BCOR 1020 Business Statistics

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BCOR 1020Business Statistics

• Lecture 23 April 15, 2008

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Overview

• Chapter 10 Two Sample Tests
• Comparing Two Means (ss unknown)
• Paired Comparisons

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Chapter 10 Comparing Two Independent Means (ss
unknown)
Format of Hypotheses

• Just as when the standard deviations are known,
  the hypotheses for comparing two independent
  population means m1 and m2 are

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Chapter 10 Comparing Two Independent Means (ss
unknown)
Test Statistic

• If the population variances s12 and s22 are
  known, then we use a standard normal distribution
  (Tuesdays notes).
• If the population variances s12 and s22 are
  unknown, then we use a students t distribution.
• There are three possible cases

Case 1 Known Variances (Last Lecture)
Use the standard normal table to define the
rejection region or calculate the p-value.
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Chapter 10 Comparing Two Independent Means (ss
unknown)
Case 2 Unknown Variances, Assumed Equal

• Since the variances are unknown, they must be
  estimated and the Students t distribution used
  to test the means.
• Assuming the population variances are equal, s12
  and s22 can be used to estimate a common pooled
  variance sp2.

• The test statistic is

With degrees of freedom n n1 n2 2.
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Chapter 10 Comparing Two Independent Means (ss
unknown)
Case 3 Unknown Variances, Assumed Unequal

• If the unknown variances are assumed to be
  unequal, they are not pooled together.

• Use the Welch-Satterthwaite test which replaces
  s12 and s22 with s12 and s22 in the known
  variance Z formula, then uses a Students t test
  with adjusted degrees of freedom.

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Chapter 10 Comparing Two Independent Means (ss
unknown)
Case 3 Unknown Variances, Assumed Unequal

• Welch-Satterthwaite test

• with degrees of freedom

• A Quick Rule for degrees of freedom is to use n
  min(n1 1, n2 1).

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Chapter 10 Comparing Two Independent Means (ss
unknown)
Steps in Testing Two Means

• Choose the level of significance, a.
• Choose the appropriate hypotheses
• Calculate the Test Statistic
• State the decision rule Based on a, determine
  the critical value(s).

• Make the decision Reject H0 if the test
  statistic falls in the rejection region(s) as
  defined by the critical value(s).

For example, for a two-tailed test for Students
t and a .05
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Chapter 10 Comparing Two Independent Means (ss
unknown)
When to Use Each Case Statistic

• If the sample sizes are equal, the Case 2 and
  Case 3 test statistics will be identical,
  although the degrees of freedom may differ.
• If the variances are similar, the two tests will
  usually agree.
• If no information about the population variances
  is available, then the best choice is Case 3.
• The fewer assumptions, the better When in
  doubt, use Case 3!

Must Sample Sizes Be Equal?

• Unequal sample sizes are common and the formulas
  still apply.

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Chapter 10 Comparing Two Independent Means (ss
unknown)

• Example
• A restaurant chain is considering closing one of
  two stores.
• In a sample of 16 randomly selected days,
  restaurant A has average daily sales of 3000
  with a standard deviation of SA 450.
• In a sample of 12 randomly selected days,
  restaurant B has average daily sales of 2700
  with a standard deviation of SB 400.
• All other things being equal, the restaurant with
  lower sales will be closed.
• Assuming that sales for restaurants within the
  same chain will have equivalent standard
  deviations, test the appropriate hypothesis to
  determine whether sales at restaurant B are
  significantly lower than sales at restaurant A at
  the 5 level of significance.

(Overhead)
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Chapter 10 Comparing Two Independent Means (ss
unknown)

• Example (continued)
• Assumptions
• Sales data are independent
• The standard deviations are unknown, but assumed
  equal.
• We will treat this as Case 2
• Hypotheses we want to determine if mA gt mB
• H0 mA lt mB vs. H1 mA gt mB (Right-tail
  test)
• Test Statistic (using pooled variance sp2)

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Chapter 10 Comparing Two Independent Means (ss
unknown)

• Example (continued)
• Test Statistic (using pooled variance sp2)

t distribution with n 16 12 2 26 d.f.
under H0.
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Clickers
What is the rejection criteria for this
problem? (A) Reject H0 in favor of H1 if T gt
1.645. (B) Reject H0 in favor of H1 if T gt
1.706. (C) Reject H0 in favor of H1 if T gt
1.701. (D) Reject H0 in favor of H1 if T gt
2.056. (E) Reject H0 in favor of H1 if T gt
1.315.
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Clickers
What is your decision? (A) Reject H0 in
favor of H1. (B) Fail to Reject H0 in favor of
H1. (C) Not enough information.
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Chapter 10 Comparing Two Independent Means (ss
unknown)

• Example (conclusion)
• Since our test statistic, T, falls in the
  rejection region, we will reject H0 in favor of
  H1.
• Based on the data collected, there is
  statistically significant evidence that mA gt mB.
• So, all other things being equal, we will close
  restaurant B.

