Inference About a Population

1
Inference About a Population

• Chapter 12

2
Introduction

• In this chapter we utilize the approach developed
  before to describe a population.
• Identify the parameter to be estimated or tested.
• Specify the parameters estimator and its
  sampling distribution.
• Construct a confidence interval estimator or
  perform a hypothesis test.

3
Introduction

• We shall develop techniques to estimate and test
  three population parameters.
• The expected value m
• The variance s2
• The population proportion p (for qualitative data)

4
12.1 Inference About a Population Mean When the
Population Standard Deviation is Unknown

• Recall By the central limit theorem, when s2
  is known is normally distributed if
• the sample is drawn from a normal population, or
  
• the population is not normal but the sample is
  sufficiently large.
• When s2 is unknown, another random variable
  describes the distribution of

5
The t - Statistic
When s2 is unknown, we use s2 instead, and the Z
statistic is then replaced by the t-statistic
t
Z
s
When the sampled population is normally
distributed, the statistic t is Student t
distributed. See next.
6
The t - Statistic
Using the t-table
t
s
The Student- t distribution is mound-shaped, and
symmetrical around zero.
The degrees of freedom determine the
distribution shape
n1 lt n2
0
7
Testing m when s is unknown

• Example 12.1 - Productivity of newly hired
  Trainees

8
Testing m when s is unknown

• Example 12.1
• 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
  (raw data is presented later in the file
  Xm12-01).

9
Testing m when s is unknown

• Example 12.1 Solution
• The problem objective is to describe the
  population of the number of packages processed in
  one hour.
• The data is quantitative.
• H0m 450 H1m gt 450
• The t statistic d.f. n - 1 49

We want to prove that the trainees reach 90
productivity of experienced workers
10
Testing m when s is unknown

• Solution - continued
• Observe H1 has the form of m gt m0, thus
• The rejection region is

After transforming into a t-statistic we
express the rejection region in terms of the
statistic t.
t ³ ta,n-1
11
Testing m when s is unknown

• Solution continued (solving by hand)The
  rejection region is t gt ta,n 1.ta,n - 1
  t.05,49

12
Testing m when s is unknown
Rejection region
1.676
1.89

• Since 1.89 gt 1.676 we reject the null hypothesis
  in favor of the alternative.
• Conclusion 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
Testing m when s is unknown
Xm12-01.xls
Using Data Analysis Plus and the p-value
approachto test the mean.
t-Test Mean t-Test Mean

Packages
Mean 460.38
Standard Deviation Standard Deviation 38.8271
Hypothesized Mean Hypothesized Mean 450
df 49
t Stat 1.8904
P(Tltt) one-tail P(Tltt) one-tail 0.0323
t Critical one-tail t Critical one-tail 1.6766
1.89
Since .02323 lt .05, 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.
14
Estimating m when s is unknown

• Confidence interval estimator of m when s2 is
  unknown

15
Estimating m when s is unknown

• Example 12.2
• An investor is trying to estimate the return on
  investment in companies that won quality awards
  last year.
• A random sample of 83 such companies is selected,
  and the return on investment is calculated had he
  invested in them.
• Construct a 95 confidence interval for the mean
  return.

16
Estimating m when s is unknown

• Solution (solving by hand)
• The problem objective is to describe the
  population of annual returns from buying shares
  of quality award-winners.
• The data is quantitative.
• Solving by hand
• From the data we determine

t.025,82_at_ t.025,80
17
Estimating m when s is unknown
Using Data Analysis Plus
t - estimate Mean Returns Mean 15.0172 Stand
ard Deviation 8.3054 LCL 13.0237 UCL 16.830
7
18
Checking the required conditions

• We need to check that the population is normally
  distributed, or at least not extremely
  non-normal.
• There are statistical methods that can be used to
  test for normality (to be introduced later in the
  book, but not discussed here).
• From the sample histograms we see

19
A Histogram for XM-11- 01
Packages
A Histogram for XM-11- 02
Returns
20
12.2 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.
• To evaluate the risk associated with different
  investments.
• To draw inference about variability, the
  parameter of interest is s2.

