Statistical Inference About Means and Proportions With Two Populations

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Statistical Inference About Means and Proportions
With Two Populations

• Estimation of the Difference between the Means of
  Two Populations Independent Samples
• Hypothesis Tests about the Difference between the
  Means of Two Populations Independent Samples
• Inference about the Difference between the Means
  of Two Populations Matched Samples (related
  populations)
• Inference about the Difference between the
  Proportions of Two Populations

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Independent Related Populations

• Independent
• Different Data
• Sources
• Unrelated
• Independent
• 2. Use Difference
• Between the 2
• Sample Means

Related 1. Same Data Source Paired or
Matched Repeated Measures(Before/After) 2.
Use Difference Between Each
Pair of Observations Dn X1n - X2n
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Estimation of the Difference Between the Means of
Two Populations Independent Samples

• Point Estimator of the Difference between the
  Means of Two Populations
• Sampling Distribution
• Interval Estimate of ????????Large-Sample Case
• Interval Estimate of ????????Small-Sample Case

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Point Estimator of the Difference Betweenthe
Means of Two Populations

• Let ?1 equal the mean of population 1 and ?2
  equal the mean of population 2.
• The difference between the two population means
  is
• ?1 - ?2.
• To estimate ?1 - ?2, we will select a simple
  random sample of size n1 from population 1 and a
  simple random sample of size n2 from population
  2.
• Let equal the mean of sample 1 and equal
  the mean of sample 2.
• The point estimator of the difference between the
  means of the populations 1 and 2 is .

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Sampling Distribution of

• The sampling distribution of has the
  following properties.
• Expected Value
• Standard Deviation
• where ?1 standard deviation of population 1
• ?2 standard deviation of population 2
• n1 sample size from population
• n2 sample size from population 2

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Hypothesis Tests About the DifferenceBetween the
Means of Two Populations Independent Samples

• Hypothesis Forms
• H0 ?1 - ?2 lt 0 H0 ?1 - ?2 gt 0
  H0 ?1 - ?2 0
• Ha ?1 - ?2 gt 0 Ha ?1 - ?2 lt 0
  Ha ?1 - ?2 ? 0
• Test Statistic
• Large-Sample Case
• Small-Sample Case

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Example Par, Inc.

• Par, Inc. is a manufacturer of golf equipment.
  Par
• has developed a new golf ball that has been
  designed to
• provide extra distance. In a test of driving
  distance
• using a mechanical driving device, a sample of
  Par golf
• balls was compared with a sample of golf balls
  made by
• Rap, Ltd., a competitor. The sample data is
  below.
• Sample 1 Sample 2
• Par, Inc. Rap, Ltd.
• Sample Size n1 120 balls n2 80 balls
• Mean 235 yards 218
  yards
• Standard Deviation s1 15 yards s2 20
  yards

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Example Par, Inc.

• Hypothesis Tests About the Difference Between the
  Means of Two Populations Large-Sample Case
• Can we conclude, using a .01 level of
  significance, that the mean driving distance of
  Par, Inc. golf balls is greater than the mean
  driving distance of Rap, Ltd. golf balls?
• ?1 mean distance for the population of Par,
  Inc.
• golf balls
• ?2 mean distance for the population of Rap,
  Ltd.
• golf balls
• Hypotheses H0 ?1 - ?2 lt 0
• Ha ?1 - ?2 gt 0

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Example Par, Inc.

• Hypothesis Tests About the Difference Between the
  Means of Two Populations Large-Sample Case
• Rejection Rule Reject H0 if z gt 2.33
• Conclusion Reject H0. We are at least 99
  confident
• that the mean driving distance of Par, Inc. golf
  balls is
• greater than the mean driving distance of Rap,
  Ltd. golf balls.

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Small Sample Case-Pooled Variance
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Small Sample Case-Unequal Variances
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Small Sample Case-Unequal Variances
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Inference About the Difference Between the Means
of Two Populations Matched Samples

• With a matched-sample design each sampled item
  provides a pair of data values.
• The matched-sample design can be referred to as
  blocking.
• This design often leads to a smaller sampling
  error than the independent-sample design because
  variation between sampled items is eliminated as
  a source of sampling error.

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Example Express Deliveries

• A Chicago-based firm has documents that must be
• quickly distributed to district offices
  throughout the
• U.S. The firm must decide between two delivery
• services, UPX (United Parcel Express) and INTEX
• (International Express), to transport its
  documents. In
• testing the delivery times of the two services,
  the firm
• sent two reports to a random sample of ten
  district
• offices with one report carried by UPX and the
  other
• report carried by INTEX.
• Do the data that follow indicate a difference in
  mean
• delivery times for the two services?

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Example Express Deliveries

• Delivery Time (Hours)
• District Office UPX INTEX Difference
• Seattle 32 25 7
• Los Angeles 30 24 6
• Boston 19 15 4
• Cleveland 16 15 1
• New York 15 13 2
• Houston 18 15 3
• Atlanta 14 15 -1
• St. Louis 10 8 2
• Milwaukee 7 9 -2
• Denver 16 11 5

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Example Express Deliveries

• Inference About the Difference Between the Means
  of Two Populations Matched Samples
• Let ?d the mean of the difference values for
  the two delivery services for
  the population of
• district offices
• Hypotheses H0 ?d 0, Ha ?d ???
• Assuming the population of difference values is
• approximately normally distributed, the t
  distribution
• with n - 1 degrees of freedom applies. With ?
  .05, t.025 2.262 (9 degrees of freedom).
• Rejection Rule Reject H0 if t lt -2.262 or if t
  gt 2.262

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Example Express Deliveries

• Inference About the Difference Between the Means
  of Two Populations Matched Samples
• Conclusion Reject H0. There is a significant
  difference between the mean delivery times for
  the two services. UPX provides faster service.

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Inference About the Difference Between the
Proportions of Two Populations

• Sampling Distribution of
• Interval Estimation of
• Hypothesis Tests about

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Sampling Distribution of

• Expected Value
• Standard Deviation
• Distribution Form
• If the sample sizes are large (n1p1, n1(1 -
  p1), n2p2,
• and n2(1 - p2) are all greater than or equal to
  5), the
• sampling distribution of can be
  approximated
• by a normal probability distribution.

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Example MRA

• MRA (Market Research Associates) is conducting
  research to evaluate the effectiveness of a
  clients new advertising campaign. Before the
  new campaign began, a telephone survey of 150
  households in the test market area showed 60
  households aware of the clients product. The
  new campaign has been initiated with TV and
  newspaper advertisements running for three weeks.
  A survey conducted immediately after the new
  campaign showed 120 of 250 households aware of
  the clients product.
• Does the data support the position that the
  advertising campaign has provided an increased
  awareness of the clients product?

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Hypothesis Testing about p1 - p2

• Hypotheses
• H0 p1 - p2 lt 0
• Ha p1 - p2 gt 0
• Test statistic
• Point Estimator of where p1 p2

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Example MRA

• Hypothesis Tests about p1 - p2
• Can we conclude, using a .05 level of
  significance, that the proportion of households
  aware of the clients product increased after the
  new advertising campaign?
• p1 proportion of the population of households
• aware of the product after the new campaign
• p2 proportion of the population of
  households
• aware of the product before the new campaign
  
• Hypotheses H0 p1 - p2 lt 0
• Ha p1 - p2 gt 0

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Example MRA

• Hypothesis Tests about p1 - p2
• Rejection Rule Reject H0 if z gt 1.645
• Conclusion Do not reject H0.