**please help me with these 3 questions**Assume you are the manager of a mustard seed factory in Colombia. Your company has received complaints that there isn’t enough mustard seed in your economy size packages, which should have 1.7 ounces. You ask your foreman and chief operating officer Juan Valdez to test the new mustard seed packaging machine you are installing. He runs a sample of 36 packages, with the results of package sizes in ounces: (your instructor will tell you which dataset to use). For a Word version of the Datasets, click on the following document: BUS6780_M3 Homework Assignment Data Set.docx (only dataset 1) Dataset one 1.5 1.7 1.8 1.9 1.7 1.75 1.75 1.66 1.81 1.65 1.77 1.75 1.8 2.2 1.7 1.65 1.88 1.6 1.7 1.7 1.6 1.65 1.65 1.65 1.8 1.91 1.7 1.59 1.76 1.65 1.3 2.0 1.91 1.65 1.8 1.8 2. Calculate a 95% confidence interval on the average weight of packaged mustard seed. Explain very carefully to the packaging workers what the 95% confidence interval numbers mean. Include your Excel output. Is there anything you would want to note for management? 3. Use the same dataset you used for the previous problem. Over a period of time, Juan determines that the true mean of the packages is in fact 1.72 ounces with standard deviation the same as the sample above. Assume that Juan has a strong consumer orientation. Juan is also a savvy businessman. Indeed, a local TV station has been checking up on Juan by periodically sampling 36 packages and figuring up the average weight. Juan tells you that he wants to be at least 95% sure that the average package has at least 1.7 ounces. Further, he wants to be sure that 95% of the time, the TV crew that is watching him will find a sample mean of at least 1.70 ounces. At the same time he doesn’t want to set the equipment to fill packages with any more product than he absolutely has to. What advice would you give the mustard seed packaging factory management as to how to calibrate their equipment? In particular, should they increase or decrease the amount they are putting in the packages and by how much? Are there any other steps they might be able to take to improve the situation?

sample_week_3__.xlsx

Unformatted Attachment Preview

Data

4.65

5.21

4.84

5.64

5.1

5.29

5.25

4.73

5.28

4.68

5.31

5.22

5.32

5.87

5.1

4.65

5.23

4.8

4.79

5.06

4.78

4.87

4.63

4.72

5.41

5.53

5.1

4.65

5.21

4.91

3.99

5.91

5.73

4.95

5.38

5.22

4.67

4.64

5.06

4.92

Mike is the manager of a flour factory in Michigan. Recently, he often receives complaints that the factory

flour packages often contain less than 5 pounds. Mike decides to do some test before calibrating the pack

He takes a sample of 40 packages and measure how many pounds of flower each package contain. He get

shown in Column A.

Question 1. Construct a 95% confidence interval on the average weight of packaged flour. Explain very ca

and to the packaging workers what the 95% confidence interval numbers mean.

Question 2. Mike’s factory has been supplying large amounts of flour to a supermarket chain. Due to the

chain decides to audit the flour packages by taking a sample of 40 packages from time to time. Mike want

that 95% of the time, the chain will find a sample mean of at least 5.0 pounds. Meanwhile, he doesn’t wan

machines to fill packages with any more product than needed.

Before the chain conducts any test, Mike took multiple samples of 40 in 2 weeks and eventually determin

mean of all packages is 5.03 pounds with a standard deviation the same as in Question 1.

What advice would you give to Mike as to how to calibrate their machines? In particular, should they incre

decrease the amount they are putting in the packages? Are there any other steps they might be able to ta

the situation?

Question 3. Mike calibrated the machines and provided training to workers. One month later, he determ

mean of all packages is 5.02 pounds with a standard deviation of 0.07 pound.

The supermarket chain is still auditing the flour packages by taking a sample of 40 packages from time to

going to find a sample mean of at least 5.0 pounds 95% of the time after the calibration and training?

ceives complaints that the factory’s 5 pounds

me test before calibrating the packaging machines.

wer each package contain. He gets the data

of packaged flour. Explain very carefully to Mike

a supermarket chain. Due to the complaints, the

ages from time to time. Mike wants to make sure

ounds. Meanwhile, he doesn’t want to adjust the

2 weeks and eventually determined that the true

as in Question 1.

es? In particular, should they increase or

ther steps they might be able to take to improve

kers. One month later, he determined the true

mple of 40 packages from time to time. Are they

r the calibration and training?

