Economics 173 Business Statistics

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Economics 173Business Statistics

• Lecture 8
• Fall, 2001
• Professor J. Petry
• http//www.cba.uiuc.edu/jpetry/Econ_173_fa01/

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Chapter 12
Inference about the Comparison ofTwo Populations
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12.1 Introduction

• Variety of techniques are presented whose
  objective is to compare two populations.
• We are interested in
• The difference between two means.
• The ratio of two variances.
• The difference between two proportions.

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12.2 Inference about the Difference b/n Two
Means Independent Samples

• Two random samples are drawn from the two
  populations of interest.
• Because we are interested in the difference
  between the two means, we build the statistic
  for each sample.

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The Sampling Distribution of

• is normally distributed if the
  (original) population distributions are normal .
• is approximately normally
  distributed if the (original) population is not
  normal, but the sample size is large.
• Expected value of is m1 - m2
• The variance of is s12/n1
  s22/n2

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• If the sampling distribution of is
  normal or approximately normal we can write
• Z can be used to build a test statistic or a
  confidence interval for m1 - m2

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• Practically, the Z statistic is hardly used,
  because the population variances are not known.

t
?
?
S22
S12

• Instead, we construct a t statistic using the
  
• sample variances (S12 and S22).

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• Two cases are considered when producing the
  t-statistic.
• The two unknown population variances are equal.
• The two unknown population variances are not
  equal.

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Case I The two variances are equal

• Calculate the pooled variance estimate by

n2 15
n1 10
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• Construct the t-statistic as follows

• Perform a hypothesis test
• H0 m1 - m2 0
• H1 m1 - m2 gt 0

or lt 0
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Case II The two variances are unequal
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Run a hypothesis test as needed, or, build an
interval estimate
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• Example 12.1
• Do people who eat high-fiber cereal for breakfast
  consume, on average, fewer calories for lunch
  than people who do not eat high-fiber cereal for
  breakfast?
• A sample of 150 people was randomly drawn. Each
  person was identified as a consumer or a
  non-consumer of high-fiber cereal.
• For each person the number of calories consumed
  at lunch was recorded.

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Calories consumed at lunch

• Solution
• The data are quantitative.
• The parameter to be tested is
• the difference between two means.
• The claim to be tested is that
• mean caloric intake of consumers (m1)
• is less than that of non-consumers (m2).

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• Identifying the technique
• The hypotheses are
• H0 (m1 - m2) 0
• H1 (m1 - m2) lt 0
• To check the relationships between the
  variances, we use a computer output to find
  the samples standard deviations. We have S1
  64.05, and S2 103.29. It appears that the
  variances are unequal.
• We run the t - test for unequal variances.

(m1 lt m2)
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Calories consumed at lunch

• At 5 significance level there is
• sufficient evidence to reject the null
• hypothesis.

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• Solving by hand
• The interval estimator for the difference between
  two means is

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• Example 12.2
• Do job design (referring to worker movements)
  affect workers productivity?
• Two job designs are being considered for the
  production of a new computer desk.
• Two samples are randomly and independently
  selected
• A sample of 25 workers assembled a desk using
  design A.
• A sample of 25 workers assembled the desk using
  design B.
• The assembly times were recorded
• Do the assembly times of the two designs differs?

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Assembly times in Minutes

• Solution
• The data are quantitative.
• The parameter of interest is the difference
• between two population means.
• The claim to be tested is whether a difference
• between the two designs exists.

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The Excel printout
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A 95 confidence interval for m1 - m2 is
calculated as follows
Thus, at 95 confidence level -0.3176 lt m1 - m2 lt
0.8616 Notice Zero is included in the
interval
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Checking the required Conditions for the equal
variances case (example 12.2)
The distributions are not bell shaped, but
they seem to be approximately normal. Since the
technique is robust, we can be confident about
the results.
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Example

• 12.20 from book
• Random samples were drawn from each of two
  populations. The data are stored in columns 1 and
  2, respectively, in file XR12-20.
• Is there sufficient evidence at the 5
  significance level to infer that the mean of
  population 1 is greater than the mean of
  population 2?

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Example 12.23

• The President of Tastee Inc., a baby-food
  producer, claims that his companys product is
  superior to that of his leading competitor,
  because babies gain weight faster with his
  product. To test this claim, a survey was
  undertaken. Mothers of newborn babies were asked
  which baby food they intended to feed their
  babies. Those who responded Tastee or the leading
  competitor were asked to keep track of their
  babies weight gains over the next two months.
  There were 15 mothers who indicated that they
  would feed their babies Tasteee and 25 who
  responded that they would feed their babies the
  product of the leading competitor. Each babys
  weight gain in ounces is recorded in XR12-23.
• Can we conclude that, using weight gain as our
  criterion, Tastee baby food is indeed superior?
• Estimate with 95 confidence the difference
  between the mean weight of the two products.
• Check to ensure the required conditions are
  satisfied.

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