Transcript Mean & Standard Deviation

Warm-up
The mean grade on a standardized test is 88 with a
standard deviation of 3.4. If the test scores are normally
distributed, what is the probability that a person scored less
than an 84?
MEAN & STANDARD
DEVIATION
Section 6.1B
Warm-up
The probability that a person fails the 9th grade is 0.22.
Let x = the number of people out of 3 who fail the 9th
grade. Write the probability distribution of x.
What about if there were 4 people?
Mean Value of Random Variable
• Describes where the probability distribution
of x is centered.
• Symbol is
x
• Where do you think the mean is located on
the problem we did as a warm-up?
Standard Deviation
• Describes the variation of the
x
distribution.
• Symbol is
x
• If it’s small, then x is close to the
mean. If it’s large then there’s more
variability.
Flip 3 coins – what’s the mean
number of heads.
x=#
heads
p(x)
0
1
1
3
2
3
8
8
3
8
1
8
1
3
3
1
 x  0    1   2    3  
8
8
8
8
x 
x 
3

8
12
8
 x  1.5
6
8

3
8
Formula
x 
x
p(x)
• It’s also known as the Expected
value and is written E(x).
Apgar scores – 1 min. after birth and again 5 min.
Possible values are from 0 to 10. Find the mean.
x
P(x)
0
1
2
3
0.002 0.001 0.002 0.005
4
5
6
7
8
9
10
0.02
0.04
0.17
0.38
0.25
0.12
0.01
What about the standard deviation?
• How do you think we find it?
Variance: 
2
x

 x  
Standard Deviation:
2
x 
p(x)

2
x
Apgar scores – Calculate the standard
deviation
x
P(x)
0
1
2
3
0.002 0.001 0.002 0.005
4
5
6
7
8
9
10
0.02
0.04
0.17
0.38
0.25
0.12
0.01
Find Mean & Standard Deviation:
x = # cars
at red
light
P(x)
0
0.13
1
0.21
2
0.28
3
0.31
4
0.07
1.
Ex.
2.
3.
x = possible
winnings
P(x)
5
0.1
7
0.31
8
0.24
10
0.16
14
0.19
Find the mean
Find the Standard Deviation
Find the probability that x is within one
deviation from the mean.
500 raffle tickets are sold at $2 each. You bought 5 tickets.
What’s your expected winning if the prize is a $200 tv.?
There are four envelopes in a box. One envelope contains a $1
bill, one contains a $5, one contains a $10, and one a $50 bill. A
person selects an envelope. Find the expected value of the
draw. What should we charge for the game for it to be fair?
A person selects a card from a deck. If it is a red card, he wins
$1. If it is a black card between or including 2 and 10, he wins
$5. If it is a black face card, he wins $10, and if it is a black ace,
he wins $100. Find the expectation of the game. What would it
be if it cost $10 to play? What should I charge to make it a fair
game?
On a roulette wheel, there are 38 slots numbered 1 through 36
plus 0 and 00. Half of the slots from 1 to 36 are red; the other
half are black. Both the 0 and 00 slots are green. Suppose that
a player places a simple $1 bet on red. If the ball lands in a red
slot, the player gets the original dollar back, plus an additional
dollar for winning the bet. If the ball lands in a different-colored
slot, the player loses the dollar bet to the casino. What is the
player’s average gain?
Homework
• Page 354 (9 – 12),15, 16, 18, 19, 28, 29, 30)