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A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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One of the bartleby experts helped me with this homework problem. I am having difficulty figuring out where 21.76 and 4.21 came from on part (a). Look at images below. Also, there is a spreadsheet on Excel with data that is used for this problem.

If there is one thing engineers like, it is a good pizza that is delivered in a timely
manner. A particular pizza chain is thinking about offering a guarantee of “less
than 30 minutes, or it’s free” on all of its deliveries. If the pizza business can pull
this off, they will impress customers across the nation, but having to walk this
promise back would be devastating to the reputation of the business. However,
the CEO will not green light this policy if it means taking a substantial hit
to profits. Accounting has run some numbers, and they have determined that
the business can only miss 5% of all deliveries before the costs of the missed
deliveries begin to impact profitability. The CEO agrees to a pilot test. 120
drivers time their deliveries with a stopwatch and log them in a spreadsheet.
The CEO then sends this spreadsheet to you for analysis.
Using all of the technical terms and symbols that you want:

(a) Should the pizza chain implement this policy?

(b) Create a graphic that visually represents the shape of the distribution
well.

(d) Describe the distribution of the data in detail (shape, center, spread, skew,
kurtosis, etc.). We have not covered how to formally calculate all of these
numbers, and that is fine – a verbal description will be sufficient in these
cases.

(e) Now that you have the details of the distribution, how did the distribution
impact your analysis?

(f) Would the analysis have been better/easier/different if the data followed a different shape?

(g) In your professional opinion, should any of the data points be removed?
Give a bit of color regarding the ethics of your decision. Are we throwing
out the data just because we want to see better symmetry and make the
math cleaner, or can we actually provide some justification for why the
data should be removed? Would such a removal change your decision of
the business case?

Here is the data from the Excel spreadsheet

Delivery Time
20.58
19.2
26.79
27.86
20.95
18.12
20.41
20.68
18.47
18.73
30.01
19.8
18.34
18.15
21.49
21.28
19.57
19.59
20.94
21.09
25.94
21.07
20.16
20.38
18.42
19.77
20.35
20.72
23.88
20.98
18.94
20.41
20.17
20.68
18.91
21.34
19.99
20.03
18.77
20.04
21.33
31.04
20.77
32.01
20.41
21
20.55
22.69
29.55
20.97
19.38
18.88
19.26
21.51
20.26
20.84
20.1
21.86
21.66
20.24
20.22
20.27
18.24
20.85
18.55
21.31
19.14
29.85
19.1
18.72
20.36
18.66
20.72
21.95
18.22
21.28
20.27
18.94
18.31
20.62
19.15
18.77
32.67
19.51
19.23
20.2
20.53
20.4
20.03
20.77
33.44
22.02
22.1
19.47
18.9
18.86
19.08
20.46
21.21
21.12
21.69
50.03
23.86
20.49
27.46
21.9
20.11
18.01
29.08
24.13
23.49
24.15
23.94
26.32
28.62
25.56
23.92
24.65
23.54
19.37

a)
x =
Ex
n
= 21.75941667
≈ 21.76
S =
Σ(-x)2
n-1
= 4.209253571
≈ 4.21
Null hypothesis: Ho: μ = 30
Alternative hypothesis: HA: μ< 30
This hypotheses constitute a one-tailed test and significance level for the test is 0.05.
Use the sample data to calculate the standard error (SE), degrees of freedom (DF), and the t statistic test
statistic (t).
S.E=
=
S.D
√n
4.21
√120
= 0.384319
≈ 0.3843
D.F=n-1

D.F=n-1
= 120 - 1
= 119
t =
x-μ
S.E
21.76 - 30
0.3843
= -21.441582
≈
21.44
The P-value for this test is the probability that a t statistic having 119 degrees of freedom is less than -21.44.
P-value = P(t < -21.44)
Use the excel formula, " =T.DIST(-21.44,119)" to determine the P-value.
P-value = 0.000
Decision:
Since the P-value (0.000) is less than the significance level(0.05), hence we have to reject the null
hypothesis.
Conclusion:
From the above test we have sufficient evidence in the favor of the claim that the pizza chain should
implement this policy.
The percentage of deliveries within 30 minutes is high enough to meet the CEO's requirement (less
than 5% missed deliveries), then implementing the policy may be feasible.

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...

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