The document discusses various methods for constructing interval estimates from sample data to describe the range of values within which the unknown population parameter is likely to fall. It provides examples of how to calculate confidence intervals for means, proportions, and percentages using information about the sample size, mean, standard deviation, and desired confidence level. Formulas for the standard error and t-distribution are also presented when the population standard deviation is unknown.

1. Estimation-2

2. Interval estimates
• An interval estimate describes a range of
values within which the population parameter
is likely to be

3. • A marketing research director needs an
estimate of the average life in months of the
car batteries his company manufactures. After
sampling 200 users we arrive at a sample
mean of 36 months.
What is the uncertainty that is likely to be
associated with this estimate? The population
s.d. is 10 months

4. • For a population with a known variance of
185, a sample of 64 individuals leads to 217 as
an estimate of the mean.
Establish the interval estimate that shauld
include the population mean 68.3% of the
time

5. • The probability that we associate with an
interval estimate is called confidence level
• The confidence interval is the range of the
estimate we are making (upper limit & lower
limit)
• A higher confidence level will produce a larger
confidence interval

6. • A social service agency is interested in
estimating the mean annual income of 700
families living in a section of the community.
We take a sample of size 50 and find that the
sample mean is $11800. The sample standard
deviation s=950
Calculate an interval estimate of the mean
income so that we are 90% confident that the
population mean falls within that interval

8. • In an automotive safety test conducted by the North
Carolina Highway Safety Research center, the average
tire pressure in a sample of 62 tires was found to be 24
psi with a standard deviation of 2.1 psi.
i) What is the estimated population s.d. for this
population? There are about a million cars registered
in NC state
ii) Calculate the estimated standard error of the mean
iii)Construct a 95% confidence interval for the population
mean

9. Interval estimates of proportion from
large samples
• The mean of the sampling distribution of the
proportion = p where p is the sample
proportion in favour
• Standard error of the proportion

10. • When a sample of 70 retail executives were
surveyed regarding the poor performance of the
retail industry, 66% believed that decreased sales
was due to unseasonably warm temperatures,
resulting in consumers delaying purchases.
i) Estimate the standard error of the proportion of
retail executives who blame warm weather for
poor sales
ii) Find the upper and lower confidence limits for
this proportion given a 95% confidence level.

11. The t-distribution
• Used when the sample size is less than 30
• Population standard deviation is not known
• We assume that the population is approx. normal
• Like the normal distribution, the t-distribution is
also symmetrical
• There is a different t-distribution for every
possible sample size

12. • The t- tables focus of the probability that the
population parameter being estimated falls
outside the confidence interval
• We must specify the d.o.f. with which we are
dealing

13. • Seven home makers were sampled and it was
determined that the distances they walked in
their housework had an average of 39.2 miles
per week and a sample standard deviation of
3.2 miles per week.
• Construct a 95% confidence interval for the
population mean

14. Determining sample size
• A university is performing a survey of the
annual earnings of last years graduates from
its business school. It knows from past
experience that the standard deviation of the
earnings of the entire population of studentsi
s $1500. How large a sample size should the
university take in order to estimate the mean
annual earnings of last years’ class within
$500 and a 95% confidence level?

15. • We want to determine what proportion of
students at a university are in favour of a new
grading system. We would like a sample size
that will enable us to be 90% certain of
estimating the true proportion of the
population of 40000 students that is in favour
of the new system within +- 0.02