Download Mini Unit Day 3 - Issaquah Connect

Name: ________________________________ Per _____ Normal Distributions and Z-Values
Assignment: Section 11.3 # 3, 6 & 11.4 # 1-6, 13
In most real life situations you collect a sample
from a population and calculate the sample mean
and standard deviation (𝑥̅ and s)
Population
Parameters:  and 
When, what you really want to know is the
population mean and standard deviation ( and )
Sample
Statistics: 𝑥̅ and s
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Standard deviation is a number that tells you how spread out the data is away from the mean in a
Normal Probability Distribution. It is a better than the either the range or the interquartile range
because it uses every piece of data in the calculation of this statistic.
Compare the graphs of several normal probability distributions that have different means and
different standard deviations. Changing the mean will change the LOCATION of the graph (shift right
or shift left). Changing the standard deviation will change the SHAPE of the graph (tall skinny graphs
for small standard deviations and short fat graphs for larger standard deviations.)
In this lesson you will learn how rare (unlikely) or how common (very likely) values of data are in a
Normal Distribution sample. We will also learn about one normal distribution, called the standard
normal distribution and how to use z-scores and the 68-95-99.7% Rule to determine how rare or
common a piece of data is in a Normal Distribution.
The 68-95-99.7% Rule
In order to find out how rare (unlikely) or how
common (very likely) a piece of data is you
need to know how many standard deviations
away from the mean the data falls.
Z-Scores
Draw vertical lines and label the x-axis to
show that you understand this rule.
Draw vertical lines to show the 68-95-99.7% Rule
A data’s z-score tells you exactly how many
standard deviations away from the mean that
piece of data is. You can find the z-score by
subtracting the mean and then dividing by the
standard deviation.
𝑧=
𝑥−𝜇
𝜎
Knowing the z-score tells you quickly how rare or common that data is.
Example A: The heights of a large group of men are distributed normally with mean 70 inches and
standard deviation 2.5 inches.
(a) Find the z-values for 67.5 inches and 72.5
inches
(b) What is the probability that a randomly
chosen member of this group has height x
between 67.5 and 72.5 inches?
(c) What is the probability that a randomly chosen member of this group has height x between 65
and 75 inches?
(d) Find an interval of x-values, symmetric about the mean, that contains 95% of the heights
(e) Find an interval of x-values, symmetric about the mean, that contains 99.7% of the heights
Example B:
Example C:
The weights of 1000 children were recorded on their first birthdays. The weights are normally
distributed with mean 10.3 kg and standard deviation 1.6 kg.

What is the probability that a randomly selected child will weigh less than 8.7 kg?

What is the probability that a randomly selected child will weigh between 8.7 kg and 11.9 kg?

How many of the 1000 children would you expect to weigh between 7.1 kg and 13.5 kg?

How many of the 1000 children would you expect to weigh more than 13.5 kg?
Example D:
The mean commuting time for a resident of a certain metropolitan area is 38 minutes, with a standard
deviation of 10 minutes. Assume that commuting times for this area are normally distributed.

Find the z-value for a 23-minute commute.

Find the z-value for a 60-minute commute.

What is the probability that a commute for a randomly chosen resident will be between 28
minutes and 58 minutes?
To show in class:
68% of the data will fall within ______ standard
deviation away from the mean.
95% of the data will fall within ______ standard
deviations away from the mean.
99.7% will fall within ______ standard deviations
away from the mean.