Lecture 3. Interval estimation

1. Lecture 3

LECTURE 3
INTERVAL ESTIMATION

2. Last Week

LAST WEEK
• We considered the point estimation of an unknown population parameter – that is,
the production of a single number that in some sense is a «good bet».
• For most practical problems, a point estimate alone is inadequate.

3.

• For instance, suppose that a check on a random sample of parts from a large
shipment leads to the estimate that 10% of all the parts are defective.
• Faced with this figure, a manager is likely to ask such questions as :
• «Can I be fairly sure that the true percentage of defectives is between 5% and 15%?» or
• «Does it then seem very likely that between 9% and 11% of all the parts are defective?»
• Questions of this kind seek information beyond that contained in a single point
estimate; they are asking about the reliability of that estimate -> interval estimate:
a range of values in which the quantity to be estimated appears likely to lie.

4. Confidence intervals

CONFIDENCE INTERVALS
• The larger the sample size, ceteris paribus, the shorter will be the interval estimates that
reflect our uncertainty about a parameter’s true value.
• An interval estimator for a population parameter is a rule for determining (based on
sample information) a range, or interval, in which the parameter is likely to fall. The
corresponding estimate is called an interval estimate.

5. Confidence intervals

CONFIDENCE INTERVALS

6.

CONFIDENCE INTERVALS

7. Confidence intervals for the mean of a normal distribution: population variance known

CONFIDENCE INTERVALS FOR THE MEAN OF A NORMAL DISTRIBUTION:
POPULATION VARIANCE KNOWN

8.

Ex: A random sample of 16 observations from a normal population with standard deviation 6
has mean 25. Find a 90% confidence interval for the population mean.

9.

10.

11.

Ex: A process produces bas of refined sugars. The weights of the content of these bags are normally distributed
with standard dev. 1.2 ounces. The content of a random sample of 25 bags had mean weight 19.8 ounces.
Find a 95% confidence interval for the true mean weight for all bags of sugar produced by the progress.

12.

PROPERTIES:
• For a given probability content and sample size, the bigger the
population’s standard deviation, the wider the confidence interval for the
population mean.
• For a given probability content and population standard deviation, the
bigger the sample size the narrower the confidence interval for the
population mean.
• For a given population standard deviation and sample size, the bigger the
probability content, the wider the confidence interval for the population
mean.

13.

14. Confidence intervals for the population mean: large sample sizes

CONFIDENCE INTERVALS FOR THE POPULATION MEAN:
LARGE SAMPLE SIZES

15.

Ex: A random sample of 172 accounting students was asked to rate the importance of particular job
characteristics on a scale from one (not important) to five (extremely important). For «job security», the
sample mean rating was 4.38 and the sample standard deviation was 0.70. Find a 99% confidence interval
for the population mean.

16. THE STUDENT’S T DISTRIBUTION

17.

18. CONFIDENCE INTERVALS FOR THE MEAN OF A NORMAL POPULATION: POPULATION VARIANCE UNKNOWN

19.

Ex: A random sample of six cars from a particular model year had the following fuel consumption figures, in
miles per gallon,
18.6
18.4
19.2
20.8
19.4
20.5
Find a 90% confidence interval for the population mean fuel consumption for cars of this model year assuming
hat the population distribution is normal.

20.

Ex: A retail clothing store is interested in the expenditures on clothes of college students in the first month of
the school year. For a random sample of 9 students, the mean expenditure was $157.82 and the sample
standard deviation was $38.89. Assuming that the population distribution is normal, find a 95% confidence
interval for the population mean expenditure.

21.

Ex: A random sample of 541 consumers was asked to respond on a scale from one (strongly disagree) to five (strongly agree) to
the statement: “A seller should be liable for a defective product even when he was exercised all possible care in its sale and
manufacture” The sample mean response was 3.81 and the sample standard deviation was 1.34.
Find a 90% confidence interval for the population mean response.

22.

Ex: A personnel manager has found that historically, the scores on aptitude tests given to applicants for entry-level positions
follow a normal distribution with standard deviation 32.4 points. A random sample of 9 test scores from the current group of
applicants had mean 187.9 points.
a. Find a 80% confidence interval for the population mean score for the current group of applicants.
b. Based on these sample results, a statistician found for the population mean a confidence interval running from 165.8
to 210.0 points. Find the probability content of this interval.