Business Statistics for Managerial Decision

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Business Statistics for Managerial Decision

• Comparing two Population Means

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Comparing Two means

• How do small businesses that fail differ from
  those that succeed?
• Business school researchers compare two samples
  of firms started in 2000, one sample of failed
  businesses and one of firms that are still going
  after two years.
• This study compares two random samples, one from
  each of two different populations.

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Two-Sample problems

• The goal of inference is to compare the responses
  in two groups
• Each group is considered to be a sample from a
  distinct population.
• The responses in each group are independent of
  those in other group

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Two-Sample problems

• Notation
• We have two independent samples, from two
  distinct populations (such as failed businesses
  and successful businesses).
• We measure the same variable (such as initial
  capital) in both samples
• We call the variable x1 in the first population
  and x2 in the second population.
• Population Variable Mean Standard deviation
• 1 x1 ??1 ?1
• 2 x2 ?2
  ?2

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Two-Sample problems

• We want to compare the two population means,
  either by giving a confidence interval for ?1-?2
  or by testing the hypothesis of no difference,
  H0?1?2.
• We base inference on two independent SRSs, one
  from each population.
• Sample Sample
• Population Sample size mean standard
  deviation
• 1 n1 s1
• 2 n2 s2

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The Two-Sample z Statistic

• The natural estimator of the difference ?1-?2 is
  the difference between the sample means
  .
• To base inference on this statistic we need to
  know its sampling distribution.
• The mean of the difference is the
  difference of the means ?1-?2.
• Because the samples are independent, their sample
  means and are independent.
• The variance of the is the sum of
  their variances which is

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The Two-Sample z Statistic

• Suppose that is the mean of a SRS of size n1
  drawn from a N(?1, ?1) population and that
  is the mean of an independent SRS of size n2
  drawn from a N(?2, ?2) population. Then the
  two-sample z statistic
• has the standard Normal (0, 1) sampling
  distribution.

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The Two-Sample t Procedures

• In practice, the two population standard
  deviations ?1 and ?2 are not known
• We estimate them by sample standard deviations s1
  and s2 from our two samples.
• The two-sample t statistic
• This statistic does not have a t distribution.
• We can approximate the distribution of the
  two-sample t statistic by using the t(k)
  distribution with an approximation for the
  degrees of freedom k.

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The Two-Sample t Procedures

• We use the approximation to find approximate
  value of t for confidence intervals and to find
  approximate P-values for significance tests.
• This can be done in two ways
• Scatterwait approximation to calculate a value of
  k from data. In general, the resulting k will not
  be a whole number.
• Use degrees of freedom k equal to the smaller of
  n1-1 and n2-1.

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The Two-Sample t Significance Test

• Draw a SRS of size n1 from a Normal population
  with unknown mean ?1 and an independent SRS of
  size n2 from another Normal population with
  unknown ?2. To test the hypothesis H0 ?1-?2 0,
  compute the two-sample t statistic
• And use P-values or critical values for the t(k)
  distribution, where the degree of freedom k are
  the smaller of n1-1 and n2-1.

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Example Is our product effective?

• A company that sells educational materials
  reports statistical studies to convince customers
  that its materials improve learning. One new
  product supplies directed reading activities
  for class room use. These activities should
  improve the reading ability of elementary school
  pupils.

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Example Is our product effective?

• A consultant arranges for a third-grad class of
  21 students to take part in these activities for
  an eight-week period. A control classroom of 23
  third-graders follows the same curriculum without
  the activities. At the end of the eight weeks,
  all students are given a Degree of Reading Power
  (DRP) test, which measures the aspects of reading
  ability that the treatment is designed to
  improve. The data appear in table 7.3.

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Example Is our product effective?
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Example Is our product effective?

• A back to back stemplot suggests that there is a
  mild outlier in the control group but no
  deviation from Normality serious enough to forbid
  use of t procedure.

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Example Is our product effective?

• The summary statistics are
• Group n s
• Treatment 21 51.48 11.01
• Control 23 41.52 17.15
• We hope to show that the treatment (group 1) is
  better than the control (group 2), therefore the
  hypotheses are
• H0 ?1 ?2
• Ha ?1 gt ?2

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Example Is our product effective?

• The two-sample t statistic is

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Example Is our product effective?

• The P-value for the one-sided test is
• The degrees of freedom k are equal to the smaller
  of n1-1 21-120 and n2-123-122 comparing t
  2.31 with entries in t-table for 20 degrees of
  freedom, we see that P lies between .02 and .01.
• Conclusion
• The data strongly suggest that directed reading
  activity improves the DRP score.

