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Chapter 6: The Normal Distribution 6.1 Probability Model for a Continuous Random Variable In the previous chapters, we have focused on discrete random variables. We now turn our attention to continuous random variables. Recall that a continuous random variable has infinitely many values that can be plotted on a line in an uninterrupted fashion. Continuous random variables are variables that result from measurement. The probability distribution of a discrete random variable can be represented by boxes on a histogram. The area of the box represented the probability of the associated value. With a continuous random variable, these boxes are “replaced” with curves. The curve is referred as the probability density curve. The mathematical function f(x) whose graph produces this curve is called the probability density function. What does this mean?? The probability that a continuous random variable is between to value can be represented as the area under the probability density function on that interval. There are lots of different kinds of distributions. Here some of the common “shapes”. Statisticians often find it useful to convenient to convert random variables to a dimensionless scale. This is done by created a new variable, Z, with mean O and sd 1. 6.2 The Normal Distribution-­Its General Features 6.3 The Standard Normal Distribution Examples
1. For the probability density function depicted at the right:
a) Find P(2<X<3)
b) find P(X>3)
c) Find the median and the quartiles.
2. Suppose Z has standard normal distribution.
a.(6 points) Find P[Z > −1.02]andP[Z ≥ −1.02]
b.(6 points) Find P[−1< Z < 2]
c. (8 points) Determine the value of a so that P[−a < Z < a] = 0.95
3. For the standard normal variable, Z, find:
a) P(Z > -1.79)
b) z such that P(-.6 < Z < z)=.5
c) The 40th percentile of the standard normal distribution.
4) a) Find the 80th percentile of the standard normal distribution.
b.) Find z such that P(-1 < Z < z) =.5
5.) Suppose Z has the standard normal distribution. Find:
a) P( Z > 1.42 )
b) P( -2.01 < Z < 0.07)
Find the z-value for the following:
c) P( -z < Z < z) =.6372
d) P(0< Z < z) =0.4099.
6.4 Probability Calculations with Normal Distributions Example 1. Given that X has the normal distribution N(60,4), find P[55<X<63]. Here the standard variable is Z=(X-­‐60)/4.
2. The meat department at a certain supermarket chain prepares “1 pound” packages of
ground beef so that there will be a variety of weights, some slightly more and some
slightly less than the 1 pound. Suppose that the weights of these 1 pound packages are
normally distributed with a mean of 1.00 pound and a standard deviation of .15 pound.
a) What is the probability that a randomly selected package of ground beef will weigh
more than 1.2 pounds?
b) What proportion of the packages will weigh between .95 and 1.10 pounds?
3. According to the Educational Testing Service, SAT scores are normally distributed
with a mean of 1000 and a standard deviation of 200.
a) Find the probability that a random test taker scores above 1324.
b) Find the probability that a random test taker scores in the range 950 to 1250.
c) Molly has scored in the 85th percentile. This means that she scored higher than 85% of
all test takers. What was her score?
4. A survey finds that students at a university use computers an average of 7 hours per
week with standard deviation 1 hour per week. Assume that the number of hours spent at
a computer is normally distributed. If a student is randomly selected, find:
a) the probability that the student uses a computer less than 4.5 hours per week.
b) the probability that the student uses a computer between 4.5 and 9.5 hours per week.
c) Find the proportion of students that use a computer more than 6 hours per week.
5. The length of a jump of a long jumper is normally distributed with a mean 28.4 feet
and variance of .09.
a) The world record long jump is 29.4 feet. What is the probability that the long jumper
will break the world record on his next jump?
b) Suppose that the long jumper’s next jump must be at least x feet to qualify for the
Olympics. If the probability the jumper qualifies is .9319, then what is x?
6. The incandescent light bulbs made by a company has a mean lifetime of 1000 hours
and standard deviation of 250 hours. Assume that the lifetime of light bulbs admits a
normal distribution.
a) The company claims that the lifetime of its light bulbs is from 750 hours to 1500
hours. What percentage of the light bulbs actually fall into this range?
b) What is the minimum lifetime for a light builb in the top 1% of the total in terms of
lifetime?
6.5 The Normal Approximation to the Binomial In a binomial experiment when the number of trials, n, the binomial probability formula
can be difficult to use. For example, suppose there are 500 trials of a binomial
experiment and we wish to compute the probability of 400 or more successes. Using the
binomial probability formula requires that we compute the following probabilities:
P(X≥400)=P(X=400)+P(X=401)+P(X=402)+…+P(X=500).
This would be time consuming to compute by hand or with a calculator.
As the number of trials, n, in a binomial experiment increases, the probability histogram
becomes more nearly symmetric and bell shaped. (See Project 3).
As the number of trials n in a binomial experiment increases, the probability distribution
of the random variable X becomes more nearly symmetric and bell shaped. As a rule of
thumb, if np ≥ 15 and n(1-p) ≥ 15, the probability distribution will be approximately
symmetric and bell shaped.
**The normal approximation to the binomial was discovered by Abraham de Moivre in
1733.
Normal Approximation to Binomial Distribution
1. Suppose 20% of the children in a town have a certain virus.
a. Estimate the probability that, in a random sample of 300 children, the number of
children with the virus will be between 50 and 70 inclusive.
b. Briefly justify your approximation procedure.
2. A survey report states that 20% of college seniors support an increase in federal
funding care of the elderly.
a) What is the approximate distribution of the random variable X=number of people that support an increase in federal funding care of the elderly?
b) If 80 college seniors are randomly selected, use the approximation of part a) to find the
probability that less than 6 of them support increased funding.
3. Suppose an operation has a 65% chance of success and is performed on a group of
individuals.
a) If the operation is performed on 6 people what is the probability that it is successful on
exactly 4 people?
b) If the operation is performed on 200 people use the normal approximation to the
binomial to find the probability that it is successful on 150 or fewer people. Round Z
values to two decimal digits so that the standard normal table look up does not require
interpolation.
4. In a certain city, 20% of the students play sports. (a) In a random sample of 20 students
from this city, what is the probability at least 5 play sports? (b) In a random sample of
200 students from this city, approximately what is the probability at most 45 play sports?
5. Only 30% of the people in a large city feel that its mass transit system is adequate.
a) what is the approximate distribution of the random variable X=numer of people that
feel the system is adequate?
b) If 70 people are selected at random, use the approximation in (a) to find the probability
that the number of people that feel the system is adequate is less than 10.
6. 60% of flights at an airport depart on time. Determine the probability that of 14
randomly selected flights:
a) at least 6 depart on time.
b) at most 5 do not depart on time.
c) exactly 8 depart on time.
d) Use the normal approximation to find the approximate probability that out of 350
randomly selected flights, less than 200 depart on time.
Central Limit Theorem
1. A software company takes on average 150 days to complete a project, with a standard
deviation of 20 days. For a random sample of 100 projects, let X denote the average time
to complete the project.
(a) What is the mean of X?
(b) What is the standard deviation of X?
(c) Estimate the probability that X lies between 145 and 155. (d) Briefly justify your
answer to (c).