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Chapter 10 Comparing Two Independent Means (ss
unknown)

• Example (revisited)
• How does our test change if we dont assume equal
  variances (Case 3)?
• We will use the same hypothesis test
• H0 mA lt mB vs. H1 mA gt mB (Right-tail
  test)
• With a different test statistic

T distribution with n min(nA 1,nB 1)
min(16 1, 12 1) 11 d.f. under H0.
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Chapter 10 Comparing Two Independent Means (ss
unknown)

• Example (revisited)
• Rejection Criteria
• For the right-tail test, we will reject H0 in
  favor of H1 if T gt ta,n.
• Decision Since T 1.861 gt ta,n t.05,11
  1.796, we will reject H0 in favor of H1.
• Just as before, based on the data collected,
  there is statistically significant evidence that
  mA gt mB.
• Our exact p-value will be a little larger in this
  case since this test makes fewer assumption and
  is therefore more conservative.

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Chapter 10 Paired Comparisons (Dependent
Samples)
Paired Data

• Data occurs in matched pairs when the same item
  is observed twice but under different
  circumstances.
• For example, blood pressure is taken before and
  after a treatment is given.
• Paired data are typically displayed in columns

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Chapter 10 Paired Comparisons (Dependent
Samples)
Paired t Test

• Paired data typically come from a before/after
  experiment on n subjects so the n observations
  of the two variable are dependent.
• In the paired t test, the difference between x1
  and x2 is measured as d x1 x2.

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Chapter 10 Paired Comparisons (Dependent
Samples)
Paired t Test

• The calculations for the mean and standard
  deviation are

• Since the population variance of d is unknown,
  use the Students t with n 1 degrees of freedom.

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Chapter 10 Paired Comparisons (Dependent
Samples)

• Selection of H0 and H1
• Remember, the conclusion we wish to test should
  be stated in the alternative hypothesis.
• Based on the problem statement, we choose from

• H0 md gt 0
• H1 md lt 0

(ii) H0 md lt 0 H1 md gt 0
(iii) H0 md 0 H1 md 0

• If the null hypothesis is true and md 0, then
  T has the students t distribution with n 1
  d.f.
• Decision Criteria (the same as any other t-test)
• We can either compare T to a critical value of
  the appropriate t distribution or calculate
  (bound) the p-value for the test.

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Chapter 10 Paired Comparisons (Dependent
Samples)

• Example
• A new cell phone battery is being considered as a
  replacement for the current one. Six college
  students were selected to try each battery in
  their usual mix of talk and standby and to
  record the number of hours until recharge was
  needed. The data is below. Using a level of
  significance of a 5, do these results show
  that the newer battery has significantly longer
  life?

(Overhead)
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Chapter 10 Paired Comparisons (Dependent
Samples)

• Example (continued)
• We can calculated the sample mean and standard
  deviation for the dis

and

• Based on the problem statement, we will test the
  hypothesis H0 md lt 0 vs. H1 md gt 0 (which
  corresponds to the battery life for the new
  battery being greater).

• The test statistic is

t distribution with n n 1 5 d.f. under H0.
or
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Chapter 10 Paired Comparisons (Dependent
Samples)

• Example (continued)
• Rejection Criteria
• For the right-tail test, we will reject H0 in
  favor of H1 if T gt ta,n.
• Decision Since T 2.86 gt ta,n t.05,5
  2.015, we will reject H0 in favor of H1.
• Based on the data collected, there is
  statistically significant evidence that the new
  battery lasts longer.

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Clickers
For this paired t test, our test statistic T
2.86 has a students t distribution with n 5
degrees of freedom. Use the t-table to find
appropriate bounds on the p-value of this
test. (A) 0.01 lt p-value lt 0.02 (B) 0.02 lt
p-value lt 0.025 (C) 0.025 lt p-value lt
0.05 (D) 0.05 lt p-value lt 0.10