Expert answer:Business Decision Making Project Part 1
Solved by verified expert:Purpose of Assignment The purpose of this assignment is to provide students the opportunity to demonstrate mastery of their ability to apply statistical concepts to business situations to inform data-driven decision-making. The project is a 3-week project, with part 1 in Week 3, part 2 in Week 4, and part 3 in Week 5. In Week 3, students identify the organization, problem, research variable, methods for collecting data, and show mastery of validity and reliability as applied to data-collection methods.Resources: Week 3 Videos; Week 3 Readings; Statistic Lab Tutorial help on Excel® and Word functions can be found on the Microsoft® Office website. There are also additional tutorials via the web offering support for Office products. Assignment Steps Identify a business problem or opportunity at a company where a team member works or with which are familiar to team. This will be a business problem you use for the assignments in Weeks 3-5. It should be a problem/opportunity for which gathering and analyzing some type of data would help each team member understand the problem/opportunity better. Identify a research variable within the problem/opportunity that could be measured with some type of data collection.Consider methods for collecting a suitable sample of either qualitative or quantitative data for the variable.Consider how you will know if the data collection method would be valid and reliable. Develop a 1,050-word analysis to describe a company, problem, and variable including the following in your submission:Identify the name and description of the selected company.Describe the problem at that company.Identify one research variable from that problem. Describe the methods you would use for collecting a suitable sample of either qualitative or quantitative data for the variable (Note: do not actually collect any data).Analyze how you will know if the data collection method would generate valid and reliable data (Note: do not actually collect any data). Format your assignment consistent with APA guidelines. Videos will be attached in a few mins
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CHAPTER 6
Continuous Random Variables and the Normal
Distribution
© Mauritius/SuperStock
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6.1 Continuous Probability Distribution and the Normal Probability Distribution
Case Study 6-1 Distribution of Time Taken to Run a Road Race
6.2 Standardizing a Normal Distribution
6.3 Applications of the Normal Distribution
6.4 Determining the z and x Values When an Area Under the Normal Distribution Curve
Is Known
6.5 The Normal Approximation to the Binomial Distribution
Appendix 6–1 Normal Quantile Plots
Have you ever participated in a road race? If you have, where did you stand in
comparison to the other runners? Do you think the time taken to finish a road race
varies as much among runners as the runners themselves? See Case Study 6-1 for the
distribution of times for runners who completed the Manchester (Connecticut) Road
Race in 2014.
Discrete random variables and their probability distributions were presented in Chapter 5.
Section 5.1 defined a continuous random variable as a variable that can assume any value in
one or more intervals.
The possible values that a continuous random variable can assume are infinite and
uncountable. For example, the variable that represents the time taken by a worker to
commute from home to work is a continuous random variable. Suppose 5 minutes is the
minimum time and 130 minutes is the maximum time taken by all workers to commute from
home to work. Let x be a continuous random variable that denotes the time taken to
commute from home to work by a randomly selected worker. Then x can assume any value in
the interval 5 to 130 minutes. This interval contains an infinite number of values that are
uncountable.
A continuous random variable can possess one of many probability distributions. In this
chapter, we discuss the normal probability distribution and the normal distribution as an
approximation to the binomial distribution.
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6.1 Continuous Probability Distribution and the Normal
Probability Distribution
In this section we will learn about the continuous probability distribution and its properties
and then discuss the normal probability distribution.
In Chapter 5, we defined a continuous random variable as a random variable whose
values are not countable. A continuous random variable can assume any value over an
interval or intervals. Because the number of values contained in any interval is infinite, the
possible number of values that a continuous random variable can assume is also infinite.
Moreover, we cannot count these values. In Chapter 5, it was stated that the life of a battery,
heights of people, time taken to complete an examination, amount of milk in a gallon
container, weights of babies, and prices of houses are all examples of continuous random
variables. Note that although money can be counted, variables involving money are often
represented by continuous random variables. This is so because a variable involving money
often has a very large number of outcomes.
6.1.1 Continuous Probability Distribution
Suppose 5000 female students are enrolled at a university, and x is the continuous random
variable that represents the height of a randomly selected female student. Table 6.1 lists the
frequency and relative frequency distributions of x.
Table 6.1 Frequency and Relative Frequency Distributions of Heights of Female Students
Height of a Female Student (inches) x
f
Relative Frequency
60 to less than 61
90
.018
61 to less than 62
170
.034
62 to less than 63
460
.092
63 to less than 64
750
.150
64 to less than 65
970
.194
65 to less than 66
760
.152
66 to less than 67
640
.128
67 to less than 68
440
.088
68 to less than 69
320
.064
69 to less than 70
220
.044
70 to less than 71
180
.036
N = 5000
Sum = 1.0
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The relative frequencies given in Table 6.1 can be used as the probabilities of the respective
classes. Note that these are exact probabilities because we are considering the population of
all female students enrolled at the university.
