Essay
1. Use the background information below to create the essay.
Case Study
Scenario
Land and Agua Insurance Company has a call center in Tempe, Arizona. The business
was originally established in Phoenix, Arizona in 1972 as a small business, and it has
grown with the population of the area. The insurance company specializes in bundling
insurance for cars, off-road vehicles, and watercraft (e.g., jet skis and boats). The
company has 150,000 clients in Arizona.
Marjorie Jones, Vice President of Operations, is concerned about customer complaints
and the amount of time representatives are taking to resolve the calls. You are part of
the team investigating the data to determine the probabilities of errors and call times.
Ms. Jones also wants to understand the approximate range around the average for call
times.
2. Answer the questions below in essay format. Your essay must include an introduction,
a body, and a conclusion. It must address all relevant parts of each question. Your
response should be a minimum of 500 words in length, and it should include your
analysis of the probability calculations. Make sure to cite any source you use. Proper
citation format for a source includes the name of the author(s), the title of the work,
the date of the publication, and the page number if you directly quote the source.
Essay: Probability
Using the Quality Summary and Call Center Data, provide a summary report for the vice
president including the following information in an essay with a minimum of 500 words:
1. Based on the probability of an error provided in the quality summary under call quality
using a sample size of 15, predict the probability of both < 2 errors or errors using the
correct discrete probability distribution. Assume calls are either correct or incorrect.
2. Using the call time mean and standard deviation from the quality sample, find the
probability of a call time < 7min, between 7 and 9 min, and > 9 min.
3. Calculate and evaluate the 95% confidence interval for the mean from the call time
data.
Answer
The central limit theorem and the typicality of the disseminations are basic components for
insights and utilized for the rest of the lessons. "The Central Limit Theorem expresses that
the example methods for huge estimated tests will be ordinarily disseminated paying little
heed to the state of their populace appropriations" (Donnelly, 2015, p. 301). For this
situation, we have to know the room for mistakes and the standard blunder are two unique
ideas. The room for give and take, or the width of the interim, is the basic z-score duplicated
by the standard mistake for the mean. Be that as it may, a bigger example lessens the
standard mistake and, along these lines, the room for give and take.
Typical conveyances utilize the z-score estimation, which you found out about in Lesson 2,
to recognize the likelihood. To additionally develop this idea, there are times when you
should discover the probabilities that are > < or some place in the middle of two distinctive
z-scores (Donnelly, 2015). In light of the likelihood of a blunder gave in the quality synopsis
under call quality utilizing an example size of 15, we have n = 15 calls. The consequent
phase is to determine the likelihood of <2 blunders. P (mistake) = 0.15. At that point we
make utilization of binomial likelihood recipe: (x < = 2) = P(x=0) + P(x=1) + P(x=2); P(x=0) =
15C0 * 0.15^0 * 0.85^15 = 0.08735; P(x=1) = 15C1 * 0.15^1 * 0.85^14 = 0.23123; P(x=2) =
15C2 * 0.15^2 * 0.85^13 = 0.28564. In this manner, we have P(X<=2) = 0.28564 + 0.23123
+ 0.08735. The last answer will be P(x < = 2) = 0.604225. The calls are perfect.
Binomial Probabilities
Data
Sample size
Probability of an event of interest
15
0.15
Statistics
Mean
Variance
Standard deviation
2.25
1.9125
1.3829
Binomial Probabilities Table
X
P(X)
0.0874
0.2312
0.2856
0.2184
0.1156
0
1
2
3
4
<2
0.3186
5
6
7
8
9
10
11
12
13
14
15
0.0449
0.0132
0.0030
0.0005
0.0001
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
>=5
0.0617
We have the quantities of test size, mean and standard deviation figured through exceed
expectations. The example estimate is 15, Mean is 12.05, Standard Deviation is 4.502. The
second inquiry is to discover P(X<7). Since (x-mean)/standard deviation=2, P(2<(712.05)/4.502)=P(z<-1.12)=P(z>1.12)=1-P(z>1.12). With a z appropriation table, the last
answer will be P(x<7) =1-0.869=0.131. The subsequent stage is to ascertain the likelihood
in the vicinity of 7 and 9. P(70.68)- 0.131=1-P(z<0.68)- 0.131=0.1169. P(x>9) =P (z> (9-12.05)/4.502) =P (z>0.68) =P (z<0.68) =0.752.
When computing the certainty interim populace standard deviation is known, we utilize the z
conveyance, with the recipe of mean +-Za /2* standard blunder. Standard mistake is a
critical piece of utilizing the central limit theorem. Standard mistake is unique in relation to
examining blunder. This measurement centers on the standard deviation versus the mean.
We require standard mistake to discover the certainty interim. Standard error= standard
deviation/sqrt test size=4.502/sqrt15= 1.16. The estimation of 1.96 depends on the way that
95% of the territory of a typical circulation is inside 1.96 standard deviations of the mean;
1.16 is the standard blunder of the mean. LCL=12.05 - (1.96) (1.16) = 9.776, UCL=12.05 +
(1.96) (1.16) = 14.32.
All in all, the likelihood of either < 2 mistakes or ≥ 5 blunders is 0.604. The likelihood of a
call time < 7 min is 0.131, the likelihood of a call time in the vicinity of 7 and 9 min is 0.1169,
and the likelihood of a call time > 9 min is 0.752. Moreover, we should know about the
meaning of the certainty interim for the mean - an interim gauge around the example imply
that gives a range of where the genuine populace means falsehoods.
•
Calculate and evaluate the 95% confidence interval for the mean from the call time
data.
o
Problem uses t critical value since you do not know population standard
deviation and you are using the sample standard deviation.
11.07 +/- (1.98)(3.38/sqrt(150)= 10.52, 11.61
Reference:
Donnelly, R. A. (2015). Business statistics (2nd ed.). Upper Saddle River, NJ:
Pearson.