• This assignment is worth 10% of your final grade.

Attempt all questions, there are 100 marks in

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Where you are asked to construct a graph, you must include it in your answer script (if you’re unsure how to copy a graph from Excel, see Help with Excel document #4 on Blackboard.) Label your graph appropri- ately.

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In questions 3 and 4 you are asked to include in your document sketches of the normal curve. One way to do this is to take photos on your phone of your sketches and paste them into your document.

When you submit your assignment it must be just one PDF
You must submit your assignment online via the link ‘Assignment 2 Sub- mission’ on the Assignments page of Blackboard (all instructions under Assignments tab). Please ensure your assignment is reasonably well spaced so that it is easier to
Late assignments will have 5% deducted from your total mark if received after the due date, but before midnight Wednesday 16 September Assignments can not be graded after this time.
In case of illness, bereavement or significant technical problems, late assignments may be accepted without incurring the 5% penalty, but you must contact us (admin193@sms.vuw.ac.nz) and we ask that you provide written
You may work together on assignments, but you must write Plagiarism is not acceptable.

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[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

[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 .
[4 marks] Ten light bulbs are randomly selected from a batch and the lengths of time they last before they fail is
[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 home- town. 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 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.

∼

[5 marks] Write down the probability distribution of X using the ‘X ‘ notation. Justify your choice of
[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 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) = …
1. exactly 10 business days ii.at least 10 business days

iii.between 6 and 16 (inclusive) business days iv.fewer than 6 business days.

[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

(d) [2 marks]

If the value of p was 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.

[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
- [3 marks] The proportion of couples with children that have had their first child before the mother is
- [3 marks] The proportion of couples with children that had their first child when the mother was over
- [2 marks] The proportion of couples with children that had their first child when the mother was between 35 and
- [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
- [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 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.
- [3 marks] P (Z < 0)
- [3 marks] P (Z > 2)
- [2 marks] P (| Z |< 2)
- [2 marks] P (| Z |> 2)
- [2 marks] The probability that a normal random variable is more than 3 standard deviations above or below the

[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

[12 marks] The course was designed to lower anger control 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.
[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

[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 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.

[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
1. 5% confidence interval 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.

[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 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.
[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?
[2 marks] Comment briefly on the suitability of the test carried out in question You may wish to include the boxplot generated by iNZight.

[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
- [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
- [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 A sample of 15 adult eels were caught and their length measured. The length of eels is known to follow a Normal distribution.