Figure 6.1 displays the histogram and polygon for the relative frequency distribution of
Table 6.1. Figure 6.2 shows the smoothed polygon for the data of Table 6.1. The smoothed
polygon is an approximation of the probability distribution curve of the continuous random
variable x. Note that each class in Table 6.1 has a width equal to 1 inch. If the width of classes
is more (or less) than 1 unit, we first obtain the relative frequency densities and then graph
these relative frequency densities to obtain the distribution curve. The relative frequency
density of a class is obtained by dividing the relative frequency of that class by the class
width. The relative frequency densities are calculated to make the sum of the areas of all
rectangles in the histogram equal to 1.0. Case Study 6-1, which appears later in this section,
illustrates this procedure. The probability distribution curve of a continuous random variable
is also called its probability density function.
Figure 6.1 Histogram and polygon for Table 6.1.
Figure 6.2 Probability distribution curve for heights.
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The probability distribution of a continuous random variable possesses the following two
characteristics.
1. The probability that x assumes a value in any interval lies in the range 0 to 1.
2. The total probability of all the (mutually exclusive) intervals within which x can assume a
value is 1.0.
The first characteristic states that the area under the probability distribution curve of a
continuous random variable between any two points is between 0 and 1, as shown in Figure
6.3. The second characteristic indicates that the total area under the probability distribution
curve of a continuous random variable is always 1.0, or 100%, as shown in Figure 6.4.
Figure 6.3 Area under a curve between two points.
Figure 6.4 Total area under a probability distribution curve.
The probability that a continuous random variable x assumes a value within a certain
interval is given by the area under the curve between the two limits of the interval, as shown
in Figure 6.5. The shaded area under the curve from a to b in this figure gives the probability
that x falls in the interval a to b; that is,
P (a ≤ x ≤ b) = Area under the curve from a to b
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Figure 6.5 Area under the curve as probability.
Note that the interval a ≤ x ≤ b states that x is greater than or equal to a but less than or
equal to b.
Reconsider the example on the heights of all female students at a university. The
probability that the height of a randomly selected female student from this university lies in
the interval 65 to 68 inches is given by the area under the distribution curve of the heights of
all female students from x = 65 to x = 68, as shown in Figure 6.6. This probability is written
as
P (65 ≤ x ≤ 68)
which states that x is greater than or equal to 65 but less than or equal to 68.
Figure 6.6 Probability that x lies in the interval 65 to 68.
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For a continuous probability distribution, the probability is always calculated for an
interval. For example, in Figure 6.6, the interval representing the shaded area is from 65 to
68. Consequently, the shaded area in that figure gives the probability for the interval
65 ≤ x ≤ 68. In other words, this shaded area gives the probability that the height of a
randomly selected female student from this university is in the interval 65 to 68 inches.
The probability that a continuous random variable x assumes a single value is
always zero. This is so because the area of a line, which represents a single point, is zero.
For example, if x is the height of a randomly selected female student from that university,
then the probability that this student is exactly 66.8 inches tall is zero; that is,
P (x = 66.8) = 0
This probability is shown in Figure 6.7. Similarly, the probability for x to assume any other
single value is zero.
Figure 6.7 The probability of a single value of x is zero.
In general, if a and b are two of the values that x can assume, then
P (a) = 0 and P (b) = 0
From this we can deduce that for a continuous random variable,
P (a ≤ x ≤ b) = P (a < x ≤ b) = P (a ≤ x < b) = P (a < x < b)
In other words, the probability that x assumes a value in the interval a to b is the same
whether or not the values a and b are included in the interval. For the example on the heights
of female students, the probability that a randomly selected female student is between 65 and
68 inches tall is the same as the probability that this female is 65 to 68 inches tall. This is
shown in Figure 6.8.
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Figure 6.8 Probability “from 65 to 68” and “between 65 and 68.”
Note that the interval “between 65 and 68” represents “65 < x < 68” and it does not
include 65 and 68. On the other hand, the interval “from 65 to 68” represents “65 ≤ x ≤ 68
and it does include 65 and 68. However, as mentioned previously, in the case of a continuous
random variable, both of these intervals contain the same probability or area under the curve.
Case Study 6-1 on the next page describes how we obtain the probability distribution curve
of a continuous random variable.
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CASE STUDY 6-1
DISTRIBUTION OF TIME TAKEN TO RUN A ROAD
RACE
The following table gives the frequency and relative frequency distributions for the time
(in minutes) taken to complete the Manchester Road Race (held on November 27, 2014)
by a total of 11,682 participants who finished that race. This event is held every year on
Thanksgiving Day in Manchester, Connecticut. The total distance of the course is 4.748
miles. The relative frequencies in the following table are used to construct the histogram
and polygon in Figure 6.9.
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Class
Frequency Relative Frequency
20 to less than 25
53
.0045
25 to less than 30
246
.0211
30 to less than 35
763
.0653
35 to less than 40
1443
.1235
40 to less than 45
1633
.1398
45 to less than 50
1906
.1632
50 to less than 55
2164
.1852
55 to less than 60
1418
.1214
60 to less than 65
672
.0575
65 to less than 70
380
.0325
70 to less than 75
230
.0197
75 to less than 80
176
.0151
80 to less than 85
186
.0159
85 to less than 90
130
.0111
90 to less than 95
113
.0097
95 to less than 100
90
.0077
100 to less than 105
33
.0028
105 to less than 110
36
.0031
110 to less than 115
9
.0008
115 to less than 120
1
.0001
∑f = 11,682
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Figure 6.9 Histogram and polygon for the road race data.