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Inference About a Population Variance

• The population variance can be estimated or its
  value tested using the sample variance s2.
• The sample variance s2 is an unbiased, consistent
  and efficient point estimator for s2.
• The inference about s2 is made by using a sample
  statistic that incorporates s2 and s2.

22
Inference About a Population Variance

• This statistic is .
• It has a distribution called Chi-squared, if the
  population is normally distributed.

23
Inference About a Population Variance
The degfrees of freedom (df)determines the
distribution shape
24
Testing the population variance Left hand tail
test

• Example 1 (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
  (Xm12-03.xls)
• Do these data support the belief that the
  variance is less than 1cc at 5 significance
  level?

25
Testing the population variance

• 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 two hypotheses are H0 s2 1
• H1 s2 lt1

We want to prove that the process is consistent
26
Testing the population variance

• Solution
• The two hypotheses are H0 s2 1
• H1 s2 lt1

27
Testing the population variance
28
Testing the population variance
2
-
Using the c2 table
78
.
20
s
)
1
n
(
2



c
,
78
.
20
2
2
s
1

c

c
2
2
.
8484
.
13
-
-
a
-
1
25
,
95
.
1
n
,
1
13.84
20.78
Since 20.78gt13.8484 do not rejectthe null
hypothesis
There is insufficient evidence to reject the
hypothesis that the variance is equal to 1cc.
29
Testing the population variance Right hand
tail test Two tail test

• A right hand tail test
• H0 s2 valueH1 s2 gt value
• Rejection region

Click
30
Testing the population variance Right hand
tail test Two tail test

• A right hand tail test
• H0 s2 valueH1 s2 gt value
• Rejection region

• A two tail test
• H0 s2 valueH1 s2 ¹ value
• Rejection region

31
Estimating the population variance

• From the following probability
  statement P(c21-a/2 lt c2 lt c2a/2) 1-awe
  have (by substituting c2 (n - 1)s2/s2.)

This is the confidence interval for s2 with 1-a
confidence level.
32
Estimating the population variance

• Example 2
• Estimate the variance of fills in example 12.3
  with 99 confidence.
• Solution
• We have (n-1)s2 20.78.From the Chi-squared
  table we havec2a/2,n-1 c2.005, 24 45.5585
  c21-a/2,n-1 c2.0995, 24 9.88623

33
Estimating the population variance

• The confidence interval is

34
12.4 Inference About a Population Proportion

• When the population consists of nominal 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
  these proportion under the binomial distribution.

35
12.4 Inference About a Population Proportion

• Statistic and sampling distribution
• the statistic used when making inference about
  p is

36
Testing and estimating the Proportion

• Test statistic for p

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

37
Testing the Proportion

• Example 12.5 (Predicting the winner in election
  day)
• Voters are asked by a certain network to
  participate in an exit poll in order to predict
  the winner on election day.
• Based on the data presented in Xm12.5.xls (where
  1Democrat, and 2Republican), can the network
  conclude that the republican candidate will win
  the state college vote?

38
Testing the Proportion

• Solution
• The problem objective is to describe the
  population of votes in the state.
• The parameter to be tested is p.
• Success is defined as Republican vote.
• The hypotheses are
• H0 p .5
• H1 p gt .5

More than 50 vote republican
39
Testing the Proportion

• Solving by hand
• The rejection region is z gt za z.05 1.645.
• From file Xm12.5.xls we count 407 success. Number
  of voters participating is 765.
• The sample proportion is
• The value of the test statistic is
• The p-value is P(Zgt1.77) .0382

40
Testing the Proportion
Using Data Analysis Plus we have
lt 0.05
There is sufficient evidence to reject the null
hypothesis in favor of the alternative
hypothesis. At 5 significance level we can
conclude that more than 50 voted Republican.
41
Estimating the Proportion

• Example (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 95 confidence
  level.
• Solution

42
Estimating the Proportion

• Solution

43
Selecting the Sample Size to Estimate the
Proportion

• Recall The confidence interval for the
  proportion is
• Thus, to estimate the proportion to within W, we
  can write

44
Selecting the Sample Size to Estimate the
Proportion

• The required sample size is

45
Sample Size to Estimate the Proportion

• 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
46
Sample Size to Estimate the Proportion

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