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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
A rural enterprise center provides an advice service to small craft
businesses on all aspects of the business, such as production
methods, marketing accounts etc.
A candle making business incurs losses as a result of candle
glasses being overfilled or under filled (hence poor quality) when
colored candle wax is poured into the glass tubes.
The wax is filled to heights that are normally distributed with µ1 =
25 cm and σ1 = 0.5 cm and the heights of the tubes are normally
distributed with µ2 = 26 cm and σ2 = 0.4 cm.
Calculate the probability that height of the gap between the top
of the tube and the wax is between 0.5 cm and 1.5 cm.
A sociologist has been studying the criminal justice system in a
large city. Among other things, she has found that over the last 5
years the length of time an arrested person must wait between
their arrest and their trial is a normally distributed variable x with
µ=210 days and σ=20 days.
Q-) What percent of these people had their trial between 160
days and 190 days after their arrest?
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
The salaries of the employees of a
corporation are normally distributed with a
mean of $25,000 and a standard deviation
of $5,000.
a.) What is the probability that a randomly
selected employee will have a starting
salary of at least $31,000?
b.) What percentage of employees has
salaries of less than $12,200?
c.) What are the minimum and the
maximum salaries of the middle 95% of
the employees?
d.) If sixty-eight of the employees have
incomes of at least $35,600, how many
individuals
are
employed
in
the
corporation?
The weights of items produced by a company
are normally distributed with a mean of 4.5
ounces and a standard deviation of 0.3
ounces.
a.) What is the probability that a randomly
selected item from the production will weigh
at least 4.14 ounces?
b.) What percentage of the items weigh
between 4.8 to 5.04 ounces?
c.) Determine the minimum weight of the
heaviest 5% of all items produced.
d.) If 27,875 of the items of the entire
production weigh at least 5.01 ounces, how
many items have been produced?