Economics%20173%20Business%20Statistics

1
Economics 173Business Statistics

• Lecture 6
• Fall, 2001
• Professor J. Petry
• http//www.cba.uiuc.edu/jpetry/Econ_173_fa01/

2
General Announcements

• For efficiency reasons, questions regarding the
  lecture materials will have to wait until after
  class, lab session, office hours or web-board.
• We will no longer require using the Z-table, nor
  any other table at the back of the book.
• The counterintuitive nature of the tables and our
  ability to use excel in its place make this
  choice optimal.
• We will provide the critical value to which you
  can compare a test statistic value, a p-value to
  which you can compare a significance level, or
  some means to find the necessary values without
  using the tables in your book.
• We will do our best to keep lectures as close to
  the schedule as possible. Regardless of the
  lecture pace, our schedule as printed in the
  Course Outline should dictate your reading
  schedule.

3
Inference About the Description of a Single
Population

• Chapter 11

4
11.1 Introduction

• In this chapter we utilize the approach developed
  before for making statistical inference about
  populations.
• Identify the parameter to be estimated or tested
  .
• Specify the parameters estimator and its
  sampling distribution.
• Construct an interval estimator or perform a test.

5

• We will develop techniques to estimate and test
  three population parameters.
• The expected value m
• The variance s2
• The population proportion p (for qualitative
  data)
• Examples
• A bank conducts a survey to estimate the number
  of times customers will actually use ATM
  machines.
• A random sample of processing times is taken to
  test the mean production time and the variance of
  production time on a production line.

6
11.2 Inference About a Population Mean When the
Population Standard Deviation Is Unknown
7
Z
Z
t
t
Z
t
t
Z
t
t
Z
t
t
Z
t
t
t
t
t
s
s
s
s
s
s
s
s
s
s
When the sampled population is normally
distributed, the statistic t is Student t
distributed.
The degrees of freedom, a function of the
sample size determines how spread
the distribution is (compared to the normal
distribution)
The t distribution is mound-shaped, and
symmetrical around zero.
d.f. n2
d.f. n1
n1 lt n2
0
8
Testing the population mean when the population
standard deviation is unknown

• If the population is normally distributed, the
  test statistic for m when s is unknown is t.
• This statistic is Student t distributed with n-1
  degrees of freedom.

9

• Example 11.1 Trainees productivity
• In order to determine the number of workers
  required to meet demand, the productivity of
  newly hired trainees is studied.
• It is believed that trainees can process and
  distribute more than 450 packages per hour within
  one week of hiring.
• Can we conclude that this belief is correct,
  based on productivity observation of 50 trainees?

10

• Solution
• The problem objective is to describe the
  population of the number of packages processed in
  one hour.
• The data are quantitative.
• H0m 450 H1m gt 450
• The t statistic d.f. n - 1 49

11

• Solving by hand
• The rejection region is t gt ta,n - 1
• ta,n - 1 t.05,49 approximately to 1.676.
  (critical value)
• From the data we have

12

• The test statistic is

Rejection region
1.676
1.89

• Since 1.89 gt 1.676 we reject the null hypothesis
  in favor of the alternative.
• There is sufficient evidence to infer that the
  mean productivity of trainees one week after
  being hired is greater than 450 packages at .05
  significance level.

13
1.68 1.89

• Since .0323 lt .05, we reject the null hypothesis
  in favor of the alternative. Or, as in last slide
  1.89 (the test statistic) is more extreme than
  1.68 (the critical value). You would be given the
  critical value in this example.
• There is sufficient evidence to infer that the
  mean productivity of trainees one week after
  being hired is greater than 450 packages at .05
  significance level.

14
Estimating the population mean when the
population standard deviation is unknown

• Confidence interval estimator of m when s is
  unknown

15
11.3 Inference About a Population Variance

• Some times we are interested in making inference
  about the variability of processes.
• Examples
• The consistency of a production process for
  quality control purposes.
• Investors use variance as a measure of risk.
• To draw inference about variability, the
  parameter of interest is s2.

