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Intro. to normal distributions (5.1)
Review Definitions:
continuous random variable=
Its probability distribution is called a continuous probability distribution.
**The most important of these type of distribution is called the
________________________________________.
Properties of a Normal Distribution
1. The mean, median, and mode are _______________.
2. Its graph is called the ____________________________ , which is ____________
shaped and is ____________________ around the mean.
3. The total area under this curve is equal to _______.
4. The curve approaches, but _____________ _______________, the x-axis as it
extends farther and farther away from the mean.
**5. The area of the region under the curve gives the ______________________ that
the random variable will have a value in the corresponding area.
Picture of the Graph
One way to find the area under the curve is the _________________________________.
In a normal distribution with mean ________ and standard deviation _______, you can
approximate areas under the normal curve as follows:
1. About _________ of the area lies between ______________ and _____________.
2. About _________ of the area lies between ______________ and _____________.
3. About _________ of the area lies between ______________ and _____________.
Examples
1. Find the interval that contains about 95% of the distribution which has a mean µ = 15 and
a standard deviation σ = 3.
2. Find the probability that x falls in the interval P(5.25 < x < 8.75), given µ = 7 and σ = 1.75.
3. Adult IQ scores are normally distributed with µ = 100 and σ = 15. Estimate the probability
that a randomly chosen adult has an IQ between 70 and 115.
4. Using the data above, estimate the probability that a randomly chosen adult has an IQ
between 85 and 145.
5. The contents of a cereal box are normally distributed with a mean weight of 20 ounces
and a standard deviation of 0.07 ounce. Determine an interval of values which 95% of the
cereal box weights will fall.
6. The time per week a student uses a lab computer is normally distributed with a mean of
6.2 hours and a standard deviation of 0.9 hour. You are planning the schedule for the
computer lab. Of 2000 students, estimate the number of students who will use a lab
computer for the given number of hours:
a) Between 5.3 hours and 7.1 hours.
b) Less than 5.3 hours.
c) More than 7.1 hours.
Section 5.1 Homework
1. The contents of a cereal box are normally distributed, with a mean weight of 20 ounces
and a standard deviation of 0.05 ounce.
Graph:
Determine an interval of values into which
a. about 95% of the bags of the cereal box weights will fall.
b. about 99.7% of the bags of the cereal box weights will fall.
2. The weights of bags of cookies are normally distributed, with a mean of 15 ounces and a
standard deviation of 0.085 ounce.
Graph:
Determine an interval of values into which
a. about 95% of the bags of cookies will fall.
b. about 68% of the bags of cookies will fall.
3. The life span of a battery is normally distributed, with a mean of 2000 hours and a
standard deviation of 30 hours. Estimate the probability that a battery’s life span is between
1970 and 2030 hours.
4. Assume the mean annual consumption of peanuts is normally distributed, with a mean of
5.9 pounds per person and a standard deviation of 1.8 pounds. Estimate the probability that
a person consumes between 2.3 pounds and 9.5 pounds per year.
5. The life span of a tire is normally distributed, with a mean of 30,000 miles and a standard
deviation of 2000 miles. Estimate the probability that a tire’s life span is between 30,000 and
34,000 miles.
6. Assume the mean annual consumption of cheese is normally distributed, with a mean of
28.4 pounds per person and a standard deviation of 9.4 pounds. Estimate the probability that
a person consumes between 19.0 pounds and 28.4 pounds in a year.