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Properties of Normal Distributions
1- The entire family of normal distribution is
differentiated by its mean µ and its standard deviation σ.
2- The highest point on the normal curve is at the mean
which is also the median and the mode of the distribution.
3- The mean of the distribution can be any numerical
value: negative, zero or positive.
4- The normal distribution is symmetric
5- The standard deviation determines how flat and wide
the curve is
6- Probabilities for the random variables are given by areas
under the curve. The total area under the curve for the
normal distribution is 1
7- Because the distribution is symmetric, the area under the
curve to the left of the mean is 0.50 and the area under the
curve to the right of the mean is 0.50
8- The percentage of values in some commonly used
intervals are;
a-) 68.3% of the values of a normal r.v are within plus or
minus one st.dev. of its mean
b-) 95.4% of the values of a normal r.v are within plus or
minus two st.dev. of its mean
c-) 99.7% of the values of a normal r.v are within plus or
minus three st.dev. of its mean
Find the area under the standard normal
curve that lies
1- to the right of z = - 0.55
2- to the left of
z = 0.84
3- to the right of z =
4- to the left of
1.69
z = - 0.74
5- between z = 0.90 and z= 1.33
6- between z = -0.29 and z= 0.59
Q-1 - MOBILE PHONE
Assume that the length of time, x, between charges of mobile
phone is normally distributed with a mean of 10 hours and a
standard deviation of 1.5 hours. Find the probability that the
mobile phone will last between 8 and 12 hours between charges.
Q-2 ALKALINITY LEVEL
The alkalinity level of water specimens collected from the River
in a country has a mean of 50 milligrams per liter and a standard
deviation of 3.2 milligrams per liter. Assume the distribution of
alkalinity levels is approximately normal and find the probability
that a water specimen collected from the river has an alkalinity
level
A-) exceeding 45 milligrams per liter.
B-) below 55 milligrams per liter.
C-) between 51 and 52 milligrams per liter.
The grades of 400 students in a statistics course
are normally distributed with mean µ = 65 and
variance σ2=100
Q: Find the probability that a student selected
randomly from this group would score within any
interval given below.
1- A grade between 60 and 65
2- A grade between 70 and 65
3- A grade between 52 and 68
4- A grade that is greater than 85
5- A grade that is less than 72
6- A grade between 70 and 78
Example : 1
A normal distribution has the mean 74.4
Find its standard deviation if 10% of the area under the
curve lies to the right of 100
Example : 2
A random variable has a normal distribution with standard
deviation 10 . Find its mean if the probability is 0.8264 that it
will take on a value less than 77.5
Example :3
For a certain random variable having the normal distribution,
the probability is 0.33 that it will take on a value less than
245 and the probability is 0.48 that it will take on a value
greater than 260.
Find the mean and standard deviation of the random
variable