Ch. 17 The Expected Value & Standard Error

• Review of box models1. Pigs – suppose there is a 40% chance of

getting a “trotter”. Toss a pig 20 times.– What does the box model look like?

2. Coin toss – 20 times- What does the box model look like?

3. Roll die – 10 times- What does the box model look like?

4. Roll die and count number of 5’s – 10 times- What does the box model look like?

Expected Value (EV)

• The expected value for a sum of draws made at random with replacement from a box = (# of draws)x(average of box).

1. Pigs tossed 20 times: EV=(20)(4/10)=8

2. Coin toss 20 times: EV =

3. 10 die rolls: EV =

4. 10 die rolls and count # of 5’s: EV =

• Sum = Expected Value + Chance Error– For example, if we toss a pig 20 times the

sum of the # of trotters = 8 + chance error

• Standard error is how large the chance error is likely to be.– The sum of draws within 1 SE of EV is

approximately 68% of the data.– The sum of draws within 2 SE of EV is

approximately 95% of the data.– The sum of draws within 1 SE of EV is

approximately 99.9% of the data.

• SE for sum =

• Example 1: Toss a coin 20 times and let the sum be the number of heads.

1. Draw the box model.2. Find the EV of the sum.3. Find the SE of the sum.

1. Find the average of the box2. Find the SD of the box3. Find the SE of the sum

4. Fill in the blanks: The number of heads in 20 coin tosses is likely to be ___ give or take ___ or so.

BOXnumber of draws SD

• Example 2: 10 die rolls1. Draw the box model.

2. Find the EV of the sum.

3. Find the SE of the sum.1. Find the average of the box

2. Find the SD of the box

3. Find the SE of the sum

4. Fill in the blanks: The sum of 10 die rolls is likely to be ___ give or take ___ or so.

• A shortcut for boxes with only 2 kinds of tickets.

Big # Small #

SD(box) = ( ) ( )( )big small fraction big fraction small

– Example 3: Roll a die 10 times and count the number of 5’s.1. Draw the box model.

2. Find the EV of the sum.

3. Find the SE of the sum.1. Find the average of the box

2. Find the SD of the box

3. Find the SE of the sum

• The number of 5’s in 10 die rolls is likely to be around ___ give or take ___.

Using the normal table

• This is a way to generalize a large number of draws with replacement.

1. Calculate the EV & SE for sum of draws

2. Repeat many times

3. Make a histogram of the sums

specific sum EV of sumz

SE of sum

• Example 4: 100 draws with replacement from

Smallest sum =

Largest sum =

Average =

EV of sum =

SD of box =

SE of sum =

We expect the sum of 100 draws to be ____

give or take ____.

1 2 3

• If we repeat this scenario many times, what percent of the time will the sum be above 220? (In other words, what is the probability of getting a sum greater than 220?)