Solution of probability problems (Poisson, Binomial, Standard normal z-score|)

Project description
some of the questions need to be answered by using R program, package Rcomr. ( there are attached files show how the assignment paper should be look and it includes examples that similar to the four question)
Also: Notice that the table in question 1 is not the correct format for entering two way table
Note: the attachment include:
the assignment problems ( assignment 2)
two examples ( chpt7 and 8 prac)

Assignment 2: Chapters 7 – 8
[
33 marks
]
Use correct probability notation in your responses to all questions. Marks
are awarded for presentation, including correct notation.
Question 1
[9 marks]
Researchers assigned newly laid water python eggs to one of three temperatures: hot, neu-
tral, or cold. Hot duplicates the warmth provided by the mother python. Neutral and cold
are cooler temperatures equivalent to when the mother is absent.
Caution:
you will need to calculate appropriate values from the table below before using
Rcmdr
since the following is
NOT
in the correct format for entering as a two-way table.
Cold Neutral Hot
Total Number of eggs
27 56 104
Number Hatched
16 38 75
Source: D. Moore, (2007)
Basic Practice of Statistics
(4 ed), Freeman
In your responses to the following use appropriate probability notation.
(a) Use
Rcmdr
to produce a table and calculate the marginal totals.
(b) Choose an egg at random from this population. Use the table to nd:
(i) The probability that the egg hatched.
(ii) The probability that the egg chosen was assigned to a cold temperature envi-
ronment.
(iii) The probability that the egg chosen was assigned to a cold environment and
hatched.
Show all calculations.
(c) Use
Rcmdr
to produce the appropriate table of percentages to nd the conditional
probability that the egg chosen was assigned to a hot environment,
given that
the
egg hatched.
Question 2
[6 marks]
Dandelions are studied for their eects on crop production and lawn growth. In one region,
the mean number of dandelions per square metre was found to be 7.0 (based on data from
Manitoba Agriculture and Food).
(M.M. Triola & M.F. Triola, 2006,
Biostatistics for the Biological and Health Sciences
, Pearson, Boston)
Using
Rcmdr
and presenting your results with correct probability notation:
(a) Find the probability of no dandelions in an area of 1 m
2
.
(b) Find the probability of at least one dandelion in an area of 1 m
2
.
(c) Find the probability of between 5 and 10 dandelions (inclusive) in an area of 1 m
2
.
Question 3
[7 marks]
The shell length of a population of the fossil snail
Calcarus porosis
is known to be normally
distributed with mean 100mm and standard deviation of 10mm. Let
X
represent the shell
lengths.
(a) Using statistical notation state the distribution of
X
.
(b) Is it likely that a randomly selected shell of length 133mm belongs to this population?
Explain your response. (You should not use
Rcmdr
)
(c) Use
Rcmdr
to nd the proportion of the population that has a length between 95mm
and 110mm?
(d) Verify the result in (c), using the standard normal distribution (Z-scores). Show all
calculations, using correct notation.
(adapted from S. McKillup & M. Dyar. (2010), Geostatistics Explained: An Introductory Guide for Earth
Scientists, CUP, Cambridge)
Question 4
[11 marks]
(a) Find an application of the binomial distribution to a problem that relates to your
degree.
(i) Give a
brief
summary of the context of the problem (no more than 100 words).
(ii) Explain why the binomial distribution is relevant. That is, what conditions
must be/are satised for this to be considered a binomial experiment? Relate
your response to the context of the problem.
(iii) Include the degree that you are enrolled in, word count and reference(s).
(b) Eighty-ve percent (85%) of a certain population has Rh positive blood. Suppose a
random sample of 10 subjects was selected from the population.
(i) Assuming that the conditions for a binomial experiment are met, nd the ex-
pected number of patients in the sample with Rh positive blood. Also nd the
standard deviation. Show all calculations and use correct probability notation.
(ii) Use
Rcmdr
to nd the probability that
(1) exactly four patients will have Rh
negative
blood.
(2) no fewer than 3 and no more than 5 patients have Rh
positive
blood.
(Source: M. Samuels and J. Witmer, Statistics for the Life Sciences (2 ed), Prentice Hall)