Expert answer:Quantitative Analysis for Managers, management ass
Answer & Explanation:There are 2 Graduate Level Assignments of Quantitative Analysis for Managers.PS：These topics are Discussion Board questions not essay.Details see: 2 Graduate Level Assignments of Quantitative Analysis for Managers.docx The reading materials listed below:1. Reading 4 – Sampling.pdf 2. Topic 2 – Sampling Example.xlsx
2_graduate_level_assignments_of_quantitative_analysis_for_managers.docx
reading_4___sampling.pdf
topic_2___sampling_example.xlsx
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There are 2 Graduate Level Assignments of Quantitative Analysis for Managers. Please
give your options, show your ethics, and discuss deeply.
PS：These topics are Discussion Board questions not essay.
Topic 1 – Devotional Question:
Please respond to the following questions:
Sampling
Read Luke 10:29-37
Since we cannot minister to all people, what is the chance God will bring someone in
need across your path? How should you respond? How do you respond? How often will
God “sample” your response?
Topic 2 – Discussion Question:
A. Read (Reading 4 – Sampling.pdf), then respond to the following question:
Why is it important to understand random sampling when drawing conclusions
about a population from a sample? Would random selection be important if you
were making a decision to accept a shipment of goods from a supplier or reject a
shipment of goods from a supplier? Why or why not? Cite a couple of examples
where the concept of random sampling is important in the business environment.
B. Prepare an Excel Spreadsheet by following the example spreadsheet attached (Topic
2 – Sampling Example.xlsx).
Illustrate a Systematic Sample of a population. Calculate the required descriptive
statistics. Be different from the example. Create a sample composed of at least 9
values. Determine a 95% Confidence Interval of the Mean.
Reading 4: Sampling Distributions & Inferential Statistics Review
(File008r reference only)
1
Sampling Distributions & Inferential Statistics Review
Inferential statistics requires the inference of something about a population from
a sample. Sampling is the process of making a selection from a population of
interest such that the sample has a high probability of representing the
population. If we were interested in the mean life of our company’s tires versus
the mean life of our competitor’s tires, we would take a sample from each
population to make that comparison and draw conclusions.
Reasons for Sampling:
1) Many times to contact the entire population is too time consuming. The
chances of a person being elected to an office are never based on the
entire population of registered voters.
2) The cost of studying the entire population may be prohibitive. To test the
possible success of marketing a new cereal you could not afford to sample
the entire population of people who have children to whom the product
might be targeted.
3) Often a sample value is just as adequate as knowing the entire population
value. The Bureau of Labor Statistics (BLS) samples grocery stores
scattered throughout the nation to determine the cost of milk, bread,
beans and other food items. Including ALL grocery stores throughout the
nation would not yield substantial differences. A sample is just as good as
a 100% survey.
4) Sampling is practical especially if you must use destructive testing to
determine the quality of a product. Example: Ammunition testing, testing
a critical part to the point of destruction.
5) Sometimes is it physically impossible to check all of the items in the
population. Examples: Fish, birds, snakes, dogs or cats.
The process of sampling must be a fair representation of the population of
interest. You cannot bias – intentionally or unintentionally – the results of your
sample and then apply inferential concepts.
Sampling error is defined as the difference in the parameter and the statistic
used to estimate it. µ – X = sampling error of the mean or σ2 – S2 is also
X and S2 are
sampling error of the variance. µ and σ2 are parameters.
statistics. We use statistics to estimate parameters.
Reading 4: Sampling Distributions & Inferential Statistics Review
(File008r reference only)
2
Two Types of Statistics:
There are two basic types of statistics – Descriptive and Inferential. Generally
descriptive statistics will describe the data set whereas inferential statistics will
infer something about the population value from taking a sample.
Descriptive statistics describes data sets.
Inferential statistics infers something about a population (larger group of data)
from a sample (smaller group of data).
When using Descriptive statistics we make use of measures of central tendency
and measures of dispersion.
Central Tendency Measures
Mean – Arithmetic average
Median – One in the middle.
Mode – One that occurs most often.
Weighted Mean – uses the probability of each possible outcome to
develop a mean.
Dispersion Measures
Range – difference between the high and the low.
Average deviation (always zero)
Variance – mean of the squared deviations around the mean.
Standard deviation – the square root of the variance.
Percentiles, quartiles, deciles – divides the data set into equal
increments.
Inter-quartile range (IQR) – the difference between the first quartile
and the third quartile.
Outliers – data set aberrations which need to be adjusted or
eliminated.
