g) Chapter 8
8.1
The standard deviation of the fat content of a brand of flapjack is known to be 0.8 gram
per 100-gram bar. The mean fat content of a random sample of 33 bars is 24.7 grams.
(a)
Construct a 95% confidence interval for the population mean fat content of the
bars of flapjack.
(b)
Construct a 99% confidence interval for the population mean fat content of the
bars of flapjack.
8.2
The burning times of aromatherapy candles are known to be normally distributed with a
standard deviation of 6 minutes. The mean burning time of a random sample of 18
candles was 3 hours 51 minutes. Set up:
(a)
A 90% confidence interval for the mean burning time of the candles.
(b)
A 99% confidence interval for the mean burning time of the candles.
8.3
To help shape her promotional strategy the General Manager of a regional airport wants
to know how far passengers travel to use the airport. The mean and standard deviation of
the distances travelled by a random sample of 61 passengers were 24.1 miles and 5.8
miles respectively.
(a)
Find a 95% confidence interval for the mean distance travelled by passengers.
(b)
If the population standard deviation were later found to be 6.3 miles, what size
sample would be needed in order to estimate the mean distance travelled by
passengers for the error to be one mile with a 95% level of confidence?
8.4
The distribution of rents of flats in a city is known to be a skewed with most flats being
relatively cheap to rent but some, in prestige blocks, very expensive. The mean rent of a
sample of 48 flats was £403 per month with a standard deviation of £65.
(a)
Construct a 90% interval estimate of the mean rent of two-bedroom flats in the
area.
(b)
The population standard deviation turns out to be £59. In the light of this new
information, determine the size of sample necessary to enable the mean rent to be
estimated to within £10, in other words for the error to be £10, with a 90% level
of confidence.
8.5
The weights of bags of pears sold to customers asking for one pound of pears at a market
stall are known to be normally distributed. The weights of bags served to a random
sample of 15 customers were measured. The sample mean weight was 1.06 lb with a
standard deviation of 0.11 lb.
(a)
Produce a 95% confidence interval for the weight of ‘one pound’ bags of pears.
(b)
Produce a 99% confidence interval for the weight of ‘one pound’ bags of pears.
8.6
A waste management consultant is investigating the waste generated by households. She
is certain that the weights of household waste are distributed normally. She selects a
random sample of 12 households and measures the weight of the waste they put in their
‘wheelie bin’ one week. The results (to the nearest kilogram) are as follows:
28 41 36 50 17 39 21 64 26 30 42 12
(a)
(b)
Use these figures to construct a 99% confidence interval for the mean weight of
waste in ‘wheelie bins’.
Does the interval suggest that the mean weight of the contents of all ‘wheelie
bins’ can be assumed to be less than 40 kilograms?
8.7
In a UK survey of a random sample of 281 working women who had children, 175
reported that they felt that having children had held back their career prospects. In a
similar study of 138 women in Denmark, 65 reported that having children had held back
their career.
(a)
Produce a 95% confidence interval for the proportion of women in the UK who
consider that having children held back their career prospects.
(b)
Produce a 95% confidence interval for the proportion of women in Denmark who
consider that having children held back their career prospects.
(c)
How do the two confidence intervals compare?
8.8
As part of a survey of a random sample of students, respondents were asked whether they
were satisfied with the Students’ Union Entertainments programme. Of the 109
respondents from the Arts School 83 reported that they were satisfied with the
Entertainments programme. Of the 116 respondents from the Business School 97
reported that they were satisfied with the Entertainments programme.
(a)
Construct a 90% confidence interval for the proportion of students in the Arts
School who are satisfied with the Entertainments programme.
(b)
Construct a 90% confidence interval for the proportion of students in the Business
School who are satisfied with the Entertainments programme.
(c)
Contrast the two confidence intervals.
8.9
A television audience research company intends to install monitoring devices in a random
sample of households in order to produce 95% interval estimates of the proportion of
households watching specific programmes. If the company wants to estimate to within
2% of the real figure, in other words so that the error is 2%, in how many households will
it have to install monitoring devices?
8.10
The standard deviation of the recovery time of young adults who contract a strain of
influenza is known to be 1.3 days. The sample mean recovery time of a random sample of
40 young adults is 12.9 days. Use this result to test the hypothesis that the mean duration
of the illness is 12 days with a 5% level of significance.
