Normal Approximation

STAT 20: Introduction to Probability and Statistics

Warmup

10:00

True or false?

One hundred draws will be made with replacement from the box: \(\boxed{\boxed{0}\quad \boxed{2} \quad \boxed{3}\quad \boxed{4} \quad\boxed{6}}\).

Which of the following statements are true? *

1. The expected value of the sum of the one hundred draws is 300, give or take 20 or so.
2. The expected value of the sum of the one hundred draws is 300.
3. The sum of the one hundred draws is 300, give or take 20 or so.
4. The sum of the one hundred draws is 300.

* From the text Statistics by Freedman, Pisani, and Purves

1. False (expected value is a constant)
2. True
3. True (value of random sum, on average, is 300, give or take one SD)
4. False

• Let \(X\) be a random variable that is the value of one ticket drawn at random from this box, and \(E(X) = 3\)

• Then \(E(X)\) is a constant, and as seen in the notes, equal to the box average, which is (0+2+3+4+6)/5 = 3.

• The expected value of the sum of 100 draws is \(100\times 3 = 300\).

• The “give or take” number given here is the SD of the sum of 100 draws, using the square root law. The SD of a single draw is 2 (\(SD(X)\)), so the SD of the sum of 100 draws is \(2\times 10 = 20\).

Concept Questions

We have two random variables: \(X \sim\) Binomial(\(10, 0.2\)) and let \(Y\) be the value of one ticket drawn at random from a box with tickets \(\fbox{0}\, \fbox{2}\, \fbox{3} \,\fbox{4} \,\fbox{6}\).

We take the sum of 100 iid random variables for each of \(X\) and \(Y\), called \(SX_{100}\) and \(SY_{100}\). The empirical distributions of \(SX_{100}\) and \(SY_{100}\) are plotted below.

Which distribution belongs to which random variable?

01:00

A is \(SY_{100}\) and B is \(SX_{100}\). They need to remember that the expected value of \(SX_{100}\) is \(100 \times 2 = 200\) and of \(SY_{100}\) is \(100 \times 3 = 300\). The spreads agree too: \(SD(SX_{100}) = 10\sqrt{1.6} \approx 12.6\) and \(SD(SY_{100}) = 10 \times 2 = 20\), so B is narrower than A.

01:00

A die will be rolled \(n\) times and the object is to guess the total number of spots in \(n\) rolls, and you choose \(n\) to be either 50 or 100. There is a one-dollar penalty for each spot that the guess is off. For instance, if you guess 200, and the total is 215, then you lose 15 dollars. Which do you prefer?*

Which do you prefer? \(n = 50\) rolls, or \(n = 100\) rolls?

* From the text Statistics by Freedman, Pisani, and Purves

Prefer 50 because the square root law tells us the spread of the distribution will be smaller for fewer rolls. If we were computing the average number of spots, then we would pick larger \(n\) since \(\sqrt{n}\) goes in the denominator of SE.

Worksheet: Normal Approximation

20:00

Break

05:00

Lab 7: Elections

25:00