Testing hypotheses about:

1

• Testing hypotheses about
• Differences between two means
• Matched populations
• Independent populations

2
Independent populationsvs.Dependent populations
(matched pairs comparisons)

• Independent groups
• The responses in each group are independent of
  those in the other group.
• Dependent groups (matched pairs comparisons)
• Two measurements on each individual

3
Comparing two means
Independent groups
Dependent groups (matched groups)
4
Decide whether the following comparisons involve
independent or matched populations

• A medical researcher is interested in the effect
  on blood pressure of added calcium in our diet.
  She conducts a randomized comparative experiment
  in which one group of subjects receives a calcium
  supplement and a control group gets a placebo.
  ___________
• To compare two treatments, each individual in a
  groups of subjects receives two treatments, A
  and B. ___________
• Measurements of the left and right hand gripping
  strength of a group of 10 left-handed writers
  were recorded. ___________
• A psychologist develops a test that measures
  social insight. He compares the social insight of
  male college students with that of female college
  students. ___________
• A bank wants to know which of two incentive plans
  will most increase the use of credit cards. It
  offers each incentive to different random sample
  of credit card customers and compares the amount
  charged during the following six months.
  ___________

independent
matched
matched
independent
independent
5
Comparing two means of matched populations
6
Example

• A manufacturer wanted to examine the quality of
  two synthetic materials, A and B, that are used
  for shoe soles. He hypothesized that Material B,
  which is was cheaper, might result in an
  increased amount of wear. He gave each of 10 boys
  a pair of shoes, one sole was made with A and the
  other with B. (Materials A and B were randomly
  chosen for left and right shoes by a flip of a
  coin).

0.8 0.6 0.3 -0.1 1.1 -0.2 0.3 0.5 0.5 0.3
Average difference 0.41 Standard deviation of
differences 0.387
7

• We assume that the differences di are
  approximately normal
• Plot the distribution of di

8

• Hypotheses
• H0µd0
• H1µdgt0
• Test statistic
• P-value
• P(t(9) 3.35)
• .001 lt Pvalue lt .005
• Decision (a1)
• Pvaluelt a ? reject H0
• Conclusion

3.35
9
Performing the test with Minitab / Excel
10
Is our height higher than our parents height?
ExcelSurvey 1100 spring 2005.xls
Survey 0200morning.MPJ Survey 0200evening.MPJ
11
Example

• A golf coach thinks players are better (lower)
  in the second round of tournament than in the
  first, because they are less nervous. Here are
  the scores (and differences) for 12 players in
  two rounds
• Do the scores support the coachs theory?
• Average difference 1.67
• Standard deviation of differences 6.2

-5 5 -2 6 5 5 -5 16 -4 -3 3 -1
12
We assume that the differences di are normally
distributed with mean 0 (under H0).

• Hypotheses
• H0µd0
• H1µdgt0
• Test statistic
• P-value
• p(t(11)0.93)?
• 0.1ltPvaluelt0.25
• Decision (a5)
• Pvaluegta ? do not reject H0

(dround1-round2)
0.93
13
Example

• A researcher wanted to examine whether a certain
  diet can help people lose at least 5 pounds.
• He took 10 people and measured their weights
  before and after the diet.
• X1- weight before diet
• X2- weight after diet

14
Hypotheses H0µd5 H1µdgt5 (µdbefore-after)
Minitab output
Paired T-Test and CI before, after Paired T for
before - after N
Mean StDev SE Mean before 10
187.30 12.35 3.90 after 10
181.80 9.35 2.96 Difference 10
5.50 5.70 1.80 T-Test of mean
difference 5 (vs gt 5) T-Value 0.28 P-Value
0.394
Decision (5) Pvaluegta ? do not reject
H0 Conclusion The diet does not helps people
loose weight
15
Robustness of the t procedures

• Results are accurate if population is normal
• Robust against nonnormality of population, except
  in case of outliers or strong skewness
• Large samples improve results when population is
  not normal (sampling distribution of is
  approximately normal)
• Rules of thumb
• one-sample t procedures can be safely used when
  n15 except in presence of outliers or strong
  skweness.
• nlt15 use t procedures if the data are close to
  normal.
• n40 can use even for skewed distributions

