Solved by verified expert:Compose a 2page essay with the given answer. This assignment requires you to answer the questions and write a 2page essay with your answers. The answer now has already been calculated, you only need to compose an essay. Please read the requirement very carefully. Thank you!NOTE: Your essay has to be different from the given answer. You can only use the calculated answer. You cant copy any sentences from the given answer.
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GBS221
Key Terms and Definitions – Lesson 6
Term
Definition
Confidence
An interval estimate around the sample mean that
Interval for
provides a range for where the true population mean lies
the Mean
Confidence
An interval estimate around the sample proportion that
Interval for
provides a range for where the true population proportion
the
lies
Proportion
Confidence The probability that the confidence interval will include the
Level population parameter; population parameter may be a
mean or proportion
Degrees of The number of values that are free to vary in equations
Freedom where certain information, such as the sample mean, is
known
Student T- The probability distribution used in place of the normal
Distribution distribution when the population standard deviation is
unknown and the sample standard deviation is used.
GBS221 – Page|1
In lesson 4,5 and 6, the most important concept is the central limit theorem. The central limit
theorem and the normality of the distributions are critical factors for statistics and used for the
remainder of the lessons. “The Central Limit Theorem states that the sample means of large-sized
samples will be normally distributed regardless of the shape of their population distributions”
(Donnelly, 2015, p. 301). In this case, we need to know the margin of error and the standard error
are two different concepts. The margin of error, or the width of the interval, is the critical z-score
multiplied by the standard error for the mean. But a larger sample reduces the standard error and,
therefore, the margin of error.
Normal distributions use the z-score calculation, which you learned about in Lesson 2, to identify
the probability. To further expand on this concept, there are times when you will need to find the
probabilities that are > < or somewhere in between two different z-scores (Donnelly, 2015). Based
on the probability of an error provided in the quality summary under call quality using a sample size
of 15, we have n = 15 calls. The next step is to find the probability of <2 errors. P (error) = 0.15.
Then we make use of binomial probability formula: (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. Therefore, we have P(X<=2) = 0.28564 + 0.23123 + 0.08735.
The final answer will be P(x < = 2) = 0.604225. The calls are correct.
We have the numbers of sample size, mean and standard deviation calculated through excel. The
sample size is 15, Mean is 12.05, Standard Deviation is 4.502. The second question is to find
P(X<7). Since (x-mean)/standard deviation=2, P(2<(7-12.05)/4.502)=P(z<-1.12)=P(z>1.12)=1P(z>1.12). With a z distribution table, the final answer will be P(x<7)=1-0.869=0.131. The next
step is to calculate the probability between 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 calculating the confidence interval population standard deviation is known, we use the z
distribution, with the formula of mean +- Za/2* standard error. Standard error is an important part
of using the central limit theorem. Standard error is different from sampling error. This statistic
focuses on the standard deviation versus the mean. We need standard error to find the confidence
interval. Standard error= standard deviation / sqrt sample size=4.502/sqrt15= 1.16. The value of
1.96 is based on the fact that 95% of the area of a normal distribution is within 1.96 standard
deviations of the mean; 1.16 is the standard error of the mean. LCL=12.05 - (1.96)( 1.16) = 9.776,
UCL=12.05 + (1.96)( 1.16) = 14.32.
In conclusion, the probability of both < 2 errors or ≥ 5 errors is 0.604. The probability of a call time
< 7 min is 0.131, the probability of a call time between 7 and 9 min is 0.1169, and the probability
of a call time > 9 min is 0.752. In addition, we need to be aware of the definition of the confidence
interval for the mean – an interval estimate around the sample mean that provides a range for where
the true population mean lies.
Reference:
Donnelly, R. A. (2015). Business statistics (2nd ed.). Upper Saddle River, NJ: Pearson.
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.
…
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