Probability and Statistics Homework Assignment: Solutions and Analysis

This document presents solutions to a probability and statistics homework assignment, addressing various statistical concepts and problem-solving techniques. The assignment includes analysis of observational studies, calculations involving standard deviation and confidence levels, and the application of z-scores. It also covers histogram interpretation and the calculation of mean and probabilities. Furthermore, the solution provides detailed answers to specific questions, including calculations and interpretations. The document incorporates the use of Excel for data analysis and includes a bibliography of relevant resources. The assignment covers topics from basic probability to more advanced statistical analysis, offering a comprehensive understanding of the subject matter.

Probability and statistics
Name
Title
31-Aug-24

Q3
This an observational study since randomly selected people are involved. There is
nothing that is control but the subjects are in their natural setting
Q4
This is an observational study. There is no criterion that is used to select the individual
subject. The test subjects are random and could be either well represented or biased.
Q11
Assumptions:1
i. 95% level of confidence
ii. The standard deviation = 0.6
Therefore;
Z= σ
√n
Where σ is the standard deviation and n is the sample size.
= 1.96 x 0.6
√ 80
= 0.1314
1 Cohen, Marina. Probability
Heiny, Katherine. Standard Deviation. New York: Vintage Books, a Division of Penguin
Random House LLC, 2018.

Q 12
Z= σ
√n
Where σ is the standard deviation and n is the sample size.
Assumed value 0f σ =0.4 and the confidence level = 95%
= 1.96 x 0. 4
√ 25 0
= 0.0.4958
Q 13
Assumed confidence level = 95% and the standard deviation = 0. 5
0.01= 1.96 x 0.5
√ n
9604 = n
n≈ 10
Q14
Assumed confidence level = 95% and the standard deviation = 0. 5
0.05= 1.96 x 0.5
√ n
384.16 = n
n≈ 384

Q15
a. ,
Using the excel, the following was produced;
Bin
Frequenc
y
1200 0
1400 6
1600 8
1800 4
2000 2
2200 4
2400 1
More 0
Histogram is as shown below
1200 1400 1600 1800 2000 2200 2400 More
0
5
10
0
6 8
4 2 4
1 0
Histogram
Frequency
Bin
Frequency
A lot of values are concentrated between 1400 and 1600. The shape of the curve could
be a sine wave whose amplitude reduces with time. 2
b. , from the excel, the mean is found to be 1654 while the standard deviation from the
mean is 335.2859. the mean gives an estimate figure of values where the one is able
2 Rotarʹ, V. I. Probability and Stochastic Modeling. Boca Raton: CRC Press, 2013.

to rough estimate how high or the values could be from the average using the
standard deviation1.
Q17
a. ,
→p( x−μ
σ ≥ 10−12.4
1.6 )
¿ p (Z≥−1.5 ¿
From the z table value = 0.0668
The probability is
=1- 0.0668
= 0.9332
b. ,
P (11,12,13) = - p (11) + p (13)
= - P ( z ≤ 11−12.4
1.6 ) + P ( z ≤ 13−12.4
1.6 )
= p(z ≤-0.875) + p(z ≤0.375)
= - p(0.31635) + P(0.6480)
= 0.4845
For the 30 years then
= 30 x 0.4845
= 14.535
3 Pagès, Gilles. Numerical Probability: An Introduction with Applications to Finance. Cham,
Switzerland: Springer, 2018.

Q 18
a. ,
= P (z ≤ 10−1 6.8
3.7 )
= p (z ≤−1.837 ¿
=P(0.0329)
b. ,
P (15,20) = - p (15) + p (20)
= - P ( z ≤ 15−16.8
3.7 ) + P (z ≤ 20−1 6.8
3.7 )
= p(z ≤-0.4864) + p(z ≤0.8649)
= - p(0.3121) + P(0.8078)
= 0.4957
For 20 years 4
= 20 x 0.4957
= 9.914
Notes
1. Cohen, Marina. Probability. New York, NY: Crabtree Pub., 2010.
4 Stine, Robert A., and Dean P. Foster. Statistics for Business: Decision Making and Analysis. (2011)

2. Rotarʹ, V. I. Probability and Stochastic Modeling. Boca Raton: CRC Press, 2013.
3. Pagès, Gilles. Numerical Probability: An Introduction with Applications to Finance. Cham,
Switzerland: Springer, 2018.
4. Stine, Robert A., and Dean P. Foster. Statistics for Business: Decision Making and Analysis.
Boston: Addison-Wesley, 2011.
5. Heiny, Katherine. Standard Deviation. New York: Vintage Books, a Division of Penguin
Random
House LLC, 2018.
Bibliography
Carlton, Matthew A., and Jay L. Devore. Probability with Applications in Engineering, Science, and
Technology. Cham, Switzerland: Springer, 2017.
Cohen, Marina. Probability. New York, NY: Crabtree Pub., 2010.
Heiny, Katherine. Standard Deviation. New York: Vintage Books, a Division of Penguin Random
House LLC, 2018.
Pagès, Gilles. Numerical Probability: An Introduction with Applications to Finance. Cham,
Switzerland: Springer, 2018.
Rotarʹ, V. I. Probability and Stochastic Modeling. Boca Raton: CRC Press, 2013.
Rowbottom, Darrell P. Probability. Cambridge, UK: Polity, 2015.
SHIRYAEV, ALBERT N. PROBABILITY. S.l.: SPRINGER, 2019.
Stine, Robert A., and Dean P. Foster. Statistics for Business: Decision Making and Analysis. Boston:
Addison-Wesley, 2011.

Suess, Eric A., and Bruce E. Trumbo. Introduction to Probability Simulation and Gibbs Sampling
with R. New York, NY: Springer New York, 2010.