Download Confidence Intervals I

Confidence Intervals I
WEEK SIX
This worksheet relates to chapter seven of the text
book (Statistics for Managers 4th Edition).
Please make sure you understand all of the
questions we go through. Do not just copy
down the answers and say that you will
work it out later… it often doesn’t happen!
DISCUSSION QUESTIONS
1.
Why are interval estimates needed in addition to point estimates?
2.
What are the differences between the t distribution and the Z
distribution? When do we use them?
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CALCULATION QUESTIONS
1.
On the west bank of Cabbage Tree Creek live a very rare, very endangered
species known as the woongoolamaloos. The harsh rains of December
attracted fierce meerkats who ate many of the woongoolamaloos. So as to
escape the wrath of the angry meerkats who suffered indigestion after their
woongoolamaloo feast, the woongoolamaloos decide to move to a new shrub
on the east bank of the river.
Historical data suggests that the time it takes this species to cross the creek
can be modeled as a normal distribution with a variance of 25 minutes. 35
woongoolamaloos were studied and the average time to travel across the creek
was found to be 55 minutes.
Find a 90% confidence interval for the true mean time it takes the
woongoolamaloo population to cross the river and find safety and security
under the new shrub. Interpret your confidence interval.
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2.
Approximately 200 people were abducted by aliens from Noosa beach
on the eve of the 2006 New Year. Tragically, their brains were tampered
with causing them to go progressively insane. Once the aliens had
completed their tests, the 200 people were brought back to earth. A
random sample of 90 of the 200 were analysed to work out how many
days it took for the aliens’ surgery to take effect and cause the abductees
to truly believe they were chickens (or other such farm animals). The
analysis showed a mean of 10 days with a standard deviation of 2.1 days.
Find a 95% confidence interval for the mean time period (in days) it
took for the alien’s brain tampering to take effect. Interpret.
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3.
A survey of 2000 men in Queen Street Mall showed that 47% preferred
blondes while 53% preferred brunettes (as for red heads… well they just
couldn’t gain any support). Find a 92% confidence interval for the
proportion of men who prefer blondes. Interpret the confidence
interval.
MULTIPLE CHOICE PRACTICE
1.
A confidence interval shows
(a)
(b)
(c)
(d)
a point estimate of the population mean
an interval estimate of where the sample mean will lie
the range which contains the population mean
an estimate of the range in which the population mean will lie
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2.
The precision of confidence intervals is related to
(a)
(b)
(c)
(d)
3.
$2567
$383
$6589
$2496
Final, June 2005
A market research company conducted a telephone survey of 765
households on behalf of a chain of new car dealers to determine the
proportion of households seeking to purchase a new car in the next two
years. An interval estimate of 0.076 and 0.112 was found. Determine the
level of confidence that can be attached to this interval.
(a)
(b)
(c)
(d)
5.
Final, June 2005
A statistical report states that the 95% confidence interval for the mean
salary of new recruits in a company is between $25 500 and $27 000
obtained from a sample of 45. Calculate the population standard
deviation.
(a)
(b)
(c)
(d)
4.
the sample size and the confidence level
the confidence level only
the mean of the population and the confidence level
the mean of the sample and the sample size
8.7%
95.6%
91.3%
insufficient information to determine
Final, June 2004
When comparing the t distribution (eg. For n = 20) with the Z
distribution it is TRUE to say
(a)
(b)
(c)
(d)
they are both normal curves
the variances for both distributions are the same
they have the same mean but the t distribution has more
probability in the tails
the t distribution has a smaller standard deviation than Z
Final, June 2003
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6.
A random sample of customers buying petrol was selected. From this
sample, the 95% confidence interval estimate for the mean amount of
petrol purchased per customer for the city was calculated to be between
55 and 78 litres. This means that:
(a)
(b)
(c)
(d)
95% of all customers bought petrol in the given range.
There is a 95% probability that the sample mean lies in this range.
If another sample was taken there is a 95% chance that the
confidence interval for the mean would be in the given range.
There is a 95% probability that the mean amount of petrol
purchased per customer for the city lies in the given range.
Final, June 1998
7.
Consider the following statements concerning confidence intervals:
A.
A disadvantage of point estimates, when compared to interval
estimates, is that the increased precision achieved through taking
a larger sample is not reflected in these estimates
B.
For a given sample size and population standard deviation, an
increase in the confidence level will make the interval narrower.
C.
The only requirement for being able to use the standard normal
variable to obtain the critical value for a confidence interval
estimate for the population proportion is that the sample size be
large
(a)
(b)
(c)
(d)
(e)
only A is true
only C is true
only A and B are true
only B and C are true
none of A, B and C are true
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8.
Consider the following of the Student’s t distribution:
A.
the variance of the t distribution decreases as the degrees of
freedom decrease.
B.
The t statistic follows the t distribution, with the degrees of
freedom being the sample size less one, provided that the random
variable which is used to construct the statistic follows the normal
distribution
C.
In practice, the t distribution can be used to determine the critical
value for a confidence interval estimate for the population mean
when the population variance is unknown, provided the sample
size is sufficiently large and the distribution of the underlying
population is not highly skewed
(a)
(b)
(c)
(d)
(e)
only B is true
only C is true
only A and B are true
only B and C are true
A, B and C are true
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notes
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