Important Concepts for Inferential Statistics:
Population – all of the observations of interest to the researcher.
Sample – a representative portion of the population.
Populations are defined by the researcher. All of the measures of central
tendency or all of the measures of dispersion can be classified as a
parameter or a statistic. A parameter is a descriptive measure of the
population (both begin with P). A statistics is a descriptive measure of the
sample (both begin with S). In other words, I can have a descriptive
Reading 4: Sampling Distributions & Inferential Statistics Review
(File008r reference only)
3
measure, say the mean, which can be called a parameter or can be called
a statistic. The mean describes a part of the data set. If reference is
made to the mean as a parameter this term refers to the population value.
If reference is made to the mean as a statistic, this term refers to the
sample value.
Variables – Quantitative and Qualitative.
Quantitative Variable are quantifiable. They take on two particular
types – discrete (whole numbers) and continuous (range of values).
Qualitative Variables are not quantifiable such as gender, religious
preference or hair color. A technique called dummy variables will
allow us to quantify this type of variable.
To work with inferential statistics, one must use a random sampling
concept. A random sample occurs when each and every element in the
population has an equal and independent chance of being selected.
Without random selection the interpretation of the results of any study are
very difficult if not impossible.
There are four basic methods of random sampling:
Random Numbers – Each and every element in the population
has an equal and independent chance of being selected.
Stratified Sampling – divide the population into strata
(homogeneous sub-sets) and then take proportionate random
samples from the population which matches the strata percentages.
Systematic Sampling – the population is placed in random order
then a starting point is selected at random. Next every 10th or 20th
or 50th or “ith” item may be selected for sampling.
Cluster Sampling – the population is divided into geographically
similar units and from those units a number of clusters are selected
at random. Once the cluster is selected 100% of that cluster will be
sampled.
If we do not select a sample that meets the condition of randomness from a
population, then we cannot apply the inferential concepts in our
interpretation. We cannot assert that the sample represents the population
if the condition of randomness is not present in our sample.
Reading 4: Sampling Distributions & Inferential Statistics Review
(File008r reference only)
4
Division of Population into Sub-Sets or Different Samples:
In inferential statistics, a sample is selected to represent the population. Let’s
assume you want to know the average age of a class of 22 people. That’s not
hard to figure out. You would go around the room and ask each person his or
her age, sum the ages and divide by 22. Let’s say we did not want to take the
time or trouble to get the average age of all the students since all students in the
class were first year college students. You could select three at random from
among the 22 and let their age represent the population age. This is letting a
sample represent the population. The problem is there are many possible
arrangements of the class of 22 into data sets three at a time. Each sample
selected would have a mean of that sample. The mean of that sample then is
supposed to represent the mean of the population, but there are a bunch (a
sophisticated statistical term) of arrangements.
Okay, you say, how do we measure a bunch?
Let’s take a simple example and determine how many sub-sets can be
developed from the 22 students 3 at a time.
There are two basic methods for dividing our population into sub-sets. One is
called permutations and the other is combinations. In using permutations
order is important, but in combinations order is not important.
nPr =
n!
(n − r )!
nCr =
n!
r!(n − r )!
Take the elements ABC and arrange them (develop the sub-sets) of three
elements three at a time. Assume 0! = 1. There are 6 arrangements of the data
set using permutations, but only 1 arrangement of the data set using
combinations.
The calculations:
3P3 =
3!
3 * 2 *1
=
=6
(3 − 3)!
0!
If order is important, then ABC is different from ACB which is different from BAC
and so forth.
Reading 4: Sampling Distributions & Inferential Statistics Review
(File008r reference only)
3C3 =
5
3!
3 * 2 *1
=
=1
3!(3 − 3)! 3 * 2 *1(0!)
If order is not important, then ABC is the same as BAC as CAB and so forth.
Let’s now turn our attention to the average age of 22 students using a sample
sub-set of 3 at a time. Determine the possible arrangement of a class size using
22 arranged 3 at a time. Assume order is not important.
22C3 =
22!
22 * 21 * 20 *19!
=
3!(22 − 3)!
3 * 2 *1(19!)
The 19! (read: 19 factorial) in both the numerator and denominator cancel out, so
there is no need to do excessive multiplication.
22C3 =
22 * 21 * 20 9,240
=
= 1,540
6
6
When order is not important, there are 1,540 possible arrangements of our data
set of 22 students arranged 3 at a time. Since we were interested in the average
age of the students in the class and wanted to take a sample of 3 at a time from
among the full population of 22, there would be 1,540 possible sample means.
