- [12 marks] In the following situations, identify the random variable of interest
(e.g. ‘Let X measure/count …’). Then state whether or not the r.v. is binomial,
justifying your answer.
(a) [4 marks] A police officer randomly selects 30 cars to find out how many do
not have a current Warrant of Fitness (WOF). She knows from experience that
the probability a car does not have a current WOF is 1
6
.
(b) [4 marks] Ten light bulbs are randomly selected from a batch and the lengths
of time they last before they fail is measured.
(c) [4 marks] Mike is repeatedly rolling two dice and will stop when he gets a
double six. He counts the number of rolls until he gets a ’success’. - [17 marks] May is interested in purchasing the local hardware store in her hometown. After examining the store accounts for the past two years, she found that the
store had been earning a gross amount of over $850 per day for 70% of the business
days it was open. A random sample of n = 20 business days is selected from the
last two years. Let X represent the number of days where the store earned over
$850 gross.
(a) [5 marks] Write down the probability distribution of X using the ‘X ∼‘
notation. Justify your choice of distribution.
(b) [8 marks] Using Excel and providing the exact Excel functions you use, give
the probability that in the sample the store will gross over $850 for the following
numbers of days. If you wish, you can use notation such as P(X < …) to write
your answers e.g. for 2(b)i you can write P(X = 10) = …
i. exactly 10 business days
ii. at least 10 business days
iii. between 6 and 16 (inclusive) business days
iv. fewer than 6 business days.
(c) [2 marks] May decides to purchase the store and records the accounts for a
period of three months from the date of purchase. She then takes a random
sample of 20 business days from those three months and finds that the store
did gross over $850 per day for fewer than 6 business days in her sample. Do
you think this should this make May suspect that p (the probability of grossing
over $850 on a business day) is now less than 0.7? Explain your answer.
(d) [2 marks]
i. If the value of p was 0.6, calculate P(X < 6) for a sample of n = 20.
ii. Suggest a value of p that might mean that May would not have been
surprised to find the store grossed over $850 for fewer than 6 business
days, briefly explaining your answer.
STAT193 2020 Trimester 2 2 Assignment 2 - [14 marks] In families consisting of a couple with children, X, the age of the
mother at the birth of her first child, follows a roughly Normal distribution with
mean µ = 27.9 years and standard deviation σ = 3.7 years. Using Excel, and
providing the exact functions you use, calculate the following. Draw a careful
sketch of the Normal curve for each of parts (a), (b), (c) and (d), showing the area
that corresponds to your answer.
(a) [3 marks] The proportion of couples with children that have had their first
child before the mother is 20.
(b) [3 marks] The proportion of couples with children that had their first child
when the mother was over 35.
(c) [2 marks] The proportion of couples with children that had their first child
when the mother was between 35 and 40.
(d) [3 marks] What is the youngest age of a mother in the oldest 10% of first-time
mothers, based on the given parameters? Give your answer to 1 decimal place.
(e) [3 marks] Calculate the standardised value (the Z-score) of a first-time mother
whose age is 20.5 years. What information does this standardised value give
about the age 20.5 in this context? - [12 marks] Find the following probabilities using the properties of the standard
normal random variable Z and Excel to calculate your answers. Give your answers
(except for part (a)) to 3 decimal places. Draw a careful sketch of the Normal curve
for each of parts (a) and (b), shading the area that corresponds to your answer.
(a) [3 marks] P(Z < 0) (b) [3 marks] P(Z > 2)
(c) [2 marks] P(| Z |< 2) (d) [2 marks] P(| Z |> 2)
(e) [2 marks] The probability that a normal random variable is more than 3
standard deviations above or below the mean.
STAT193 2020 Trimester 2 3 Assignment 2 - [16 marks] Twelve inmates in a particular prison took part in an anger management
course. Before the course began, the median score on an anger control assessment
of the group was 42 (out of a maximum of 60). After the six-month course, the
participants’ scores on the same assessment were:
35 41 40 44 57 36 39 47 58 39 42 39
(a) [12 marks] The course was designed to lower anger control scores. Perform a
sign test on whether the course was successful using α = 0.05. Clearly show all
steps of the hypothesis testing process, including how you obtained your p-value.
(b) [4 marks] Enter the data in a single column in Excel by typing the values into
successive cells in the column. Construct a boxplot of the data. Based on the
boxplot, explain briefly why a sign test is appropriate to use in this case. - [21 marks] A Waikato couple have decided to improve stock management practices
on their dairy farm as they are concerned about the amount of pollution that the
farm has caused in nearby waterways. They are particularly concerned about the
amount of nitrate run-off coming from their dairy herd. Until recently, nitrate levels
in their local river were often well above the MAV (maximum acceptable value) for
drinking water of 50mg/litre.
Once the final improvements on the farm were made, the farmers took frequent
measurements of nitrate levels in the river over a period of 6 months. They then
took a random sample of 15 of those measurements. These data are available from
Blackboard in the files NitrateLevels.csv and also NitrateLevels.xlsx.
(a) [8 marks] Using Excel and the data provided, give the margin of error (to 3
d.p.) and confidence interval (to 2 d.p.) for the
i. 95% confidence interval
ii. 99% confidence interval
of the nitrate levels in the river in mg/litre (to get the required accuracy you
will need to give your value of s to at least 3 dp). Explain briefly what the
95% CI tells us.
(b) [8 marks] The farmers wish to test their hypothesis that the improvements
they have made have caused nitrate levels in the river to drop below the MAV.
Imagine that they have asked you to make use of your statistical knowledge.
Using iNZight and the data provided, conduct a one sample t-test (t-test of a
mean) using a significance level of α = 0.01. Clearly show all steps in your test,
including both hypotheses, the relevant information obtained from iNZight and
of course the conclusion.
(c) [3 marks] What would have been the changes to your answer to question
6b if the farmers had wanted to test whether nitrate levels were significantly
different from the MAV?
(d) [2 marks] Comment briefly on the suitability of the test carried out in question
6b. You may wish to include the boxplot generated by iNZight.
STAT193 2020 Trimester 2 4 Assignment 2 - [8 marks] In each of the scenarios described below, state what the appropriate test
to perform would be. Justify each answer in one or two sentences.
(a) [4 marks] A study is to be conducted to determine whether one kind of cough
medicine is more effective than another in increasing sleep. 20 people with
colds are given medicine A the first night and then medicine B the second
night. Their hours of sleep each night are recorded. The hours of sleep are
known to follow a skewed distribution.
(b) [4 marks] A study is to be carried out to determine whether the mean length
of adult fresh water eels in a lake has changed since a new dairy farm was
established nearby. A sample of 15 adult eels were caught and their length
measured. The length of eels is known to follow a Normal distribution.
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