Math 1710 Class 12 CLT Normal Arising in Dr. Allen Back Averages - - PowerPoint PPT Presentation

Math 1710 Class 12 V1c Normal Prob

• n a Calc

Sampling Dist CLT Normal Arising in Averages

Math 1710 Class 12

• Dr. Allen Back
• Sep. 21, 2016

Math 1710 Class 12 V1c Normal Prob

• n a Calc

Sampling Dist CLT Normal Arising in Averages

From Z-scores to Probabilities on the Calculator

On something like a TI-84 2nd → DISTR → normalcdf (−10, 1) will give you P(Z < 1). Here −10 is a substitute for −∞, good enough for 8 decimal places. Please always compute your Z-scores if using the calculator even though the calculator has a way of bypassing this.

Math 1710 Class 12 V1c Normal Prob

• n a Calc

Sampling Dist CLT Normal Arising in Averages

From Z-scores to Probabilities on the Calculator

On something like a TI-89: catalog → F3 → 2nd alpha N → normalcdf (−10, 1)

Math 1710 Class 12 V1c Normal Prob

• n a Calc

Sampling Dist CLT Normal Arising in Averages

Sampling Distribution

Suppose that the chance of a ‘yes’ answer from each individual in a poll is p and each of the responses of the n people participating are independent.

Math 1710 Class 12 V1c Normal Prob

• n a Calc

Sampling Dist CLT Normal Arising in Averages

Sampling Distribution

Suppose that the chance of a ‘yes’ answer from each individual in a poll is p and each of the responses of the n people participating are independent. Then if we take one such poll of size n, we will get k yes responses and compute an observed proportion of yeses of ˆ p = k n.

Math 1710 Class 12 V1c Normal Prob

• n a Calc

Sampling Dist CLT Normal Arising in Averages

Sampling Distribution

Then if we take one such poll of size n, we will get k yes responses and compute an observed proportion of yeses of ˆ p = k n. If we have a (typically large) population, each sample gives rise to a number ˆ

• p. Each sample also has a certain probability. So

we can view ˆ p as a random variable with a probability distribution reflecting the chance of all the different values of ˆ p showing up. This random variable is what we mean by the sampling distribution of ˆ p.

Math 1710 Class 12 V1c Normal Prob

• n a Calc

Sampling Dist CLT Normal Arising in Averages

Sampling Distribution

If we have a (typically large) population, each sample gives rise to a number ˆ

• p. Each sample also has a certain probability. So

we can view ˆ p as a random variable with a probability distribution reflecting the chance of all the different values of ˆ p showing up. This random variable is what we mean by the sampling distribution of ˆ p. It is a probability based “theoretical” idea, though we can explore it by simulation.

Math 1710 Class 12 V1c Normal Prob

• n a Calc

Sampling Dist CLT Normal Arising in Averages

Sampling Distribution

Similarly, for quantitative data following some model, we can think about all possible samples of size n, and keep track of the probabilities of all the different ¯ x′s showing up. This is the sampling distribution of ¯ x.

Math 1710 Class 12 V1c Normal Prob

• n a Calc

Sampling Dist CLT Normal Arising in Averages

Sampling Distribution

One doesn’t actually need a population to do this. If one has a random variable describing one individual’s response, (perhaps X ∼ Bernoulli(p) in the polling (categorical with 2 responses) case or a more general X in the quantitative case, we can just consider n independent copies of X (keeping track of the n responses in a sample of size n) to determine our summary statistic ˆ p or ¯ x.

Math 1710 Class 12 V1c Normal Prob

• n a Calc

Sampling Dist CLT Normal Arising in Averages

Three CLT problems

(1) 80% of all cars on the interstate speed. Randomly sample 50. What proportion of speeders might we see?

Math 1710 Class 12 V1c Normal Prob

• n a Calc

Sampling Dist CLT Normal Arising in Averages

Three CLT problems

(1) 80% of all cars on the interstate speed. Randomly sample 50. What proportion of speeders might we see? (e.g.: A policeman has a quota of at least 35 speeders to ticket. What is the chance he’ll meet his quota in such a sample?)