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The Two Sample t Confidence Interval

• The same ideas that we used for the two-sample t
  significance test can apply to give us two-sample
  t confidence interval.
• Draw a SRS of size n1 from a Normal population
  with unknown mean ?1 and an independent SRS of
  size n2 from another Normal population with
  unknown mean ?2. The confidence interval for ?1-
  ?2 given by
• t is the value for t(k) density curve with area
  C between t and t. The value of the degrees
  of freedom k is approximated by software or we
  use the smaller of n1-1 and n2-1.

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ExampleHow much improvement?

• We will find a 95 confidence interval for the
  mean improvement in the entire population of
  third-graders. The interval is
• Using t(20) distribution, t-table gives t
  2.086
• We estimate the mean improvement in DRP scores to
  be about 10 point, but with a margin of error of
  almost 9 points.

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The Pooled Two-sample t Procedures

• There is one situation in which a t statistic for
  comparing two means has exactly a t distribution.
• Suppose that the two Normal population
  distribution have the same standard deviation.
• Call the common standard deviations ?. Both
  sample variances s12 and S22 estimate ?2.
• The best way to combine these two estimates is to
  average them with weights equal to their degrees
  of freedom.
• The resulting estimate of ?2 is

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The Pooled Two-sample t Procedures

• Sp2 is called the pooled estimator of ?2.
• When both populations have variance ?2, the
  addition rule for variance says that
  has variance equal to the sum of the individual
  variances
• Now we can substitute sp2 in the test statistic,
  and the resulting t statistic has a t
  distribution.

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The Pooled Two-sample t Procedures

• Draw a SRS of size n1 from a Normal population
  with unknown mean ?1 and an independent SRS of
  size n2 from another Normal population with
  unknown mean ?2. Suppose that the two populations
  have the same unknown standard deviation. A level
  C confidence interval for ?1- ?2 is
• Here t is the value for the t(n1n2 -2) density
  curve with area C between -t and t.

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The Pooled Two-sample t Procedures

• To test the hypothesis H0 ?1?2, compute the
  pooled two-sample t statistic
• and use P-values from the t(n1 n2 - 2)
  distribution.

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Healthy Companies versus Failed Companies

• In what ways are companies that fail different
  from those that continue to do business?
• To answer this question, one study compared
  various characteristics of 68 healthy and 33
  failed firms.
• One of the variables was the ratio of current
  assets to current liabilities.
• The data appear in table 7.4.

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Healthy Companies versus Failed Companies
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Healthy Companies versus Failed Companies

• First lets Look at the data.
• Histograms for the two groups of firms
  superimposed with a Normal curve with mean and
  standard deviation equal to the sample values is
  given.
• The distribution for the healthy firms looks more
  Normal than the distribution for the failed firms.

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Healthy Companies versus Failed Companies

• The back to back stemplot confirms our findings
  from the previous plots that there are no
  outliers or strong departure from Normality that
  prevent us from using the t procedure for these
  data.

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Example Do mean asset/liability ratio differ?

• Take group 1 to be the firms that were healthy
  and group 2 to be those that failed. The question
  of interest is whether or not the mean ratio of
  current assets to current liabilities is
  different for the two groups. We therefore test
• H0?1 ?2
• Ha?1 ? ?2

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Example Do mean asset/liability ratio differ?

• Here are the summary statistics
• Group Firm n s
• 1 Health 68 1.7256 .6393
• 2 Failed 33 0.8236 .4811
• The sample standard deviations are fairly
  close.We are willing to assume equal population
  standard deviations. The pooled sample variance
  is
• and

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Example Do mean asset/liability ratio differ?

• The pooled two-sample t statistic is
• The P-value is
• Where t has t(99) distribution. In t-table we
  have entries for100 degrees of freedom. We will
  use the entries for 100. Our calculated value of
  t is larger than the t-value corresponding to p
  .0005 entry in the table. Doubling 0.0005 , we
  conclude that the two sided P-value is less than
  .001.

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Example How different are mean Asset/liability
ratios?

• P-value is rarely a complete summary of a
  statistical analysis. To make a judgment
  regarding the size of the difference between the
  two groups of firms, we need a confidence
  interval.
• The difference in mean current assets to current
  liabilities ratios for healthy versus failed
  firms is

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Example How different are mean Asset/liability
ratios?

• For a 95 margin of error we will use the
  critical value t 1.984 from the t(100)
  distribution. The margin of error is
• This will gives the following 95 confidence
  interval

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Example How different are mean Asset/liability
ratios?

• We report that the successful firms have current
  assets to current liabilities ratio that average
  1.15 higher than failed firms, with margin of
  error 0.25 for 95 confidence.
• Alternatively, we are 95 confident that the
  difference is between 0.90 and 1.40.