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Ch. 17 The Expected Value & Standard Error

Ch. 17 The Expected Value & Standard Error. Review of box models Pigs – suppose there is a 40% chance of getting a “trotter”. Toss a pig 20 times. What does the box model look like? Coin toss – 20 times What does the box model look like? Roll die – 10 times

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Ch. 17 The Expected Value & Standard Error

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  1. Ch. 17 The Expected Value & Standard Error • Review of box models • Pigs – suppose there is a 40% chance of getting a “trotter”. Toss a pig 20 times. • What does the box model look like? • Coin toss – 20 times • What does the box model look like? • Roll die – 10 times • What does the box model look like? • Roll die and count number of 5’s – 10 times • What does the box model look like?

  2. 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). • Pigs tossed 20 times: EV=(20)(4/10)=8 • Coin toss 20 times: EV = • 10 die rolls: EV = • 10 die rolls and count # of 5’s: EV =

  3. 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.

  4. SE for sum = • Example 1: Toss a coin 20 times and let the sum be the number of heads. • Draw the box model. • Find the EV of the sum. • Find the SE of the sum. • Find the average of the box • Find the SD of the box • Find the SE of the sum • Fill in the blanks: The number of heads in 20 coin tosses is likely to be ___ give or take ___ or so.

  5. Example 2: 10 die rolls • Draw the box model. • Find the EV of the sum. • Find the SE of the sum. • Find the average of the box • Find the SD of the box • Find the SE of the sum • Fill in the blanks: The sum of 10 die rolls is likely to be ___ give or take ___ or so.

  6. A shortcut for boxes with only 2 kinds of tickets. Big # Small # SD(box) =

  7. Example 3: Roll a die 10 times and count the number of 5’s. • Draw the box model. • Find the EV of the sum. • Find the SE of the sum. • Find the average of the box • Find the SD of the box • Find the SE of the sum • The number of 5’s in 10 die rolls is likely to be around ___ give or take ___.

  8. Using the normal table • This is a way to generalize a large number of draws with replacement. • Calculate the EV & SE for sum of draws • Repeat many times • Make a histogram of the sums

  9. 1 2 3 • 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 ____.

  10. 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?)

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