To derive the probability distribution curve for these data, we calculate the relative
frequency densities by dividing the relative frequencies by the class widths. The width of
each class in the above table is 5. By dividing the relative frequencies by 5, we obtain the
relative frequency densities, which are recorded in the next table. Using the relative
frequency densities, we draw a histogram and a smoothed polygon, as shown in Figure
6.10. The curve in this figure is the probability distribution curve for the road race data.
Figure 6.10 Probability distribution curve for the road race data.
Note that the areas of the rectangles in Figure 6.9 do not give probabilities (which are
approximated by relative frequencies). Rather here the heights of the rectangles give the
probabilities. This is so because the base of each rectangle is 5 in this histogram.
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Consequently, the area of any rectangle is given by its height
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multiplied by 5. Thus the total area of all the rectangle in Figure 6.9 is 5.0, not 1.0.
However, in Figure 6.10, it is the areas, not the heights, of rectangles that give the
probabilities of the respective classes. Thus, if we add the areas of all the rectangles in
Figure 6.10, we obtain the sum of all probabilities equal to 1.0. Consequently the total
area under the curve is equal to 1.0.
Class
Relative Frequency Density
20 to less than 25
.00090
25 to less than 30
.00422
30 to less than 35
.01306
35 to less than 40
.02470
40 to less than 45
.02796
45 to less than 50
.03264
50 to less than 55
.03704
55 to less than 60
.02428
60 to less than 65
.01150
65 to less than 70
.00650
70 to less than 75
.00394
75 to less than 80
.00302
80 to less than 85
.00318
85 to less than 90
.00222
90 to less than 95
.00194
95 to less than 100
.00154
100 to less than 105
.00056
105 to less than 110
.00062
110 to less than 115
.00016
115 to less than 120
.00002
The probability distribution of a continuous random variable has a mean and a standard
deviation, which are denoted by µ and σ, respectively. The mean and standard deviation
of the probability distribution given in the above table and Figure 6.10 are 50.873 and
14.011 minutes, respectively. These values of µ and σ are calculated by using the data on
all 11,682 participants.
Source: This case study is based on data published on the official Web site of the Manchester
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Road Race.
6.1.2 The Normal Distribution
The normal distribution is one of the many probability distributions that a continuous
random variable can possess. The normal distribution is the most important and most widely
used of all probability distributions. A large number of phenomena in the real world are
approximately normally distributed. The continuous random variables representing heights
and weights of people,
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scores on an examination, weights of packages (e.g., cereal boxes, boxes of cookies), amount
of milk in a gallon, life of an item (such as a light-bulb or a television set), and time taken to
complete a certain job have all been observed to have an approximate normal distribution.
A normal probability distribution or a normal curve is a bell-shaped (symmetric)
curve. Such a curve is shown in Figure 6.11. Its mean is denoted by µ and its standard
deviation by σ. A continuous random variable x that has a normal distribution is called a
normal random variable. Note that not all bell-shaped curves represent a normal
distribution curve. Only a specific kind of bell-shaped curve represents a normal curve.
Figure 6.11 Normal distribution with mean µ and standard deviation σ.
Normal Probability Distribution A normal probability distribution, when plotted,
gives a bell-shaped curve such that:
1. The total area under the curve is 1.0.
2. The curve is symmetric about the mean.
3. The two tails of the curve extend indefinitely.
A normal distribution possesses the following three characteristics:
1. The total area under a normal distribution curve is 1.0, or 100%, as shown in Figure 6.12.
Figure 6.12 Total area under a normal curve.
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2. A normal distribution curve is symmetric about the mean, as shown in Figure 6.13.
Consequently, 50% of the total area under a normal distribution curve lies on the left side
of the mean, and 50% lies on the right side of the mean.
Figure 6.13 A normal curve is symmetric about the mean
3. The tails of a normal distribution curve extend indefinitely in both directions without
touching or crossing the horizontal axis. Although a normal distribution curve never
meets the horizontal axis, beyond the points represented by µ − 3σ and µ + 3σ it
becomes so close to
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this axis that the area under the curve beyond these points in both directions is very small
and can be taken as very close to zero (but not zero). The actual area in each tail of the
standard normal distribution curve beyond three standard deviations of the mean is
.0013. These areas are shown in Figure 6.14.
Figure 6.14 Areas of the normal curve beyond µ ± 3σ.
The mean, µ, and the standard deviation, σ, are the parameters of the normal
distribution. Given the values of these two parameters, we can find the area under a normal
distribution curve for any interval. Remember, there is not just one normal distribution curve
but a family of normal distribution curves. Each different set of values of µ and σ gives a
normal distribution curve with different height and/or spread. The value of µ determines the
center of a normal distribution curve on the horizontal axis, and the value of σ gives the
spread of the normal distribution curve. The three normal distribution curves shown in
Figure 6.15 have the same mean but different standard deviations. By contrast, the three
normal distribution curves in Figure 6.16 have different means but the same standard
devi ...
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