything else constant (ceteris

paribus)

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MLR Assumption 1 Linear in Parameters MLR.1

Defines POPULATION model The dependent variable y is

related to the independent variables x and the error (or disturbance)

are k unknown population parameters

u is unobservable random error

uxxxy kk ...

MLR.1 Assumption

22110

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MLR Assumption 2 Random Sampling Use a random sample of size

n, {(xi,yi): i=1,2,…,n} from the population model

Allows redefinition of MLR.1. Want to use DATA to estimate our parameters

All are k population parameters to be estimated

},...,2,1:),{(

MLR.2 Assumption

niyx ii

niuxxxy iikkii ,...,2,1,...22110

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MLR Assumption 3

No Perfect Collinearity In the sample (and therefore in the population),

none of the independent variables is constant, and there are no exact linear relationships among the independent variables

With collinearity, there is no way to get ceteris paribus relationship.

utotspendspendBspendA

3210voteA

totspendspendBspendA whereip,relationshlinear of Example

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MLR Assumption 4

Zero Conditional Mean For a random sample,

implication is that NO independent variable is correlated with ANY unobservable (remember error includes unobservable data)

0],...,,|[

MLR.4 Assumption

21 kxxxuE

n1,2,...,i allfor

0]|[

Sample RANDOM aFor

ii xuE

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Regression Estimators As I have repeatedly said, in the multiple

regression case, we cannot use the same methods for calculating our estimates as before.

We MUST control for the correlation (or relationship) between different values for X

To get the values for our estimators of Beta we are actually regressing each X variable against ALL OTHER X variables first… Y is not involved in the calculation.

Each Beta estimated with this method CONTROLS for other x’s when being calculated.

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Regression Estimators As I have repeatedly said, in the multiple regression case,

we cannot use the same methods for calculating our estimates for Beta as before.

We MUST control for the correlation (or relationship) between different values for X

on... so and ,,on

of regression thefrom residuals theis Where

ˆ

ˆˆ

case MLRin for Estimator

4321

1

21

11

1

1

xxxx

r

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n

ii

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1 Lecture 3 Ordinary Least Squares Assumptions, Confidence Intervals, and Statistical Significance.

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Transcript of 1 Lecture 3 Ordinary Least Squares Assumptions, Confidence Intervals, and Statistical Significance.