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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 7
You are advised to spend 40 minutes answering the questions in this booklet.
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Question One
A cell phone company is launching a new phone called the zPhone.
(a)
A random sample of 400 current customers found that 30% are planning on buying the zPhone.
Calculate a 95% confidence interval for the proportion of all the company’s current customers
who are planning on buying the zPhone.
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(b)
Before the launch of the phone, a total of 754 pre-orders had been made for the zPhone by current
customers of the cell phone company.
Another random sample of 160 current customers made just before the launch of the phone found
that 76 had pre-ordered the zPhone.
Use these results to calculate a 95% confidence interval for the total number of current customers
of the cell phone company.
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(c)
One week before launching the new zPhone, the cell phone company takes another survey of their
current customers to find out how many are planning on buying the zPhone.
They randomly sampled 100 current customers and found that the standard error for the estimate
was twice the size of the standard error determined in (a). They also found that the proportion of
current customers who are planning on buying the zPhone had increased, compared to the sample
results in (a).
Use these results to determine how many of the 100 customers sampled are planning on buying
the zPhone.
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Question Two
The cell phone company offers two plans for customers who mostly use their phones to send texts: a
super plan and a texter plan.
The super plan costs $40 and includes 400 free texts per month, where as the texter plan costs $30 and
includes 200 free texts per month.
They used data from February this year to produce the table below:
Sample mean number of
texts sent
Sample standard
deviation
Sample size (number of
customers)
Customers on
super plan
Customers on
texter plan
386.4
267.4
84.8
69.6
204
259
Let μ1 be the mean number of texts sent for all customers on the super plan over one month and μ2 be the
mean number of texts sent for all customers on the texter plan over one month.
(a)
Calculate a 99% confidence interval for μ1 - μ2.
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(b)
Does this confidence interval suggest that customers on the super plan send on average 200 more
texts per month than customers on the texter plan (the difference in the allocated free texts for the
two plans)? You must justify your answer with statistical reasoning.
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Data collected over the previous year indicated that the mean number of texts sent per month for all
customers on the super plan is 400.8.
(c)
Using the standard deviation of 84.8, calculate the probability of obtaining a sample mean as small
as 386.4 for a sample of 204 customers, and interpret what this suggests about this particular
sample of customers and the number of texts they send.
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Question Three
The manufacturers of the zPhone claim that the battery for the phone will last for at least 5 hours of
talking it will need recharging. This is called the talk time of a phone.
(a)
A random sample of 50 zPhones was taken. For the sample of phones, the mean talk time was 5.4
hours, and the standard deviation for the talk times was 0.4 hours.
Calculate a 96% confidence interval for the mean talk time of all zPhones.
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(b)
The manufacturers want to estimate the mean talk time for all zPhones to within 6 minutes of the
true value with 96% confidence.
Assuming a standard deviation for talk times of 0.4 hours, calculate the minimum number of
zPhones they should sample.
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(c)
The weights of completely packaged zPhones (including the box, manuals and packing materials)
have a mean weight of 410 g and a standard deviation of 4 g.
zPhones are shipped in cartons, where each carton contains 100 zPhones.
Calculate a 96% confidence interval for T, the true total weight of a carton of 100 zPhones.
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