Download Finding Z - scores - Newfane Central School

Finding Z – scores
& Normal Distribution
Using the Standard Normal Distribution
Week 9
Chapter’s 5.1, 5.2, 5.3
Normal Distribution
• Normal Distribution - is a very important
statistical data distribution pattern occurring in
many natural phenomena, such as height, blood
pressure, grades, IQ, baby birth weights, etc.
• Normal Curve - when graphing the normal
distribution as a histogram, it will create a bellshaped curve known as a normal curve.
• It is based on Probability! You’ll see!
Normal Distribution Curve:
What is this curve all about?
• The shape of the curve is bell-shaped
• The graph falls off evenly on either side of the
mean. (symmetrical)
• 50% of the distribution lies on the left of the mean
• 50% lies to the right of the mean. (above)
• The spread of the normal distribution is controlled
by the standard deviation.
• The mean and the median are the same in a normal
distribution. (and even the mode)
Features of Standard Normal Curve
•
•
•
•
•
Mean is the center
68% of the area is within one S.D.
95% of area is within two S.D.’s
99% of area is within 3 S.D.’s
As each tail increases/decreases, the graph
approaches zero (y axis), but never equals zero
on each end.
• For each of these problems we will need pull-out
table IV in the back of text
What is a Z – Score?
• Z-score’s allow us a method of converting,
proportionally, a study sample to the whole
population.
• Z-Score’s are the exact number of standard
deviations that the given value is away from
the mean of a NORMAL CURVE.
• Table IV always solves for the area to the left
of the Z-Score!
Finding the area to the left of a Z
(Ex. 1) – Find the area under the standard
normal curve that lies to the left of Z=1.34.
Finding the area to the right of a Z
(Ex. 2) - Find the area under the standard normal
curve that lies to the right of Z = -1.07.
Finding the area in-between two Z’s
(Ex. 3) - Find the area under the standard
normal curve that lies between Z=-2.04 and
Z=1.25.
Formula:
Z
xi  
x = data value
u = population mean

Practice examples:
• For each of the following examples, Look
for the words "normally distributed" in a
question before using Table IV to solve
them.
• Don’t forget - Table IV always solves for
the area to the left of the Z-Score!
Finding Probabilities
The shaded area under the curve is equal to the
probability of the specific event occurring.
• Ex (4) - A shoe manufacturer collected data
regarding men's shoe sizes and found that the
distribution of sizes exactly fits a normal
curve. If the mean shoe size is 11 and the
standard deviation is 1.5.
(a)What is the probability of randomly selecting a
man with a shoe size smaller than 9.5?
(b)If I surveyed 40 men, how many would be
expected to wear smaller than 9.5?
How did we get that answer:
Z
xi  

9.5  11
Z
 1
1.5
This is how
many SD’s
from the mean
• -1.00 is a Z-score (# of S.D.’s from the mean) that refers
to the area to the left of that position. Find it in Table IV.
• -1.00 = .1587
•We want the area to the left of that curve, so, this is the
answer. Table IV gives us the answer for area to the left of
the curve.
(b) .1587(40) = 6.3 = 6
Ex (5) – Gas mileage of vehicles follows a normal
curve. A Ford Escape claims to get 25 mpg
highway, with a standard deviation of 1.6 mpg. A
Ford Escape is selected at random.
(a) What is the probability that it will get more than
28 mpg?
(b) If I sampled 250 Ford Escapes, how many
would I expect to get more than 28 mpg?
How did we get that answer:
xi   Z  28  25  1.875
Z

1. 6
•1.875 is a Z-score (# of S.D.’s from the mean) that
refers to the area to the left of that position.
1.875 = .9696
•We want the area to the right of that curve, thus
1- .9696 = .0304
Ex (6) – This past week gas prices followed a
normal distribution curve and averaged $3.73
per gallon, with a standard deviation of 3
cents. What percentage of gas stations charge
between $ 3.68 and $ 3.77?
Ex (7) – This week gas prices followed a normal
distribution curve and averaged $3.71 per
gallon, with a standard deviation of 3 cents.
(a) What percent of stations charge at least $3.77?
(b) What percent will charge less than $3.71?
(c) What percent will charge less than $3.69?
(d) What percent will charge in-between $3.67and
$3.75 per gallon?
(e) If I sampled 30 gas stations, how many would
charge between $3.67and $3.75 per gallon?
NOW – Going back from
probabilities to Z-Scores:
Chapter 5.3
– Finding Z-scores from probabilities
– Transforming a Z-score to an X-value
• Look up .9406 on Table 4
• What Z-score corresponds to this area?
Finding Z-score’s of area to the left
(Ex. 8)
(a) Find the Z-score so that the area to the left is 10.75%
(b) Find the Z-score that represents the 75th percentile?
(c) Find the Z-score so that the area to the left is .88
(d) Find the Z-score so that the area to the left is .9880
Transforming a Z-score to an x-value
x    z
Look for the three ingredients to solve for x:
Population mean, standard deviation, and you
will need the Z-score that corresponds to the
given percent (or probability)
Try: 90th percentile
  45   2.5
Finding Z-score’s of area to the left
(Ex. 9) – The national average on the math portion of
the SAT is a 510 with a standard deviation of 130.
SAT scores follow a normal curve.
(a) What score represents the 90th percentile?
(b) What score will place you at the 35th percentile?
Finding Z-score’s of area to the Right
(Ex. 10) – Find the Z score so that the area
under the standard normal curve to the right
is .7881
Find the Z score of area to the Right
(Ex. 11) – A batch of Northern Pike at a local
fish hatchery has a mean length of a 8 inches
just as they are released to the wild. Their
lengths are normally distributed with a
standard deviation of 1.25 inches.
What is the shortest length that could still be
considered part of the top 15% of lengths?
Find the Z score of area in-between
two Z’s
(Ex. 12) – Find the Z score that divides the
middle 90% of the area under the standard
normal curve.
Critical Two-tailed Z value:
Z
2
  1 %
- Used to find the remaining
percent on the outside of the
area under the curve.
- 90% is equal to 1-.90=.10
- .10/2 = .05 = 5%
M&M Packaging
Ex (13) – A bag of M&M’s contains 40 candies
with a Standard deviation of 3 candies. The
packaging machine is considered un-calibrated
if it packages bags outside of 80%, centered
about the mean. What interval must the candies
be between for sale?
The Central Limit Theorem:
• As the sample sizes increases, the sampling
distribution becomes more accurate in
representation of the entire population.
• Thus, As additional observations are added
to the sample, the difference of the Sample
mean and the population mean approaches
ZERO. ( No difference)