Math 1710 Class 12 V1c Normal Prob

• n a Calc

Sampling Dist CLT Normal Arising in Averages

Three CLT problems

(2) At birth, babies average 7.8 pounds with a standard deviation of 2.1 pounds. 34 Babies born near a large possibly polluting factory average 7.2 pounds.

Math 1710 Class 12 V1c Normal Prob

• n a Calc

Sampling Dist CLT Normal Arising in Averages

Three CLT problems

(2) At birth, babies average 7.8 pounds with a standard deviation of 2.1 pounds. 34 Babies born near a large possibly polluting factory average 7.2 pounds. Is that unusually low?

Math 1710 Class 12 V1c Normal Prob

• n a Calc

Sampling Dist CLT Normal Arising in Averages

Three CLT problems

(3) Suppose SAT’s have a mean of 500, standard deviation of 100 and are approximately normal. Form means of samples of 20 students.

Math 1710 Class 12 V1c Normal Prob

• n a Calc

Sampling Dist CLT Normal Arising in Averages

Three CLT problems

(3) Suppose SAT’s have a mean of 500, standard deviation of 100 and are approximately normal. Form means of samples of 20 students. What distribution do these follow? What is the chance of a sample averaging over 600? Over 550?

Math 1710 Class 12 V1c Normal Prob

• n a Calc

Sampling Dist CLT Normal Arising in Averages

Proportion Case of the CLT

X ∼ Binomial(n,p) X approximated by normal RV Y ∼ N(np, √npq). Suppose we keep track of the observed proportion ˆ p = X n here.

Math 1710 Class 12 V1c Normal Prob

• n a Calc

Sampling Dist CLT Normal Arising in Averages

Proportion Case of the CLT

X ∼ Binomial(n,p) X approximated by normal RV Y ∼ N(np, √npq). Suppose we keep track of the observed proportion ˆ p = X n here. Then ˆ p is an RV approximated by Y /n ∼ N(p, SD(ˆ p)). where SD(ˆ p) = pq n .

Math 1710 Class 12 V1c Normal Prob

• n a Calc

Sampling Dist CLT Normal Arising in Averages

Proportion Case of the CLT

X ∼ Binomial(n,p) X approximated by normal RV Y ∼ N(np, √npq). Suppose we keep track of the observed proportion ˆ p = X n here. Then ˆ p is an RV approximated by Y /n ∼ N(p, SD(ˆ p)). where SD(ˆ p) = pq n . Thus N(p,

• pq

n ) is the sampling distribution of ˆ

p.

Math 1710 Class 12 V1c Normal Prob

• n a Calc

Sampling Dist CLT Normal Arising in Averages

Proportion Case of the CLT

Some books uses the notation SD(ˆ p) = pq n .

Math 1710 Class 12 V1c Normal Prob

• n a Calc

Sampling Dist CLT Normal Arising in Averages

Proportion Case of the CLT

Some books uses the notation SD(ˆ p) = pq n . SD(ˆ p) stands for Standard Deviation of the Sampling Distribution of ˆ p.

Math 1710 Class 12 V1c Normal Prob

• n a Calc

Sampling Dist CLT Normal Arising in Averages

Proportion Case of the CLT

Some books uses the notation SD(ˆ p) = pq n . SD(ˆ p) stands for Standard Deviation of the Sampling Distribution of ˆ p. And the notation SE(ˆ p) =

• ˆ

pˆ q n . SE(ˆ p) stands for the Standard Error of ˆ p.

Math 1710 Class 12 V1c Normal Prob

• n a Calc

Sampling Dist CLT Normal Arising in Averages

Proportion Case of the CLT

Others, including your textbook, I believe, are happy to refer to both of these as standard errors of ˆ

• p. One is exact (but hard to

know in practice) while the other is only an estimate, but feasible to produce.

Math 1710 Class 12 V1c Normal Prob

• n a Calc

Sampling Dist CLT Normal Arising in Averages

Quantitative Case of the CLT

Suppose X is a random variable with mean µ and standard deviation σ. Let X1, X2,. . . ,Xn be n independent copies of X. Set ¯ X = X1 + X2 + . . . + Xn n . Then

1 The mean of ¯

X is µ.

