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

2. Sampling Terminology Parameter Statistic Variability
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

3. Parameter vs. Statistic
A 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.
From Seeing Through Statistics, 2nd Edition by Jessica M. Utts.

4. 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.
The true proportion of a population with a certain trait is denoted by p – this is a parameter.
From Seeing Through Statistics, 2nd Edition by Jessica M. Utts.
The proportion of a sample with a certain trait is denoted by (“p-hat”) – this is a statistic. is used to estimate p.

5. The Law of Large Numbers
Consider 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 )

6. The Law of Large Numbers Gambling
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

7. Figure 10.1: Odor Threshhold

8. 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 spread
we will discuss the distribution of the sample mean (x-bar).

9. Does This Wine Smell Bad?
Case Study
Does 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.

10. Does This Wine Smell Bad?
Case Study
Does This Wine Smell Bad?
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!!!

11. 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?

12. 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

13. Does This Wine Smell Bad?
Case Study
Does This Wine Smell Bad?
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 x-bar values is on the next slide.

14. Does This Wine Smell Bad?
Case Study
Does This Wine Smell Bad?

15. Mean and Standard Deviation of Sample Means
If numerous samples of size n are taken from a population with mean m and standard deviation  , then the mean of the sampling distribution of is m (the population mean) and the standard deviation is: ( is the population s.d.)

16. Mean and Standard Deviation of Sample Means
Since the mean of is m, we say that is an unbiased estimator of m
Individual observations have standard deviation , but sample means from samples of size n have standard deviation Averages are less variable than individual observations.

17. Sampling Distribution of Sample 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)
“If measurements in the population follow a Normal distribution, then so does the sample mean.”

18. (Population distribution)
Case Study
Does This Wine Smell Bad?
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)

20. is approximately N(µ, / )
Central Limit Theorem
If a random sample of size n is selected from ANY population with mean m and standard deviation  , then when n is “large” the sampling distribution of the sample mean is approximately Normal:
is approximately N(µ, / )
“No matter what distribution the population values follow, the sample mean will follow a Normal distribution if the sample size is large.”

21. 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)

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

23. Statistical Inference
Provides methods for drawing conclusions about a population from sample data
Confidence Intervals
What is the population mean?
Tests of Significance
Is the population mean larger than 66.5?
This would be ONE-SIDED

24. Inference about a Mean Simple Conditions-will be relaxed
SRS from the population of interest
Variable has a Normal distribution N(m, s) in the population
Although the value of m is unknown, the value of the population standard deviation s is known

25. Confidence Interval A level C confidence interval has two parts
An interval calculated from the data, usually of the form: estimate ± margin of error
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.

26. NAEP Quantitative Scores
Case Study
NAEP Quantitative Scores

27. NAEP Quantitative Scores
Case Study
NAEP Quantitative Scores
The rule indicates that and m are within two standard deviations (4.2) of each other in about 95% of all samples.

28. NAEP Quantitative Scores
Case Study
NAEP Quantitative Scores
So, if we estimate that m lies within 4.2 of , we’ll be right about 95% of the time.

29. Confidence Interval Mean of a Normal Population
Take an SRS of size n from a Normal population with unknown mean m and known std dev. s. A level C confidence interval for m is:

30. Confidence Interval Mean of a Normal Population
LOOKING FOR z*

31. NAEP Quantitative Scores
Case Study
NAEP Quantitative Scores
Using the 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
We are 95% confident that the average NAEP quantitative score for all adult males is between and

32. 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. n
Sample means, n subjects
Population, x individual subjects m
If the population is normally distributed N(µ,σ), so will the sampling distribution N(µ,σ/√n).

33. This reasoning is the essence of statistical inference.
Ninety-five percent of all sample means will be within roughly 2 standard deviations (2*s/√n) of the population parameter m.
Because distances are symmetrical, this implies that the population parameter m must be within roughly 2 standard deviations from the sample average , in 95% of all samples.
Red dot: mean value
of individual sample
This reasoning is the essence of statistical inference.

34. Summary: Confidence Interval for the Population Mean

35. Hypothesis Testing Start by explaining when σ is known
Move to unknown σ should be straightforward

36. Stating Hypotheses Null Hypothesis, H0
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.

37. Stating Hypotheses Alternative Hypothesis, Ha
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.

38. 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

39. The P-value
The packaging process has a known standard deviation s = 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.

40. 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…?

41. P =
P =
Significant P-value ???
P =
P = 0.05
P =
P = 0.01
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.

42. The significance level a
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.

43. THE WHOLE POINT OF THIS!!!! Implications m
We don’t need to take lots of random samples to “rebuild” the sampling distribution and find m at its center.
THE WHOLE POINT OF THIS!!!! n
Sample
Population m
All we need is one SRS of size n, and relying on the properties of the sample means distribution to infer the population mean m.

44. 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. ^

45. Conditions for Inference about a Mean
Data are from a SRS of size n.
Population has a Normal distribution with mean m and standard deviation s.
Both m and s are usually unknown.
we use inference to estimate m.
Problem: s unknown means we cannot use the z procedures previously learned.

46. Standard Error
When we do not know the population standard deviation s (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

47. One-Sample t Statistic
When we estimate s 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.

48. 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 s more accurately as the sample size increases.

49. The t Distributions

50. Critical Values from
T-Distribution

52. 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
Ex. For a 98% confidence level, z*=t=2.326
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) = − (= neg. z*)

53. Excel TDIST(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.

54. 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) = × Hours, suggesting that if a person spent 10 hours studying their grade would be 70 points.

55. 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.

56. Sampling Distribution of ̂0 and ̂1 Based on Simulation
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 values
Then use it to conduct confidence intervals and hypothesis tests

57. Sampling Distribution of ̂0 and ̂1 Based on Simulation
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

58. Sampling Distribution of ̂0 and ̂1 Based on Simulation
Based on this simulation we can make the following statements
Estimated 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.

59. The Linearity of the OLS Estimators
A linear estimator satisfies the condition that it is a linear combination of the dependent variable
The estimator for the population slope is
Known as the
ordinary least
squares (OLS)
estimator.

60. 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

61. 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

62. Hypothesis Testing
The farther ̂1 is from a the more plausible the alternative hypothesis
Formalized via the T-statistic T
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

63. 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

64. Hypothesis Testing Alternative approach
Calculate p-values rather than comparing a z- or t-statistic to the relevant critical values
Start 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

65. 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)

66. Interpretation of the Regression Coefficients
The value of the dependent variable will change by j units with a one unit change in the explanatory variable
Holding everything else constant (ceteris paribus)

67. 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)
b0,b1,b2,...,bk are k unknown population parameters
u is unobservable random error

68. 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 b0,b1,...,bk are k population parameters to be estimated

69. 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.

70. 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)

71. 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.

72. 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