Download Unit 7 Memories Section 1

Probability and Statistics – Mrs. Leahy
Unit 7 Memories
Section 1: Graphs of Normal Probability Distribution
Ex: The yearly wheat yield per acre on a particular farm is
normally distributed with a mean µ=35 bushels and a standard
deviation σ=8 bushels. What is the probability that the yield
will be between 19 and 35 bushels per acre?
Section 2: Standard Units and Areas Under the Standard Normal Distribution
standard score: z 
x

convert standard score to raw score: x  z  
z score tells the number of standard deviations that x is from the mean.
Ex: Robert figures it takes an average (mean) of 17 minutes with a standard
deviation of 3 minutes to drive from home, park the car, and walk to an earlymorning class. One day it took Robert 21 minutes to get to class. How many
standard deviations from the average is that?
What commuting time corresponds to a standard score of z= -2.5 ? Could Robert
count on making it to class in this amount of time or less?
Comparisons of Standard Scores:
Normal Distribution 1:
Bob’s History Quiz
µ = 50, σ=3
Bob’s score: 55
Normal Distribution 2:
Bob’s Algebra Quiz
µ=35, σ=15
Bob’s score: 55
On which quiz
did Bob do better
compared to the
rest of his class?
Standard Normal Distribution Table:
Tells you the probability that a raw score x falls to the left of a specified z value.
Ex. P( z  1.38)
Ex. P( z  0.45)
Ex. P( 1.25  z  1.78)
Working Backwards: Find z if you know the probability
Use table to identify the closest probability.
Read up and left to identify the corresponding z value
Reading Table:
Left Tail
Centered on Mean
P( z  #)  A = read from table directly
P(  #  z  #)  P( between z &  z ) 
Right Tail
P( z  #) = 1  P( z  #) = 1  A
1 A
2
Ex. Find z such that 8% of the standard normal curve lies to the
left of z.
Ex. Find z such that 42% of the standard normal curve lies to
the right of z.
Ex. Find z such that 70% of the standard normal curve lies
between –z and z.
Section 3: Areas Under ANY Normal Curve
Convert raw scores (x) to standard scores (z), then use the standard
normal distribution table to find the corresponding probabilities.
Ex. P( x  95);   80;  25
Working Backwards: Find x if you know the probability
Use table to identify the closest probability.
Read up and left to identify the corresponding z value
Convert the z value into the x value using x  z  
Ex. The average life of a refrigerator is 14 years with a standard
deviation of 2.5 years. A company has a guarantee program that
will replace the product if it breaks. How many years (round to
nearest tenth) should they guarantee the product if they don’t
want to replace more than 5% of the refrigerators?
Section 4: Central Limit Theorem
Use for finding the probability of an average for a sample (mean of a sample) instead of a single value.
Probability of a single value x
̅
Probability of an average 𝒙
Use
µ = mean
σ = standard deviation
x = single value
Use
µ = mean
σ = standard deviation
𝑥̅ = average of the sample
n = sample size
Convert to standard score:
Convert to standard score:
z
z
x

x

n
Use table to find probability.
Use table to find probability.
Ex. In mountain country, major highways sometimes use tunnels instead of long, winding roads over high passes.
However, too many vehicles in the tunnel at the same time can cause a hazardous situation. Traffic engineers are
studying a long tunnel in Colorado. If x represents the tme for a vehicle to go through the tunnel, it is known that
the x distribution has a mean of 12.1 minutes and a standard deviation of 3.8 minutes under normal traffic
conditions. The shape of the x distribution is mound shaped symmetrical.
Engineers have calculated that on average, vehicles should spend from 11 to 13 minutes in the tunnel -- less than
11 minutes means traffic is moving too fast, more than 13 minutes creates a problem of bad air quality.
a) If a single vehicle is selected at random,
what is the probabilty that it will be in the
tunnel for less than 10 minutes?
b) If a sample of 50 vehicles is selected,
what is the probability that the mean time
for the 50 vehicles will be less than 10 minutes?
c) If a single vehicle is selected at random,
what is the probability that it will be in the
tunnel from 11 to 13 minutes?
d) If a sample of 50 vehicles is selected,
what is the probability that the mean time
for the 50 vehicles will be from 11 to 13
minutes?