Each sample mean would then be representative of the population mean as long
as the rule of randomness is not violate.
The distribution of these means is referred to as the sampling distribution of
sample means. It is important to know that the Central Limit Theorem (which
was introduced in the last reading) allows us to assume the distribution of these
sample means is itself normally distributed as long as “n” (the sample size) is
sufficiently large (usually 30 or greater). When this occurs, we can then use the
Z-process to determine probabilities for the sampling distribution of sample
means.
Standard Deviation, Standard Error and Sampling Error:
Variance and the variance of the stand error are essentially the same thing. Both
are variances. There is one difference. The variance is associated with a
sample and the variance of the standard error is associated with the sampling
distribution of sample means.
Sampling error is, however, a different term. Sampling error is the difference
between the population mean and the sample mean. ( µ less X )
Reading 4: Sampling Distributions & Inferential Statistics Review
6
(File008r reference only)
Any time n is greater than 1, the sampling distribution of sample means is the
distribution of interest. Samples greater than 1 make all calculations using the
standard error (square root of the variance of the standard error). A single
sample, however, has a standard deviation and not a standard error. When
comparing the parameter (mean of the population) to the statistic (mean of the
sample), the difference will be sampling error.
Even though this was discussed in the review on probability, let’s review the
concept one more time. In the review on probability, we looked at the normal
distribution and the Z-process. Remember the Z-process is used when working
with normal distributions and continuous data sets.
Z=
X −µ
σ
This Z approach would be used if “n” is unspecified or n = 1.
If “n” is greater than 1, the standard error is used.
calculated as follows:
Z=
X −µ
σx
where the standard error is
Here the Z-process is
σx =
σ
n
As “n” increases the standard error will decrease.
For example:
Using the Z formula for standard error just above and given a standard
deviation of 2, determine the standard error for the following values of n.
n
σx
4
16
25
36
Try to make these calculations before looking at the solution below:
Don’t peek yet!!!
Reading 4: Sampling Distributions & Inferential Statistics Review
(File008r reference only)
7
Solution:
n
σx
4
16
25
36
1.00
0.50
0.40
0.33
Notice that as the sample size increases (n goes up) the standard error (not
sampling error, but standard error) decreases. Technically if we survey 100% of
the population, we will have no standard error. We will still have a standard
deviation, but no standard error. This inverse relationship is quite useful.
One of the most important theorems in all of inferential statistics is the Central
Limit Theorem (CLT). As demonstrated above, as the sample size increases the
underlying distribution of sample means will approach a normal distribution.
This occurs when “n” is equal to or greater than 30. This is important because
when the distribution is normal the Z-distribution can be used to determine
probabilities. As you should recall from your statistics course, if the sample size
is less than 30, the t-distribution must be used instead of the Z-distribution.
Two techniques are useful when working with sampling error (difference between
X and µ). One is confidence intervals and the other is hypothesis testing.
Confidence Intervals measure how confident we are that the true, unknown
population mean is between and upper and lower confident limit. Hypothesis
testing measures the hypothesized mean of the population against a sample
value to determine if the null hypothesis is true or not true. This course will not
concern itself with the details of either of those two techniques. They are simply
mentioned here as a reminder.
Forecasting Using Samples:
In reading 3, some time was spent on the ideas of distinguishable (not parallel,
different or not similar) and indistinguishable (parallel, not different or
similar) data sets. A newspaper vendor was used as an example. Going one
step further, suppose the vendor wanted to know how many newspapers the
vendor could sell. To do this, the vendor believes that the use of historical data
would best forecast future results (sales in the future). From reading 3, the basic
data set is repeated below. This provides the starting point for the historic
forecast.
