Download Properties of the Normal Distribution

8.2 Properties of the Normal Distribution
- Most continuous distributions are distributed ______________________________
about the mean, forming a distinctive _____________________.
- This bell-shaped distribution is the ‘normal’ distribution.
- This symmetry makes both the mode and median equal to the mean. This allows one to
use ________________, and the _____________________________ to describe
the data.
- Some distinctive patterns can be found within the normal distribution.
  3
  2
 

 
  2
  3
- The mean,  , lies at the _____________________________________________,
so there is an equal number of data values of X ____________________ the mean.
- Approximately ____% of the data values of X will lie within the range ________ and
_________.
- Approximately ____% of the data values of X will lie within the range ________ and
________.
- Approximately ____% of the data values of X will lie within the range ________ and
________.
Properties and Predictions
Ex/ The weights of 10 000 students are normally distributed. The mean of the weights
is 55 kg and the standard deviation is 5 kg.
a) Find the number of students weighing between 50 and 60 kg.
b) Find the number of students weighing less than 45 kg.
c) Find a range of weights that would include 99% of the measures.
a)
b)
c)
- The area under the curve can be found by using _____________, where z 
x 

because these scores are also have a normal distribution with mean 0 and standard
deviation 1.
- The z-score refers to the ___________________________________________
_______________________________________________________.
- To determine a probability using a z-score:
1) Calculate the z-score using the formula:
z 
x 

2) Using a z-score table, find the probability value for P (X  x )  P (Z  a )
- A positive z-score means the value lies ___________________________, a negative
value is _______________________. See Pg. 606-7 for a z-score table.
Ex/ The times that people wait in a grocery checkout line is normally distributed. The
mean waiting time is 10 minutes, with a 2.5 minute standard deviation. What is the
probability that Jack will be through the line in less than 5 minutes.
Ex/ A region receives an average of 5100 mm of rain annually. If the annual rainfall is
assumed to be normally distributed, with a standard deviation of 150 mm, find the
probability that
a) More than 5500 mm of rain will fall in a year.
b) Less than 5000 mm of rain will fall in a year.
Using Percentiles
- A percentile refers to the set of values _______________________________ of
the normal distribution.
i.e. The 90% percentile of a distribution means that a given value is ________
than or _____________ 90% of the values in the data set.
- You are looking for the value that satisfies the equation _____________________
Ex/ Resting pulse rates of humans are normally distributed. The mean pulse rate is 72
beats per minute and the standard deviation is 10.
a) What is the percentile rank of an athlete with a resting pulse rate of 96?
b) How many, of 1 million people, would have a rate less than 60 beats per minute?
a) P (X <80)
b) P (X <60)
Cutoff Values
Ex/ a) What heart rate would the top 10% of the population have?
b)How about the top 10% of athletes?
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