8.11
Office workers at a large organization are told that they should not work for more than
one and a half hours at their PCs before taking a break. A survey of a random sample of
37 employees found that the mean time that they worked at their PCs before taking a
break one morning was one hour 43 minutes with a standard deviation of 18 minutes.
Test the hypothesis that on average the employees of this organization are working for no
more than one and a half hours before taking a break using a 1% level of significance.
8.12
In an advertisement it is claimed that a brand of domestic air freshener will last on
average at least 70 days. The mean and standard deviation of the lifetimes of a sample of
19 air fresheners were 69.3 days and 2.9 days respectively.
(a)
Test the claim made for the product using a 5% level of significance.
(b)
What assumption must be made for the test to be valid?
8.13
An IT Manager considers that the average number of hits per week on the company
intranet pages should be 75. The mean number of hits per week on a random sample of
38 pages is 71.4 with a standard deviation of 31.9.
(a)
Test a suitable hypothesis at the 10% level of significance to assess the
plausibility of the IT Manager’s assertion.
(b)
It later emerges that 13 of these pages are often visited by non-company users,
and should therefore not be included in the sample. The mean and standard
deviation of the remaining 25 pages are 65.8 and 28.3 respectively. Test the same
hypothesis again at the 10% level. What assumption must you make for the test to
be valid?
8.14
A bus company claims that the mean journey time for the service from Paglashonny to
Gorley is 53 minutes. A random sample of 35 journeys was chosen and each journey
timed. The mean journey time of these journeys was 57 minutes with a standard deviation
of 4.6 minutes.
(a)
Test the bus company’s claim at the 5% level of significance and comment on the
result.
(b)
It appears that for nine of these journeys the regular driver was absent from work
through illness and relief drivers, who were unfamiliar with the route, had to be
used. The mean journey time for the remaining journeys was 55 minutes with a
standard deviation of 3.1 minutes. Test the bus company’s claim again at the 5%
level of significance using these figures and comment on the result.
(c)
What do you need to assume about the population of journey times for the results
from the second test to be valid?
8.15
A Trade Union official wants to find out if the wages in the Deshovy region of the
country are lower than the wages in the Bugatty region. The mean and standard deviation
of the basic weekly wages of a random sample of 58 workers in the Deshovy region were
£315 and £47. The mean and standard deviation of the basic weekly wages of a random
sample of 39 Bugatty workers were £361 and £53. Test the hypothesis that on average
workers in Deshovy are paid no less than workers in Bugatty. Use a 1% level of
significance.
8.16
A garage offers a ‘clean and green engine makeover’, which it promises will reduce the
amount of CO2 emissions from cars. A random sample of ten cars was tested for CO2
emissions before and after the service. The emission figures (in grams of CO2 per
kilometre travelled) were:
Car
1
2
3
4
5
6
7
8
9
10
Before
174
160
196
214
219
149
292
158
186
200
After
169
158
183
210
204
148
285
155
179
183
Test the hypothesis that after the service emission levels are at least as high as before
against the alternative that emission levels are lower. Use a 1% level of significance.
8.17
Last year 61% of total job applications received by a national recruitment agency were
from female applicants. Out of a random sample of 192 applications received this year,
105 were from females. Test the hypothesis that the proportion of applications from
females has not changed using a 10% level of significance.
8.18
A cinema manager claims that at least one quarter of customers buy popcorn during their
visit to the cinema. If 38 out of a random sample of 176 customers buy popcorn, is the
manager right? Use a 5% level of significance.
8.19
The number of pairs of shoes owned by adult women is assumed to follow a skewed
distribution. A random sample of 10 women was taken and each woman was asked how
many pairs of shoes she owned. Their responses were:
7 24 15 7 18 15 9 13 8 10
Using this data test the hypothesis that the median number of pairs of shoes owned by
adult women is 12. Apply the sign test with a 5% level of significance.
8.20
Select the appropriate definition from
on the left-hand side.
(a)
point estimate
(i)
(b)
z interval
(ii)
(c)
error
(iii)
(d)
confidence
(iv)
(e)
t interval
(v)
(f)
confidence interval
(vi)
(g)
null hypothesis
(vii)
(h)
significant
(viii)
the list on the right-hand side for the terms listed
the likelihood that an estimate is accurate
a result that refutes a null hypothesis
another name for an interval estimate
an interval estimate based on a small sample
a single figure estimate
an interval estimate based on a large sample
what to add and subtract to create an interval
estimate
a claim, to be tested, about a population measure