2 The standard deviation of ¯

X is σ √n.

3

¯ X is approximately normal as n gets large.

Math 1710 Class 12 V1c Normal Prob

• n a Calc

Sampling Dist CLT Normal Arising in Averages

Quantitative CLT - Why?

¯ X = X1 + X2 + . . . + Xn n .

1 The mean of ¯

X is µ.

2 The standard deviation of ¯

X is σ √n.

3

¯ X is approximately normal as n gets large.

Math 1710 Class 12 V1c Normal Prob

• n a Calc

Sampling Dist CLT Normal Arising in Averages

Quantitative CLT - Why?

¯ X = X1 + X2 + . . . + Xn n .

1 The mean of ¯

X is µ.

2 The standard deviation of ¯

X is σ √n.

3

¯ X is approximately normal as n gets large. (1) is immediate from E(X1 + X2 + . . . + Xn) = nE(X) so E( ¯ X) = nE(X) n = E(X).

Math 1710 Class 12 V1c Normal Prob

• n a Calc

Sampling Dist CLT Normal Arising in Averages

Quantitative CLT - Why?

¯ X = X1 + X2 + . . . + Xn n .

1 The mean of ¯

X is µ.

2 The standard deviation of ¯

X is σ √n.

3

¯ X is approximately normal as n gets large. (2) is immediate from Var(X1 + X2 + . . . + Xn) = nVar(X) so Var( ¯ X) = nVar(X) n2 = Var(X) n . And so the std. dev of ¯ X is

σ √n.

Math 1710 Class 12 V1c Normal Prob

• n a Calc

Sampling Dist CLT Normal Arising in Averages

Quantitative CLT - Why?

¯ X = X1 + X2 + . . . + Xn n .

1 The mean of ¯

X is µ.

2 The standard deviation of ¯

X is σ √n.

3

¯ X is approximately normal as n gets large. (3) is much deeper! We will illustrate, but it’s only barely possible to prove in even an advanced undergraduate course. Your text uses the notation SD(¯ x) = σ √n for the exact standard deviation of the sampling distribution of ¯ x and SE(¯ x) =

s √n for its approximation, the standard error of

¯ x.

Math 1710 Class 12 V1c Normal Prob

• n a Calc

Sampling Dist CLT Normal Arising in Averages

Underweight Babies?

(2) At birth, babies average 7.8 pounds with a standard deviation of 2.1 pounds. 34 Babies born near a large possibly polluting factory average 7.2 pounds.

Math 1710 Class 12 V1c Normal Prob

• n a Calc

Sampling Dist CLT Normal Arising in Averages

Underweight Babies?

(2) At birth, babies average 7.8 pounds with a standard deviation of 2.1 pounds. 34 Babies born near a large possibly polluting factory average 7.2 pounds. Is that unusually low?

Math 1710 Class 12 V1c Normal Prob

• n a Calc

Sampling Dist CLT Normal Arising in Averages

Underweight Babies?

(2) At birth, babies average 7.8 pounds with a standard deviation of 2.1 pounds. 34 Babies born near a large possibly polluting factory average 7.2 pounds. Is that unusually low? Solution: In fact 34 is not large enough to have confidence that an average of 34 copies will be approx. normal. Let’s work the problem under the assumption that it is in this case.

Math 1710 Class 12 V1c Normal Prob

• n a Calc

Sampling Dist CLT Normal Arising in Averages

Underweight Babies?

(2) At birth, babies average 7.8 pounds with a standard deviation of 2.1 pounds. 34 Babies born near a large possibly polluting factory average 7.2 pounds. Is that unusually low? If so, then an RV X describing the avg of 34 observations would follow: ¯ X ∼ N(7.8, 2.1 √ 34 ) = N(7.8, .36).

Math 1710 Class 12 V1c Normal Prob

• n a Calc

Sampling Dist CLT Normal Arising in Averages

Underweight Babies?