Reading 4: Sampling Distributions & Inferential Statistics Review
(File008r reference only)
Newspaper Demand Data Set (Original Before Analysis):
Date
24
25
26
27
28
29
30
31
1
2
3
4
5
6
7
8
9
10
Day
Sold Date Day
Sold
Wednesday 70
11 Sunday
41
Thursday
73
12 Monday
59
Friday
68
13 Tuesday
43
Saturday
56
14 Wednesday 46
Sunday
58
15 Thursday
49
Monday
71
16 Friday
52
Tuesday
65
17 Saturday
39
Wednesday 55
18 Sunday
40
Thursday
47
19 Monday
44
Friday
54
20 Tuesday
59
Saturday
42
21 Wednesday 52
Sunday
44
22 Thursday
41
Monday
61
23 Friday
56
Tuesday
51
24 Saturday
44
Wednesday 48
25 Sunday
40
Thursday
50
26 Monday
54
Friday
54
27 Tuesday
46
Saturday
45
Adjusting for Distinguishable Data (different – grey shade) and
Indistinguishable Data (similar – un-shaded area):
Date
24
25
26
27
28
29
30
31
1
2
3
4
5
6
7
8
9
10
Day
Sold Date Day
Sold
Wednesday 70
11 Sunday
41
Thursday
73
12 Monday
59
Friday
68
13 Tuesday
43
Saturday
56
14 Wednesday 46
Sunday
58
15 Thursday
49
Monday
71
16 Friday
52
Tuesday
65
17 Saturday
39
Wednesday 55
18 Sunday
40
Thursday
47
19 Monday
44
Friday
54
20 Tuesday
59
Saturday
42
21 Wednesday 52
Sunday
44
22 Thursday
41
Monday
61
23 Friday
56
Tuesday
51
24 Saturday
44
Wednesday 48
25 Sunday
40
Thursday
50
26 Monday
54
Friday
54
27 Tuesday
46
Saturday
45
8
Reading 4: Sampling Distributions & Inferential Statistics Review
9
(File008r reference only)
There is a difference in demand between weekday and weekend sales
(distinguishable data or different data). There was an environmental
change (unmatched price increase) at the first of the month which
rendered the last 8 days of the previous month different (distinguishable
from the next month).
After adjusting the data set, there remained two separate indistinguishable
(similar) data sets. The weekdays are indistinguishable from each other
and the weekends are indistinguishable from other weekends. Using
these two data sets a forecast of future sales for the weekdays and a
separate forecast of future sales for the weekends may be accomplished.
The forecast for the weekday is shown in the following table.
Indistinguishable Weekday Data Set
Weekday Date
Demand
1 Thursday
2 Friday
5 Monday
6 Tuesday
7 Wednesday
8 Thursday
9 Friday
12 Monday
13 Tuesday
14 Wednesday
15 Thursday
16 Friday
47
54
61
51
48
50
54
59
43
46
49
52
Weekday
Date
19 Monday
20 Tuesday
21 Wednesday
22 Thursday
23 Friday
26 Monday
27 Tuesday
Demand
44
59
52
41
56
54
46
Mean Average = 50.84
Standard Deviation = 5.65
Standard Error =
σ
n
=
5.65
= 1.30
19
The mean sales is 50.84 (rounded to 51). The standard deviation is 5.65
with a standard error of 1.30 papers. The expected future sales of
weekdays is 51 papers. However, how is the standard error useful, you
might ask?
The vendor will use the 51 newspapers sold on the weekdays as a point
estimate of future weekday sales. However, the point estimate is an
estimate of the sales at a given point. From the basic statistics course,
point estimates are just that – point estimates. It is quite difficult if not
impossible to have the forecasted point estimate of the sales, in this
instance, equal the actual sales. It is more practical to place around the
point estimate an interval which then can be interpreted using either the
Empirical Rule or Chebyshev’s Theorem.
Reading 4: Sampling Distributions & Inferential Statistics Review
(File008r reference only)
10
Several rules apply for the use of the Empirical Rule. First the data set
must be continuous data (range of values) and the distribution must be
normal. In this case, the data set is discrete (whole numbers) and there is
not evidence of a normal distribution. Does this mean the forecast cannot
have an interval estimate, you ask? Of course not, I reply. Whatever the
interpretation (Empirical Rule or Chebyshev’s Theorem), the calculation
will be the same.
X ± 1S X
X ± 2S X
X ± 3S X
Around the point estimate of 51 an interval is calculated which is based on
1, 2 or 3 times the standard error. 1, 2 or 3 is the number of standard
errors. Here the assumption is that the sampling distribution of sample
means is being encountered (n ≥ 30).
Arbitrarily selecting 2, the interval estimate is calculated to be the
following:
51 ± 2 (1.30)
The standard error comes from the table.
51 ± 2.6
48.4
to
53.6
48
to
54
Rounded
The calculation is rather simple. However, how is it interpreted?
Since the data set is discrete and non-normal, the Empirical Rule
interpretation cannot be used. Chebyshev’s Theorem applies to any data
set (discrete or continuous). Chebyshev’s formula is as follows:
1−
1
K2
Where K = the number of standard deviations (standard error here).
K must also be greater than 1 (K > 1). If 1 is used in the formula,
the result is “zero”, which is impractical for an interpretation.
Reading 4: Sampling Distributions & Inferential Statistics Review
(File008r reference only)
11
Since K was selected to be 2, two can be sub …
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