Data

4.65

5.21

4.84

5.64

5.1

5.29

5.25

4.73

5.28

4.68

5.31

5.22

5.32

5.87

5.1

4.65

5.23

4.8

4.79

5.06

4.78

4.87

4.63

4.72

5.41

5.53

5.1

4.65

5.21

4.91

3.99

5.91

5.73

4.95

5.38

5.22

4.67

4.64

5.06

4.92

Question 1:

Data

Mean

Standard Error

Median

Mode

Standard Deviation

Sample Variance

Kurtosis

Skewness

Range

Minimum

Maximum

Sum

Count

Confidence Level(95.0%)

5.0575

0.061477

5.08

4.65

0.388816

0.151178

0.56402

0.040889

1.92

3.99

5.91

202.3

40

0.12435

Then continue to calculate the 95% confidence interval as below:

lower end = mean – Confidence Level (95.0%) =

4.93

upper end = mean + Confidence Level (95.0%) =

5.18

Answer to Question 1:

The population is all the 5 pounds flour packages produced in the factory

recently. The sample mean is 5.06 pounds (rounded from 5.0575). It is a point

estimate of the population mean. Is this point estimate accurate? How much

confidence do we have in it?

The confidence interval means that we can reasonably expect that 95% of the

means of all similar samples of 40 can fall between 4.93 to 5.18 pounds.

other words, if we take many samples of 40 packages again and again, we will

get many sample means. These means may differ from each other slightly. The

95% confidence interval tells us that 95% of these sample means are expected

to fall between 4.93 to 5.18 pounds. Therefore, we can conclude that

95% confident that the MEAN of all flour packages would fall between 4.93 to

5.18 pounds. It doesn’t mean that 95% of all flour packages fall between 4.93

– 5.18. We can see that 95% of the data do not necessarily fall in this interval.

The mode, mean and median are not close. The range is huge. These indicate a

relatively wide distribution of products and the producing process may not be

in control. The manager may need to bring the process into control by

calibrating the machines.

Go to Data tab, click Data Analysis, then to Descriptive Statistics, then

OK. For the input range, cursor over the entire data set (Cells A1:A41).

Check Label in First Row. The data need to be grouped by columns.

Select a blank cell as the starting cell of the Output Range. Check

Summary Statistics, and make sure that the Confidence Level for the

Mean is 95%, then click OK. The output should look something like the

left.

ced in the factory

m 5.0575). It is a point

accurate? How much

expect that 95% of the

3 to 5.18 pounds. In

gain and again, we will

each other slightly. The

le means are expected

conclude that we are

uld fall between 4.93 to

kages fall between 4.93

ily fall in this interval.

huge. These indicate a

process may not be

nto control by

Question 2:

According to the given information,

40

sample size

n=

5.03

population mean (mean of sample means)

m=m =

s=

0.388816

Standard deviation of population

Calculate Standard deviation of sample means (standard error of the mean)

σ = σ/√n = 0.388816/√40 =

0.0614772

Calculate probability:

P(drawn sample mean < 5.0) =
P(drawn sample mean >= 5.0) =

0.31278

1 – 0.313 =

See textbook Black (2010) Pages 230

68.7%

Since the sample size n is 40 (big enough), the central limit theorem may be applied. As a

result, it may be assumed that the sample means are normally distributed. This normal

distribution has a mean of 5.03 and a standard deviation of 0.061477. Based on these data,

we can calculate the probability of getting a sample mean no less than 5.0 (cutoff point X)

on this normal distribution. It can be completed by using the NORM.DIST formula in Excel

shown below:

Use Excel Formula NORM.DIST (enter the values and text shown in bold):

Formulas – Insert Function – NORM.DIST – OK – X 5.0

Mean 5.03

Standard_dev 0.061477

Cumulative true

The result is 0.312. It means that P(drawn sample mean < 5.0) = 0.313.
Then P(drawn sample mean >= 5.0) = 1 – P(drawn sample mean < 5.0) = 1 – 0.313 = 0.687 =
68.7%.
This result indicates that only 68.7% of randomly drawn samples will have a mean equal to
or greater than 5.0 pounds. That is, only for 68.7% of the time, the supermarket chain will
find a sample mean of at least 5.0 pounds. This doesn't meet Mike's expectation, which is to
make sure that 95% of the time, the chain will find a sample mean of at least 5.0 pounds.
Short Term Solution: The short term solution is to calibrate the machines to add a bit more
flour to each package so that the whole distribution can be pushed to the right. This way,
we will ensure that 95% of the time, the chain will find a sample mean of at least 5.0
pounds.
Long Term Solution: Given the relatively wide distribution of product, the process may not
be in control. The larger issue is to bring the process into control.
0.313
0.687 or 68.7%
x=5.0 µ =5.03
σ =0.06
Question 3:
According to the given information,
40
sample size
n=
5.02
population mean (mean of sample means)
m=m =
s=
0.07
Standard deviation of population
Calculate Standard deviation of sample means (standard error of the mean)
σ = σ/√n = 0.388816/√40 =
0.011068
Calculate probability:
P(drawn sample mean < 5.0) =
P(drawn sample mean >= 5.0) =

