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Comparing Means of Two Populations: Inference and Estimation, Assignments of Mathematics

An activity for students to learn about inferential methods for comparing means of two populations. Learning objectives, criteria for success, resources, and a model for carrying out the activity. The activity involves analyzing data from two different production methods for scanners and a pizza delivery comparison. Students will work in teams to complete exercises, assess their work, and discuss their results.

Typology: Assignments

Pre 2010

Uploaded on 08/05/2009

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Statistical Applications ACTIVITY 5: Inference on Means - Two Populations
Why
Now that we have established the basic inferential methods confidence intervals and tests of
significance for means and proportions, we are ready to extend these to many other situations.
Our first extension is to comparisons between two populations. This frees us from the need to
have a “known” or “established” value for comparison especially useful for experiments.
LEARNING OBJECTIVES
1. Work as a team, using the team roles
2. Understand the use of inferential tools for comparing means of two populations.
3. Distinguish the “test” situation from “estimation”, and be able to carry out inference (including
calculations) for either.
4. Unbderstand the distinction between “paired samples” situation and the “independent samples”
situation.
CRITERIA
1. Success in completing the exercises.
2. Success in answering questions about the model
3. Success in working as a team
RESOURCES
1. The document “Inference (hypothesis tests and estimation): Differences between means, proportions
in two populations” handed out Monday
2. The document “Hypothesis testing - generalities” distributed last week
3. Your Text - especially sections 10.1 through 10.3
4. The model below
5. Your Calculator
6. 40 minutes
PLAN
1. Select roles, if you have not already done so, and decide how you will carry out steps 2 and 3 (5
minutes)
2. Work through the exercises given here - be sure everyone understands all results (30 minutes)
3. Assess the team’s work and roles performances and prepare the Reflector’s and Recorder’s reports
including team grade (5 minutes).
4. Be prepared to discuss your results.
MODELS
1. A company making plastic optical components for bar codes scanners was studying inconsistencies
in an optical measurement called “tilt” - knowing that scanners work best when tilt is small. Two
different types of locking mechanism in the mold (for forming the body of the scanner) produced the
following results:
Value of “Tilt”
Taper Locks Locking Pins
¯x1.062 0.761
s0.297 0.307
n 20 15
Do we have evidence that there is a difference in average tilt for scanners produced by the two
1
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Statistical Applications ACTIVITY 5: Inference on Means - Two Populations

Why

Now that we have established the basic inferential methods — confidence intervals and tests of significance — for means and proportions, we are ready to extend these to many other situations. Our first extension is to comparisons between two populations. This frees us from the need to have a “known” or “established” value for comparison — especially useful for experiments.

LEARNING OBJECTIVES

  1. Work as a team, using the team roles
  2. Understand the use of inferential tools for comparing means of two populations.
  3. Distinguish the “test” situation from “estimation”, and be able to carry out inference (including calculations) for either.
  4. Unbderstand the distinction between “paired samples” situation and the “independent samples” situation.

CRITERIA

  1. Success in completing the exercises.
  2. Success in answering questions about the model
  3. Success in working as a team

RESOURCES

  1. The document “Inference (hypothesis tests and estimation): Differences between means, proportions in two populations” handed out Monday
  2. The document “Hypothesis testing - generalities” distributed last week
  3. Your Text - especially sections 10.1 through 10.
  4. The model below
  5. Your Calculator
  6. 40 minutes

PLAN

  1. Select roles, if you have not already done so, and decide how you will carry out steps 2 and 3 ( minutes)
  2. Work through the exercises given here - be sure everyone understands all results (30 minutes)
  3. Assess the team’s work and roles performances and prepare the Reflector’s and Recorder’s reports including team grade (5 minutes).
  4. Be prepared to discuss your results.

MODELS

  1. A company making plastic optical components for bar codes scanners was studying inconsistencies in an optical measurement called “tilt” - knowing that scanners work best when tilt is small. Two different types of locking mechanism in the mold (for forming the body of the scanner) produced the following results:

Value of “Tilt” Taper Locks Locking Pins x ¯ 1.062 0. s 0.297 0. n 20 15 Do we have evidence that there is a difference in average tilt for scanners produced by the two

methods? What is our estimate of the difference in mean tilt for the two locking methods? The first question is a “Do we have evidence of a difference?” question - a Test. The second calls for an estimate of the difference.

I Populations - scanners produced by the different methods. Variable XT = tilt measurement on a tabular lock scanner XL = tilt measurement on a locking pin scanner Test is on difference of means. These are independent samples. Test is : H 0 : μT = μL Ha : μT 6 = μL

II Test statistic issample t = x¯T − x¯L − 0 √ s^2 xT nT +^

(^22) XL nL

with df =

s^2 T nT +^

s^2 L nL

1 nT − 1

( (^) s 2 T nT

  • (^) nL^1 − 1

( (^) s 2 L nL

III Critical value approach: Reject H 0 if sample t > t. 025 or if sample t < −t. 025

Here df =

. 2972 20 +^ . 3072 15

1 19

20

15

) 2 = 29.^75 [We use the 29 row in the t-table] so^ t.^025 = 2.^045

IV t =

. 2972 20 +^ . 3072 15

V Reject H 0 and support Ha VI The sample does give evidence (p <. 01 ) at the .05 level that there is a difference in mean tilt values for the two production methods. [Note: Computer gives p = .007]

The second asks for an estimate of the difference. We have to decide which way to subtract; since no confidence level is given, we’ll use 95%

The estimate will be x¯T − x¯L ± E with E = t. 025

s^2 XT nT

s^2 XL nL

df calculated above as 29.75 (round to

29 to use table) so t. 025 = 2. 045. Our error allowance is E = 2. 045

=. 216 and x¯T − x¯L =

  1. 062 − 0 .761 =. 301. With 95% confidence, we say the mean tilt value for Taperlock-produced scanners is between. and .517 greater than the mean value for Locking pin-produced scanners. (NOTE: We found a difference with the test (at the .05 level), and our estimate (with 95% confidence) for the difference does not include 0. We expect this correspondence.)

  2. The pizza restaurant across the street from a college dorm advertises “Quicker delivery than Geron- imo’s”. To test, this, a group of students orders a pizza from the local restaurant and, at the same time, a pizza from the local Geronimo’s. Thus they obtain matched samples of data for delivery times, and can obtain better results (higher power) in their test and smaller error allowance in their estimates. They obtain the following data for ten orders:

Time (minutes) order to delivery Order # Local Geronimo’s difference(local − G’s) 1 16.8 22.0 -5. 2 11.7 15.2 -3. 3 15.6 18.7 -3. 4 16.7 15.6 1. 5 17.5 20.8 -3. 6 18.1 19.5 -1. 7 14.1 17.0 -2. 8 21.8 19.5 2. 9 13.9 16.5 -2. 10 20.8 24.0 -3. A check of the data shows that the differences are not badly skewed and there are no outliers [so it is reasonable to assume the population distribution is approximately normal - We can use t]. We have d¯ = − 2. 180 , sd = 2. 264 For the test:

READING ASSIGNMENT (in preparation for next class) Read Chapter 10 (10.4) - Inference on the difference of population proportions

SKILL EXERCISES:Use your calculator or Minitab for number -crunching [Minitab will carry out hypoth- esis tests when you have actual data to work with] but you have to write the hypotheses and conclusion. p.407 #13–14 p.413 #26–27, p.425 #42-