Statistical decisionmaking when m is unknown

1
Statistical decision-making when m is unknown
2
Hypothesis testing when m is unknown

• Dependent samples t test for correlated groups
• Independent samples t test for independent
  groups.

3
Tests that you know so far.

• Single sample tests used with one set of scores
• One sample z, when m and s are known
• One sample t when m is known and s must be
  estimated with s.
• The sign test used with two sets of scores that
  are paired
• data collected from the same group of people
  under two different conditions
• known as repeated measures, within-subjects,
  or correlated measures designs.

4
You can also use a t test for two sets of scores.

• Use t when you have two sets of scores and
  neither m nor s are known.
• Almost everything about these tests is identical
  to what we have done except for
• H0 and HA are slightly different.
• The formula for calculating t (and df) depends on
  the groups
• When the scores are paired use the t test for
  correlated groups.
• When the scores are from two separate groups use
  the t test for independent groups.
• In the t test for independent groups, the
  sampling distribution is the sampling
  distribution of the difference between two means.

5
t tests for two samples

• t test for correlated samples
• A little theory (less than 5 minutes, I promise).
• An example problem.
• t test for independent samples
• Minimal theory read chapter 14 for MC questions
  on final!
• An example problem.
• More examples (time permitting).

6
t tests for correlated samples theory

• Data for t test for correlated samples are of the
  same type as data we have been testing with the
  sign test paired scores collected under 2
  conditions.
• Prefer t test rather than sign test because t
  test uses more information (magnitude of
  difference as well as sign) t test is more
  powerful test
• Like sign test, t test is concerned with
  difference between the conditions.
• Unlike sign test, t test involves the mean of the
  difference scores.
• H0 and HA of t test will involve means (not the
  probability of a plus).
• Mechanics of correlated samples t test are
  virtually identical to one sample t test.

7
Correlated t test example to help understand H0
and HA

• My Ph.D. research investigated human color
  perception. I believed I had a way of training
  people to see pictures as colored even when no
  color was there. Even more exciting, I thought
  that my training technique could make people see
  green in a picture when the picture was actually
  pink! To test my theory, I invited 18 subjects
  into my office and asked them to identify 25 pink
  and 25 green pictures as either pink or
  green. Then I trained them using my technique.
  After training I showed them the same 50
  pictures again. Before and after the training I
  counted the total number of times (out of 50
  possible) that they said green.

8
Correlated t test example to help understand H0
and HA

• Before training, if everyone always perfectly
  identified the color of a picture, how many times
  would they say green?
• If the training had no effect (H0 is true) and
  everyone could still perfectly identify the color
  of a picture after training, how many times would
  they say green?
• Value of D for any subject if D
  (Scoreafter-Scorebefore)?
• If the training had the desired effect (HA is
  true) and everyone saw more green pictures after
  training, how many times would they say green?
• Value of D for any subject if D
  (Scoreafter-Scorebefore)?

9
Correlated t test example to help understand H0
and HA

• For correlated samples t test, H0 and HA will
  always be expressed in terms of mD, the
  population mean of the difference scores.
• Most often, mD under H0 will be 0, as in
• H0 mD 0 and HA mD 0
• or
• H0 mD 0 OR H0 mD 0
  HA mD
• Only in very rare instances, where a particular
  mD is hypothesized, would it be non-zero.
• Notes Difference scores reduce two sets of
  scores to one (D), making the correlated sample t
  test virtually identical to the one-sample t
  test, using the values of D as the sample and the
  sampling distribution of the mean as the sampling
  distribution. As with one-sample t, df N-1,
  where N is the number of difference scores.

10
Decision-making steps the pink and green test
problem

• 1. Define problem Does training increase number
  of green pictures?
• 2. Define hypotheses with respect to mD (D
  Greenafter Greenbefore)
• H0
• HA
• 3. Define experiment 18 people label pictures
  before/after training.
• 4. Define statistic Correlated samples t.
• 5. Define acceptable probability of Type I error
  a
• 6. Define value of statistic upon which your
  decision hinges t17
• 7. Perform experiment/collect data
• 8. Compare observed statistic to critical value.
• 9. Decide
• 10. Draw conclusion using at least one complete
  sentence

11
Subject green green Difference before
training after training (after-before)
1 25 48 23 2 23 32 9
3 27 39 12 4 26 34 8
5 28 42 14 6 18 47 29 7 21 23
2 8 30 31 1 9 27 38 11 10 25 45
20 11 23 21 -2 12 24 31 7 13 29 46
17 14 24 35 11 15 28 32
4 16 21 29 8 17 25 32 7 18 22 29
7
Dobt 10.44 SD 7.89
12
t tests for independent samples theory

• Can be differentiated from other t tests because
  there are two, independent (separate, different)
  groups of people.
• Each group experiences one condition (one level
  of the IV). For example
• IV is drug dose, levels placebo and active drug
  conditions
• IV is brain injury, levels actual and sham
  brain injury
• IV gender, levels men and women
• Under H0, m1 m2 under HA, m1 is , or m2.
• Assumptions of the independent samples t test
• Homogeneity of variance is assumed.
• Always, s1 is assumed to be equal to s2
• Additive effect of IV.
• Sampling distribution of the difference between
  two means is normally distributed (the two
  populations are normally distributed).

Note df N1 N2 - 2
13
Independent samples t test

• A social psychologist is investigating the
  development of generosity in preschool children
  and would like to see if girls and boys differ in
  their generosity at age 4. Each child at a
  day-care center is given 16 small pieces of candy
  and is asked to put some in a bag for your very
  best friend. The number of candies set aside
  for friends for the 12 girls and 10 boys
  follow. Determine if there is a gender
  difference for generosity in these samples.

