Mathematics and Statistics 91267 (2.12) Exam

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Credits: Four

You should answer ALL parts of ALL questions in this booklet.

You should show ALL your working.

If you need more space for any answer, use the page(s) provided at the back of this booklet and clearly number the question.

Check that this booklet has pages 2–12 in the correct order and that none of these pages is blank.

YOU MUST HAND THIS BOOKLET TO YOUR TEACHER AT THE END OF THE ALLOTTED TIME.

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You are advised to spend 60 minutes answering the questions in this booklet.

Question one: normal distribution

(a)  Research has shown that the weights of newborn lambs in Southland are normally distributed with a mean of 1.5 kg and a standard deviation of 0.125 kg. Use this model to answer the questions below.

(i)  What is the probability that a newborn lamb, in Southland, chosen at random, weighs between 1.5 kg and 1.7 kg?

(ii)  What percentage of newborn lambs would Sam, a farmer on a Southland farm, expect to weigh more than 1.75 kg.

(iii)  Newborn lambs in Southland that weigh less than 1.25 kg are underweight and likely to die. Sam expects that he will have 6 400 lambs born this year on his farm.

How many newborn lambs would Sam expect to die from being underweight?

(iv)  What birth weight is exceeded by 30 % of the newborn lambs?

(v)  What is the range for the central 70 % of weights of newborn lambs?

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(b)  Ali has a farm in Southland. She records the weights of 32 lambs born on her farm. The results are shown on the histogram below.

(i)  What proportion of the lambs weighs less than 1.25 kg?

(ii)  Is Ali justified in thinking that the weights of her lambs are normally distributed.

Use statistical terms to explain your answer.

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(c)  Bill, a farmer who lives in Northland, records the weights of his 130 newborn lambs. His results are shown on the histogram below.

Compare the distribution of the weight of newborn lambs on Bill’s farm with the distribution of the weights of newborns lambs on Ali’s farm in Southland.

Use statistical terms to explain your answer.

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Question two: risk

(a)  People with a serious skin rash are treated with EITHER Medical Cream A OR Moisturising Cream B. Moisturising Cream B has no medical content.

The table below gives the number of people treated with each type of cream.

(i)  What proportion of the people treated with EITHER Medical Cream A OR Moisturising Cream B were cured?

(ii)  What percentage of people who were cured used Medical Cream A?

(iii)  What proportion of the people in the trial were not cured and had used Medical Cream A?

(iv)  If a pilot study was conducted with 2 000 participants to test Medical Cream A, how many could be expected to be cured?

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(v)  A newspaper headline states that by using Medical Cream A, a person is twice as likely to be cured as a person using Moisturising Cream B. State whether you agree with this headline, giving full reasons and showing calculations.

(b)  Of the 535 people who were treated, 274 of them were males and 93 males were cured. 50% of the males were using Medical Cream A and 75 of these males were cured.

(i)  What is the probability that a male using Moisturising Cream B is not cured?

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(ii)  Some researchers claimed that the relative risk of a man not being cured is two times as great for a man who uses Moisturising Cream B, compared to a man who uses Medical Cream A. Give statistical reasons supported by calculations for this claim.

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Question three: probability trees

A factory manufactures parts for cars. Two different machines, A and B, run continuously when the factory is working.

Sammi, Tama, and Jake work on Machines A and B.

The company are investigating the performance of their machines and their workers.

Machine A is used to produce 66% of the parts.

Some of this information is shown on the probability tree below.

(a)  What is the probability that a part selected at random is produced by Machine B?

(b)  42 % of Machine A parts are produced by Sammi. Jake and Tama each produce an equal number of the remaining parts that are made on Machine A.

(i)  What is the probability that a randomly selected part is produced by Jake using
Machine A?

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(ii)  Raju only works on Machine B and produces 45% of the parts from that machine. Sammi and Tama each produce 15% of the parts on Machine B. Jake produces the remainder.

What is the probability that a part selected at random was produced by Jake on
Machine B?

(iii)  If 12 000 parts are produced in a day, how many parts would Jake produce?

(c)  The probability that any part Sammi makes on Machine A will be faulty is 0.01 and the probability that any part he makes on Machine B will be faulty is 0.005.

(i)  A part that Sammi has made is found to be faulty. What is the probability that Sammi produced the faulty part on Machine A?

(ii)  The probability that any of the operators on Machine B will produce a faulty part is 0.005.

A part made on Machine B is faulty. What is the probability that it was made by Raju?

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(iii)  On one day 0.8% of the 12 000 parts were found to be faulty. 24 of these parts had been produced by Raju.

What percentage of the parts Raju produced that day were faulty? Compare this result with the overall production of parts that were faulty.

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