Download Credits: Three - Selwyn College

Selwyn College
Level 3 Statistics and Modelling
90642 Calculate confidence intervals for population
parameters.
Credits: Three
Answer ALL questions in the spaces provided in this booklet.
Show ALL working for ALL questions.
For Assessor’s use only
Achievement Criteria
Achievement
Calculate confidence intervals
for population parameters.
Achievement with Merit
Demonstrate an understanding of
confidence intervals.
Overall level of Performance
Achievement with Excellence
Demonstrate an understanding of
the theory behind confidence
intervals.
Page 2 of 5
You are advised to spend 35 minutes answering the questions in this booklet.
Show ALL working
Question One
(a)
Coffee stains are a common form of damage to books at the library. A random sample of 200
books at the library found that 26% of the books had coffee stains.
Calculate a 99% confidence interval for the proportion of all books at the library that have coffee
stains.
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(b)
The library wants to estimate the proportion of all its books which have at least one page missing
to within 10%, at a 98% level of confidence.
What is the minimum sample size needed?
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(c)
The shelves of the library are designed to hold different weights of books.
(i)
A random sample of 150 young adult fiction books at the library gave a mean weight of
340g and a standard deviation of 52g.
Construct a 95% confidence interval for the mean weight of all young adult fiction books
at the library.
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(ii)
50 of the books from the sample in (c)(i) were randomly selected to be placed on a shelf.
Construct a 90% confidence interval for T, the total weight of 50 randomly selected books
from the sample in (c)(i).
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Question Two
(a)
The library is considering the popularity of its digital media service. A random sample of the
amounts spent on borrowing digital media was taken from the library’s issues database. The
amount spent on borrowing music and movies was compared, and the results are summarised in
the table below:
Music
Movies
40
50
Sample mean ($)
314.50
424.80
Sample standard deviation ($)
287.20
298.90
Sample size
(i)
Let µ1 = the mean amount of money spent on borrowing music and µ2 = the mean amount
of money spent on borrowing music.
Construct a 95% confidence interval for µ1 - µ2
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(ii)
Explain what the confidence interval constructed in (i) indicates about whether music is
more popular than movies from the library’s digital media service.
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(b)
The library sends out a newsletter each month. Last month’s newsletter contained an article titled
“Last month teenagers borrowed more books per visit than adults”.
The article explains that when asked how many books they had borrowed on their last visit, a
random sample of 60 teenagers gave a sample mean of 3.5 books per visit and a random sample of
60 adults gave a sample mean of 2.8 books per visit.
(i)
If the library used a 99% confidence interval as the basis for the claim for their article, give
the range of possible values for the standard error of the estimated difference.
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(ii)
The library analysed the number of issues per visit for all teenagers on its issues database
and discovered the mean number of books borrowed per visit is actually 2.9.
If standard deviation for the number of books borrowed per visit for teenagers is 2.3,
calculate the probability of obtaining a sample mean of 3.5 or higher using the same
sample size as used by the library and discuss what this indicates about the original sample
of teenagers obtained.
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