Boys Girls Mean 2.7 6.5 SS 12.1 45.0 N 10 12
14
Decision-making steps

• 1. Define problem Is there a gender difference
  in generosity at age 4?
• 2. Define hypotheses with respect to the two ms
  
• H0 mboys mgirls
• HA mboys mgirls
• 3. Define experiment 10 boys 12 girls given
  candy to save for a friend.
• 4. Define statistic Independent samples t.
• 5. Define acceptable probability of Type I error
  a
• 6. Define value of statistic upon which your
  decision hinges t20 2.086
• 7. Perform experiment/collect data
• 8. Compare observed statistic to critical value.

N1 N2 2 10 12 2 20
Boys Girls Mean 2.7 6.5 SS 12.1 45.0 N 10 12
15
Decision-making steps

• 1. Define problem Is there a gender difference
  in generosity at age 4?
• 2. Define hypotheses with respect to the two ms
  
• H0 mboys mgirls
• HA mboys mgirls
• 3. Define experiment 10 boys 12 girls given
  candy to save for a friend.
• 4. Define statistic Independent samples t.
• 5. Define acceptable probability of Type I error
  a
• 6. Define value of statistic upon which your
  decision hinges t20 2.086
• 7. Perform experiment/collect data
• 8. Compare observed statistic to critical value.

N1 N2 2 10 12 2 20
Boys Girls Mean 2.7 6.5 SS 12.1 45.0 N 10 12
X1 X2 (m1 m2)
SS1 SS2 1 1 n1n2 - 2 n1 n2
16
Decision-making steps

• 1. Define problem Is there a gender difference
  in generosity at age 4?
• 2. Define hypotheses with respect to the two ms
  
• H0 mboys mgirls
• HA mboys mgirls
• 3. Define experiment 10 boys 12 girls given
  candy to save for a friend.
• 4. Define statistic Independent samples t.
• 5. Define acceptable probability of Type I error
  a
• 6. Define value of statistic upon which your
  decision hinges t20 2.086
• 7. Perform experiment/collect data
• 8. Compare observed statistic to critical value.

N1 N2 2 10 12 2 20
Boys Girls Mean 2.7 6.5 SS 12.1 45.0 N 10 12
X1 X2
SS1 SS2 1 1 n1n2 - 2 n1 n2
17
Decision-making steps
Decision-making steps

• 1. Define problem Is there a gender difference
  in generosity at age 4?
• 2. Define hypotheses with respect to the two ms
  
• H0 mboys mgirls
• HA mboys mgirls
• 3. Define experiment 10 boys 12 girls given
  candy to save for a friend.
• 4. Define statistic Independent samples t.
• 5. Define acceptable probability of Type I error
  a
• 6. Define value of statistic upon which your
  decision hinges t20 2.086
• 7. Perform experiment/collect data
• 8. Compare observed statistic to critical value.

N1 N2 2 10 12 2 20
Boys Girls Mean 2.7 6.5 SS 12.1 45.0 N 10 12
2.7 6.5
12.1 45.0 1 1 1012 - 2 10 12
18
Decision-making steps

• 1. Define problem Is there a gender difference
  in generosity at age 4?
• 2. Define hypotheses with respect to the two ms
  
• H0 mboys mgirls
• HA mboys mgirls
• 3. Define experiment 10 boys 12 girls given
  candy to save for a friend.
• 4. Define statistic Independent samples t.
• 5. Define acceptable probability of Type I error
  a
• 6. Define value of statistic upon which your
  decision hinges t20 2.086
• 7. Perform experiment/collect data
• 8. Compare observed statistic to critical value.
  t
• 9. Decide
• 10. Conclusion

Boys Girls Mean 2.7 6.5 SS 12.1 45.0 N 10 12
19
Does the medication change blood pressure?

• Researchers studied the effectiveness of a new
  medication for high blood pressure. They
  recorded the systolic and diastolic blood
  pressure for 12 patients before giving them the
  medication and again four weeks after daily
  medication. The data for the systolic blood
  pressure readings taken before and after use of
  the medication are shown. Was there a
  significant difference in systolic blood pressure
  before and after use of this medication?

Before After 140 125 163 121 182 131 154 118 162 1
23 177 134 157 123 167 150 144 134 168 154 181 166
179 132
20
Does the medication change blood pressure?

• Researchers studied the effectiveness of a new
  medication for high blood pressure. They
  recorded the systolic and diastolic blood
  pressure for 24 patients before giving 12 of them
  placebo and 12 of them the active medication for
  four weeks. The data for the systolic blood
  pressure readings taken after the four weeks of
  drug exposure are shown. Was there a significant
  difference in systolic blood pressure caused by
  medication administration?

Placebo Active 140 125 163 121 182 131 154 118 162
123 177 134 157 123 167 150 144 134 168 154 181 1
66 179 132
21
Does attractiveness take the years off?

• Research has suggested that attractive defendants
  in court cases are given less punishment for
  their crimes than unattractive defendants. Some
  political scientists provided a written case
  account of a crime to 20 senior university
  students. For 10 students, a picture of an
  attractive defendant was provided with the
  written case account, and for the other 10
  subjects a picture of an unattractive defendant
  was provided. The subjects were asked to assign
  punishment by suggesting the number of years the
  defendant should serve in prison for the crime.
  Was their a significant decrease in the suggested
  years of imprisonment for attractive defendants?

Attractive Unattractive 2 3 3 2 1 5 2 3 2 2 2
3 3 2 4 3 1 4 1 2