See textbook Black (2010) P

1.0003

0.035380

1 – 0.035 =

96.5%

Since the sample size n is 40 (big enough), the central limit theorem may be applied. As a result, it

may be assumed that the sample means is normally distributed. This normal distribution has a

mean of 5.02 and a stardard deviation of 0.011068. Based on these data, we can calculate the

probablity of getting a sample mean no less than 5.0 (cutoff point X) on this normal distribution.

It can be completed by using the NORM.DIST formula in Excel shown below:

Use Excel Formula NORM.DIST (enter the values and text shown in bold):

Formulas – Insert Function – NORM.DIST – OK – X 5.0

Mean 5.02

Standard_dev 0.011068

Cumulative true

The result is 0.035. It means that P(drawn sample mean < 5.0) = 0.035.
Then P(drawn sample mean >= 5.0) = 1 – P(drawn sample mean < 5.0) = 1 – 0.035 = 0.965 =
96.5%.
This result indicates that 96.5% of randomly drawn samples will have a mean equal to or greater
than 5.0 pounds. This meets Mike's expectation, which is to make sure that 95% of the time, the
chain will find a sample mean of at least 5.0 pounds. It indicates that the calibration is successful.
0.035
0.035
x=5.0
0.965 or 96.5%
µ =5.02
σ =0.011
Data 1
1.5
1.75
1.8
1.7
1.8
1.3
1.7
1.66
2.2
1.7
1.91
2
1.8
1.81
1.7
1.6
1.7
1.91
1.9
1.65
1.65
1.65
1.59
1.65
1.7
1.77
1.88
1.65
1.76
1.8
1.75
1.75
1.6
1.65
1.65
1.8
1. Assume you are the manager of a musta
has received complaints that there isn't en
packages, which should have 1.7 ounces. Y
officer Juan Valdez to test the new mustar
He runs a sample of 36 packages, with the
instructor will tell you which dataset to use
on the following document:
dataset 1 only - answer the question in the
2. Calculate a 95% confidence interval on t
mustard seed. Explain very carefully to the
95% confidence interval numbers mean. In
there anything you would want to note fo
question in the same format as the 'Quest
3. Use the same dataset you used for the p
of time, Juan determines that the true mea
ounces with standard deviation the same a
that Juan has a strong consumer orientatio
businessman. Indeed, a local TV station has
periodically sampling 36 packages and figu
tells you that he wants to be at least 95% s
has at least 1.7 ounces. Further, he wants t
the TV crew that is watching him will find a
ounces. At the same time he doesn't want
packages with any more product than he a
would you give the mustard seed packagin
to calibrate their equipment? In particular,
decrease the amount they are putting in th
Are there any other steps they might be ab
situation? answer the question in the same
e you are the manager of a mustard seed factory in Colombia. Your company
ved complaints that there isn't enough mustard seed in your economy size
s, which should have 1.7 ounces. You ask your foreman and chief operating
an Valdez to test the new mustard seed packaging machine you are installing.
a sample of 36 packages, with the results of package sizes in ounces: (your
r will tell you which dataset to use). For a Word version of the Datasets, click
llowing document: BUS6780_M3 Homework Assignment Data Set.docx answer the question in the same format as the 'Question 1' tab
ate a 95% confidence interval on the average weight of packaged
seed. Explain very carefully to the packaging workers what the
fidence interval numbers mean. Include your Excel output. Is
ything you would want to note for management? answer the
n in the same format as the 'Question 2' tab
e same dataset you used for the previous problem. Over a period
uan determines that the true mean of the packages is in fact 1.72
ith standard deviation the same as the sample above. Assume
has a strong consumer orientation. Juan is also a savvy
man. Indeed, a local TV station has been checking up on Juan by
lly sampling 36 packages and figuring up the average weight. Juan
that he wants to be at least 95% sure that the average package
ast 1.7 ounces. Further, he wants to be sure that 95% of the time,
ew that is watching him will find a sample mean of at least 1.70
At the same time he doesn't want to set the equipment to fill
with any more product than he absolutely has to. What advice
u give the mustard seed packaging factory management as to how
te their equipment? In particular, should they increase or
the amount they are putting in the packages and by how much?
e any other steps they might be able to take to improve the
answer the question in the same format as the 'Question 3' tab
...
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