(2) At birth, babies average 7.8 pounds with a standard deviation of 2.1 pounds. 34 Babies born near a large possibly polluting factory average 7.2 pounds. Is that unusually low? ¯ X ∼ N(7.8, 2.1 √ 34 ) = N(7.8, .36). The z score of 7.2 is −.6

.36 = −1.67 and P(Z < −1.67) = .0485.

Math 1710 Class 12 V1c Normal Prob

• n a Calc

Sampling Dist CLT Normal Arising in Averages

Underweight Babies?

(2) At birth, babies average 7.8 pounds with a standard deviation of 2.1 pounds. 34 Babies born near a large possibly polluting factory average 7.2 pounds. Is that unusually low? ¯ X ∼ N(7.8, 2.1 √ 34 ) = N(7.8, .36). The z score of 7.2 is −.6

.36 = −1.67 and P(Z < −1.67) = .0485.

So (under our additional assumption of approximate normality), about 5% of the time a sample of 34 babies would average as low as 7.2 pounds just by chance. The evidence is not conclusive.

Math 1710 Class 12 V1c Normal Prob

• n a Calc

Sampling Dist CLT Normal Arising in Averages

Quantitive CLT Problem

(3) Suppose SAT’s have a mean of 500, standard deviation of 100 and are approximately normal. Form means of samples of 20 students.

Math 1710 Class 12 V1c Normal Prob

• n a Calc

Sampling Dist CLT Normal Arising in Averages

Quantitive CLT Problem

(3) Suppose SAT’s have a mean of 500, standard deviation of 100 and are approximately normal. Form means of samples of 20 students. What distribution do these follow? What is the chance of a sample averaging over 600? Over 550?

Math 1710 Class 12 V1c Normal Prob

• n a Calc

Sampling Dist CLT Normal Arising in Averages

SAT Scores of Group Averages

(3) Suppose SAT’s have a mean of 500, standard deviation of 100 and are approximately normal. Form means of samples of 20 students.

Math 1710 Class 12 V1c Normal Prob

• n a Calc

Sampling Dist CLT Normal Arising in Averages

SAT Scores of Group Averages

(3) Suppose SAT’s have a mean of 500, standard deviation of 100 and are approximately normal. Form means of samples of 20 students. What distribution do these follow? What is the chance of a sample averaging over 600? Over 550?

Math 1710 Class 12 V1c Normal Prob

• n a Calc

Sampling Dist CLT Normal Arising in Averages

SAT Scores of Group Averages

(3) Suppose SAT’s have a mean of 500, standard deviation of 100 and are approximately normal. Form means of samples of 20 students. What distribution do these follow? What is the chance of a sample averaging over 600? Over 550? Solution: n = 20 is too small to assume an average is automatically

• normal. But since the sum of independent normal RV’s is

normal, in this case we do have approximate normality.

Math 1710 Class 12 V1c Normal Prob

• n a Calc

Sampling Dist CLT Normal Arising in Averages

SAT Scores of Group Averages

(3) Suppose SAT’s have a mean of 500, standard deviation of 100 and are approximately normal. Form means of samples of 20 students. What distribution do these follow? What is the chance of a sample averaging over 600? Over 550? Solution: Thus the RV ¯ X describing means of 20 students satisfies X ∼ N(500, 100 √ 20 ) = N(100, 22.36).

Math 1710 Class 12 V1c Normal Prob

• n a Calc

Sampling Dist CLT Normal Arising in Averages

SAT Scores of Group Averages

(3) Suppose SAT’s have a mean of 500, standard deviation of 100 and are approximately normal. Form means of samples of 20 students. What distribution do these follow? What is the chance of a sample averaging over 600? Over 550? Solution: Thus the RV ¯ X describing means of 20 students satisfies X ∼ N(500, 100 √ 20 ) = N(100, 22.36). The z score of 600 is

100 22.36 = 4.47.

P( ¯ X > 600) = P(Z > 4.47) < .0001.

Math 1710 Class 12 V1c Normal Prob

• n a Calc

Sampling Dist CLT Normal Arising in Averages

SAT Scores of Group Averages

(3) Suppose SAT’s have a mean of 500, standard deviation of 100 and are approximately normal. Form means of samples of 20 students. What distribution do these follow? What is the chance of a sample averaging over 600? Over 550? Solution: The z score of 600 is

100 22.36 = 4.47.