16

• Independent samples

17
Comparing two means of independent populations
18
Test statistic for2 independent samples

• s1, s2 known
• s1, s2 unknown
• Assume s1, s2 are equal and compute a pooled
  estimator Sp

Test statistic
19
CI for µ1-µ2
1. s1, s2 known 2. s1, s2 unknown (but
assumed equal)
20
Example

• The following data were obtained in an
  experiment conducted by an amateur gardener whose
  object was to discover whether a change in the
  fertilizer mixture applied to his tomato plants
  would result in an improved yield. He had 11
  plants set out in a single row 5 were given the
  standard fertilizer mixture A, and the remaining
  6 were fed a supposedly improved mixture B. The
  As and Bs were randomly applied to the
  positions in the row to give the design shown
  below.

21
(No Transcript)
22
Reorganize the data
The population standard deviations s1, s2, are
unknown. We assume that they are equal and
compute Sp
23
Hypotheses H0 µA-µB0 H1 µA-µBlt0
B should be better, so A-Blt0
Test statistic
24
P-value P(t(9)-.44) Pvaluegt.1 Decision
(a5) Pvaluegt0.05 ? Do not reject
H0. Conclusion Fertilizer B does not improve
the yield.
t(9)
0.44
25
Data
26
Choose Stat gt Basic Statistics gt 2-Sample t
27
Pick the alternative hypothesis in the options
window
28
Results in session window
Two-Sample T-Test and CI Fertilizer A,
Fertilizer B Two-sample T for Fertilizer A vs
Fertilizer B N Mean
StDev SE Mean Fertilizer A 5 20.84
7.25 3.2 Fertilizer B 6 22.53
5.43 2.2 Difference mu Fertilizer A
- mu Fertilizer B Estimate for difference
-1.69 T-Test of difference 0 (vs lt) T-Value
-0.44 P-Value 0.334 DF 9 Both use Pooled
StDev 6.30
29
Minitab Use the survey data to examine whether
the average GPA is different for males and
females students.
Survey 0200morning.MPJ Survey 0200evening.MPJ
30
Example

• An experiment was conducted for comparing two
  teaching methods for a statistics course. Two
  groups of 25 U.S. college students in each were
  randomly selected and taught by the same
  instructor. The material covered in both groups
  was identical, except that in group I teaching
  was accompanied with computer demonstrations. The
  average grade in group I was 72.9 with a standard
  deviation of 6.7. The average grade in group II
  was 69.6 with a standard deviation of 8.8.
• Is there sufficient evidence to suggest that
  using the computer improves students
  achievements in the course?

31
Hypotheses H0 µ1-µ20 H1 µ1-µ2gt0

• Test statistic
• Sp7.821
• t1.49
• P.Value
• p.vp(t(48)1.49)0.0714
• Decision (a5)
• Pvaluegt0.05 ? do not reject H0.
• Conclusion
• The computer does not improve the achievements

32

• P-value
• 2P(t(28)-1.53)
• 2(0.05 to 0.1)
• 0.1ltPvaluelt0.2
• Decision (a1)
• Pvaluegt0.01 ? do not reject H0.
• Conclusion
• The two brands of soda do not differ in the mean
  number of calories.

t(28)
-1.53
1.53
33
Example

• A researcher wanted to examine whether a certain
  component in a diet helps people lose more
  weight. The researcher examined two groups of 15
  obese people in each. The two groups were given
  the same low-calorie diet, but group 1 also
  obtained the special component in their diet.
  After one month, the researcher recorded the
  weight loss (in pounds) of the people in the two
  groups. Following are two Minitab outputs for
  analyzing the data Output 1 Output 2

34
Output 1
Output 2
35

• Which of the outputs should be used?_____
• What is the mean weight loss in each of the two
  groups?__________
• What is the value of the test statistic?______
• Do the results suggest that the chemical helps
  lose weight (a1)?

36
Flow diagram
..\flow diagram.pdf