16

• The sample variance s2 is an unbiased, consistent
  and efficient point estimator for s2.
• The statistic has a
  distribution called Chi-squared, if the
  population is normally distributed.

d.f. 1
d.f. 10
d.f. 5
17

• Example 11.3 (operation management application)
• A container-filling machine is believed to fill 1
  liter containers so consistently, that the
  variance of the filling will be less than 1 cc
  (.001 liter).
• To test this belief a random sample of 25 1-liter
  fills was taken, and the results recorded.
• Do these data support the belief that the
  variance is less than 1cc at 5 significance
  level?

18

• Solution
• The problem objective is to describe the
  population of 1-liter fills from a filling
  machine.
• The data are quantitative, and we are interested
  in the variability of the fills.
• The complete test is H0 s2 1
• H1 s2 lt1

We want to prove that the process is consistent
19

• Solving by hand
• Note that (n - 1)s2 S(xi - x)2 Sxi2 - Sxi/n
• From the sample (data is presented in units of
  cc-1000 to avoid rounding) we can calculate Sxi
  -3.6, and Sxi2 21.3.
• Then (n - 1)s2 21.3 - (-3.6)2/25 20.8.
• The complete test is shown next

There is insufficient evidence to reject the
hypothesis that the variance is equal to 1cc, in
favor of the hypothesis that it is smaller.
20
P-value .3485
a .05
Rejection region
13.8484
20.8
Do not reject the null hypothesis
21
11.4 Inference About a Population Proportion

• When the population consists of qualitative or
  categorical data, the only inference we can make
  is about the proportion of occurrence of a
  certain value.
• The parameter p was used before to calculate
  probabilities using the binomial distribution.

22

• Statistic and sampling distribution
• the statistic employed is

23

• Test statistic for p

• Interval estimator for p (1-a confidence level)

24

• Example 11.5 (marketing application)
• For a new newspaper to be financially viable, it
  has to capture at least 12 of the Toronto
  market.
• In a survey conducted among 400 randomly selected
  prospective readers, 58 participants indicated
  they would subscribe to the newspaper if its cost
  did not exceed 20 a month.
• Can the publisher conclude that the proposed
  newspaper will be financially viable at 10
  significance level?

25

• Solution
• The problem objective is to describe the
  population of newspaper readers in Toronto.
• The responses to the survey are qualitative.
• The parameter to be tested is p.
• The hypotheses are
• H0 p .12
• H1 p gt .12

We want to prove that the newspaper is
financially viable
26

• Solving by hand
• The rejection region is z gt za z.10 1.28. (
  critical value)
• The sample proportion is
• The value of the test statistic is
• The p-value is P(Zgt1.54) .0618 alpha 0.10

There is sufficient evidence to reject the null
hypothesis in favor of the alternative
hypothesis. At 10 significance level we can
argue that at least 12 of Torontos readers
will subscribe to the new newspaper.
27
(No Transcript)
28

• Example 11.6 (marketing application)
• In a survey of 2000 TV viewers at 11.40 p.m. on a
  certain night, 226 indicated they watched The
  Tonight Show.
• Estimate the number of TVs tuned to the Tonight
  Show in a typical night, if there are 100 million
  potential television sets. Use a 95 confidence
  level.
• Solution

29
Selecting the Sample Size to Estimate the
Proportion

• The interval estimator for the proportion is
• Thus, if we wish to estimate the proportion to
  within W, we can write
• The required sample size is

30

• Example
• Suppose we want to estimate the proportion of
  customers who prefer our companys brand to
  within .03 with 95 confidence.
• Find the sample size needed to guarantee that
  this requirement is met.
• Solution
• W .03 1 - a .95,
• therefore a/2 .025,
• so z.025 1.96

Since the sample has not yet been taken, the
sample proportion is still unknown.
We proceed using either one of the following two
methods
31

• Method 1
• There is no knowledge about the value of
• Let , which results in the largest
  possible n needed for a 1-a
  confidence interval.
• If the sample proportion does not equal .5, the
  actual W will be narrower than .03.