P( ¯ X > 600) = P(Z > 4.47) < .0001. Very tiny. An individual has a good chance of getting over 600 but an average of 20 randomly chosen people does not.

Math 1710 Class 12 V1c Normal Prob

• n a Calc

Sampling Dist CLT Normal Arising in Averages

SAT Scores of Group Averages

(3) Suppose SAT’s have a mean of 500, standard deviation of 100 and are approximately normal. Form means of samples of 20 students. What distribution do these follow? What is the chance of a sample averaging over 600? Over 550? Solution: The z score of 550 is

50 22.36 = 2.23.

P( ¯ X > 550) = P(Z > 2.23) = P(Z < −2.23) = .0129.

Math 1710 Class 12 V1c Normal Prob

• n a Calc

Sampling Dist CLT Normal Arising in Averages

SAT Scores of Group Averages

(3) Suppose SAT’s have a mean of 500, standard deviation of 100 and are approximately normal. Form means of samples of 20 students. What distribution do these follow? What is the chance of a sample averaging over 600? Over 550? Good intuition why the standard deviation of an average ¯ X is less than that of the RV X describing one individual:

Math 1710 Class 12 V1c Normal Prob

• n a Calc

Sampling Dist CLT Normal Arising in Averages

SAT Scores of Group Averages

(3) Suppose SAT’s have a mean of 500, standard deviation of 100 and are approximately normal. Form means of samples of 20 students. What distribution do these follow? What is the chance of a sample averaging over 600? Over 550? Good intuition why the standard deviation of an average ¯ X is less than that of the RV X describing one individual: Since 600 is one standard deviation above the mean, there is about a 16% chance for one individual to be above 600.

Math 1710 Class 12 V1c Normal Prob

• n a Calc

Sampling Dist CLT Normal Arising in Averages

SAT Scores of Group Averages

(3) Suppose SAT’s have a mean of 500, standard deviation of 100 and are approximately normal. Form means of samples of 20 students. What distribution do these follow? What is the chance of a sample averaging over 600? Over 550? Good intuition why the standard deviation of an average ¯ X is less than that of the RV X describing one individual: Since 600 is one standard deviation above the mean, there is about a 16% chance for one individual to be above 600. But for an average of 20 to be over 600, most of the 20 will likely have to achieve this 16% challenge.

Math 1710 Class 12 V1c Normal Prob

• n a Calc

Sampling Dist CLT Normal Arising in Averages

SAT Scores of Group Averages

(3) Suppose SAT’s have a mean of 500, standard deviation of 100 and are approximately normal. Form means of samples of 20 students. What distribution do these follow? What is the chance of a sample averaging over 600? Over 550? Good intuition why the standard deviation of an average ¯ X is less than that of the RV X describing one individual: Since 600 is one standard deviation above the mean, there is about a 16% chance for one individual to be above 600. But for an average of 20 to be over 600, most of the 20 will likely have to achieve this 16% challenge. The multiplication rule says this is quite unlikely.

Math 1710 Class 12 V1c Normal Prob

• n a Calc

Sampling Dist CLT Normal Arising in Averages

SAT Scores of Group Averages

(3) Suppose SAT’s have a mean of 500, standard deviation of 100 and are approximately normal. Form means of samples of 20 students. What distribution do these follow? What is the chance of a sample averaging over 600? Over 550? Good intuition why the standard deviation of an average ¯ X is less than that of the RV X describing one individual: Since 600 is one standard deviation above the mean, there is about a 16% chance for one individual to be above 600. But for an average of 20 to be over 600, most of the 20 will likely have to achieve this 16% challenge. The multiplication rule says this is quite unlikely. Our RV calculation σ ¯

X = σ

√n makes this intuition precise.

Math 1710 Class 12 V1c Normal Prob

• n a Calc

Sampling Dist CLT Normal Arising in Averages

CLT Simulation Notes

Start by realizing 500 trials from a uniform [0,1] distribution. (Mean=.5, Standard Deviation=.sqrt(1/12)=.289)

Math 1710 Class 12 V1c Normal Prob

• n a Calc

Sampling Dist CLT Normal Arising in Averages

CLT Simulation Notes

Math 1710 Class 12 V1c Normal Prob

• n a Calc

Sampling Dist CLT Normal Arising in Averages

CLT Simulation Notes

Probability Histograms - area of blue rectangle is the fraction

• f the results falling within the range at the base.

Math 1710 Class 12 V1c Normal Prob

• n a Calc

Sampling Dist CLT Normal Arising in Averages

CLT Simulation Notes

A Theoretical Uniform Distribution

Math 1710 Class 12 V1c Normal Prob

• n a Calc

Sampling Dist CLT Normal Arising in Averages

CLT Simulation Notes

Total area 1 and shaded area (gray) equal to

• prob. of a range of values (green) along the x-axis.

A Theoretical Uniform Distribution

Math 1710 Class 12 V1c Normal Prob

• n a Calc

Sampling Dist CLT Normal Arising in Averages

CLT Simulation Notes

Now square each value to get the simulation of a new

• distribution. (Mean=.333, Standard Deviation=.298)

Math 1710 Class 12 V1c Normal Prob

• n a Calc

Sampling Dist CLT Normal Arising in Averages

CLT Simulation Notes

Now square each value to get the simulation of a new

• distribution. (Mean=.333, Standard Deviation=.298)

Compare with a normal distribution N(.333,.298).

Math 1710 Class 12 V1c Normal Prob

• n a Calc

Sampling Dist CLT Normal Arising in Averages

CLT Simulation Notes

Math 1710 Class 12 V1c Normal Prob

• n a Calc

Sampling Dist CLT Normal Arising in Averages

CLT Simulation Notes

A poor match!

Math 1710 Class 12 V1c Normal Prob

• n a Calc

Sampling Dist CLT Normal Arising in Averages

CLT Pictures

Simulation of a Uniform Distribution

Math 1710 Class 12 V1c Normal Prob

• n a Calc

Sampling Dist CLT Normal Arising in Averages

CLT Pictures

Simulation of Square of a Uniform Distribution

Math 1710 Class 12 V1c Normal Prob

• n a Calc

Sampling Dist CLT Normal Arising in Averages

CLT Pictures

Now do the above four times and compare the averages of each

• f 500 sets of four values with N(.333,.298/2).

Math 1710 Class 12 V1c Normal Prob

• n a Calc

Sampling Dist CLT Normal Arising in Averages

CLT Pictures

Then 25 times and compare the averages of each of 500 sets of 25 values with N(.333,.298/5).

Math 1710 Class 12 V1c Normal Prob

• n a Calc

Sampling Dist CLT Normal Arising in Averages

CLT Pictures

As we compare the different averages, note that the horizontal scale is decreasing.

Math 1710 Class 12 V1c Normal Prob

• n a Calc

Sampling Dist CLT Normal Arising in Averages

CLT Pictures

For averages of four values, the part of the graph shown is from below 0 to .78.

Math 1710 Class 12 V1c Normal Prob

• n a Calc

Sampling Dist CLT Normal Arising in Averages

CLT Pictures

For averages of 25 values, the part of the graph shown is from .25 to .43.

Math 1710 Class 12 V1c Normal Prob

• n a Calc

Sampling Dist CLT Normal Arising in Averages

CLT Pictures

Now 100 times and compare the averages of each of 500 sets

• f 100 values with N(.333,.298/10).

Math 1710 Class 12 V1c Normal Prob

• n a Calc

Sampling Dist CLT Normal Arising in Averages

CLT Pictures

1600 times.

Math 1710 Class 12 V1c Normal Prob

• n a Calc

Sampling Dist CLT Normal Arising in Averages

CLT Pictures

A normal by comparison.

Math 1710 Class 12 V1c Normal Prob

• n a Calc

Sampling Dist CLT Normal Arising in Averages

CLT Pictures

A normal by comparison. This suggests the parts we see not matching are more due to simulation than non-normality.

Math 1710 Class 12 V1c Normal Prob

• n a Calc

Sampling Dist CLT Normal Arising in Averages

CLT Pictures

Fit near the ends.