Patient Score before counseling(B) Score after counseling(A) Jan12176 Tom93 Diane10564 Becky115117 Fred13082 John9880 Craig14279

Tests Involving Paired Differences ( Dependent Samples )

Many statistical applications use paired data samples to draw conclusions about the difference between two population means. Data pairs occur very naturally in “before and after” situations, where the same object or item is measured both before and after a treatment.



For example : A psychologist has developed a series of exercises called the Instrumental Enrichment (IE) Program, which he claims is useful in overcoming cognitive deficiencies in mentally handicapped children. In one experiment, a random sample of 10 – year – old students with IQ scores below 80 was selected. An IQ test was given to these students before they spent 2 years in an IE Program, and an IQ test was given to the same students after the program.



For example : A psychologist has developed a series of exercises called the Instrumental Enrichment (IE) Program, which he claims is useful in overcoming cognitive deficiencies in mentally handicapped children. In one experiment, a random sample of 10 – year – old students with IQ scores below 80 was selected. An IQ test was given to these students before they spent 2 years in an IE Program, and an IQ test was given to the same students after the program.

If 20 students are included in the random sample, you would have 20 data pairs.


HOW TO TEST PAIRED DIFFERENCES USING STUDENTS

Requirements :a) Obtain a simple random sample of matched data pairs A, B.



Requirements :a) Obtain a simple random sample of matched data pairs A, B.b) Let be a random variable representing the difference between the values in a

matched data pair




matched data pairc) Compute the sample mean and sample standard deviation




matched data pairc) Compute the sample mean and sample standard deviation d) If you can assume that has a normal distribution or simply has a mound –

shaped, symmetric distribution, then any sample size will work. If you can not assume this, then use a sample size



Procedure :1. Use the null hypothesis of no difference, In the context of the application choose alternate hypothesis




2. Find the sample test statistic :

with





with

3. Use the student’s – distribution and the type of test, one – tailed or two – tailed, to find ( or estimate ) the corresponding to the test statistic





with


4. Conclude the test. If then reject . If then do not reject





with


4. Conclude the test. If then reject . If then do not reject

5. Interpret your conclusion in the context of the application

Tests Involving Paired Differences ( Dependent Samples )EXAMPLE : It has been known that patients who undergo corrective heart surgery have a dangerous build up of anxiety before their scheduled operations. A psychiatrist at the hospital started a new counseling program intended to reduce this anxiety. A test of anxiety is given to patients who know the must undergo heart surgery. Then each patient participates in a series of counseling sessions with a staff psychiatrist. At the end of the counseling sessions, each patient is retested. From the given data ( below ) can we conclude that the counseling sessions reduced anxiety ?Use

Patient Score before counseling(B)

Score after counseling(A)

Jan 121 76

Tom 93 93

Diane 105 64

Becky 115 117

Fred 130 82

John 98 80

Craig 142 79

Allyson 118 67

Hailey 125 89




Jan 121 76 45

Tom 93 93 0

Diane 105 64 41

Becky 115 117 -2

Fred 130 82 48

John 98 80 18

Craig 142 79 63

Allyson 118 67 51

Hailey 125 89 36

1. Find




Jan 121 76 45

Tom 93 93 0

Diane 105 64 41

Becky 115 117 -2

Fred 130 82 48

John 98 80 18

Craig 142 79 63

Allyson 118 67 51

Hailey 125 89 36

1. Find 2. Using a calculator, I found




Jan 121 76 45

Tom 93 93 0

Diane 105 64 41

Becky 115 117 -2

Fred 130 82 48

John 98 80 18

Craig 142 79 63

Allyson 118 67 51

Hailey 125 89 36

1. Find 2. Using a calculator, I found 3. and




Jan 121 76 45

Tom 93 93 0

Diane 105 64 41

Becky 115 117 -2

Fred 130 82 48

John 98 80 18

Craig 142 79 63

Allyson 118 67 51

Hailey 125 89 36


4. with




Jan 121 76 45

Tom 93 93 0

Diane 105 64 41

Becky 115 117 -2

Fred 130 82 48

John 98 80 18

Craig 142 79 63

Allyson 118 67 51

Hailey 125 89 36


4. with 5. ( right - tailed ) between 0.005 and 0.0005




Jan 121 76 45

Tom 93 93 0

Diane 105 64 41

Becky 115 117 -2

Fred 130 82 48

John 98 80 18

Craig 142 79 63

Allyson 118 67 51

Hailey 125 89 36


4. with 5. ( right - tailed ) between 0.005 and 0.00056. Since the interval containing our we reject




Jan 121 76 45

Tom 93 93 0

Diane 105 64 41

Becky 115 117 -2

Fred 130 82 48

John 98 80 18

Craig 142 79 63

Allyson 118 67 51

Hailey 125 89 36


4. with 5. ( right - tailed ) between 0.005 and 0.00056. Since the interval containing our we reject

At the 1% level, we conclude that the counseling sessions reduce the anxiety level of patients about to undergo corrective heart surgery

Tests Involving Paired Differences ( Dependent Samples )Let’s try another :Do educational toys make a difference in the age at which a child learns to read ? To study this, one group of preschool children spent 2 hours per day in a room, well supplied with educational toys such as alphabet blocks, puzzles, ABC readers, etc. A control group spent two hours per day ( for 6 months ) on a “noneducational” toy room. It was anticipated that IQ differences and home environment might be uncontrollable unless identical twins could be used. Therefore, six pairs of identical twins of preschool age was randomly selected. And also, each twin was randomly placed in a group. For each twin, the data item recorded is the age in months at which the child began reading at the primary level. Assume the distribution is mound – shaped and symmetric.

We are looking to test that the experimental group learned to read at a different age ( either younger or older ) Use

The table appears on the next page with the steps to follow…

Tests Involving Paired Differences ( Dependent Samples )Reading Ages for Identical Twins ( in months )

Twin Pair Exper. GroupB = reading age

Control GroupA = reading age

Difference

1 58 60 -2

2 61 64 -3

3 53 52 1

4 60 65 -5

5 71 75 -4

6 62 63 -1

1. Find




Difference

1 58 60 -2

2 61 64 -3

3 53 52 1

4 60 65 -5

5 71 75 -4

6 62 63 -1

1. Find 2. Using a calculator :




Difference

1 58 60 -2

2 61 64 -3

3 53 52 1

4 60 65 -5

5 71 75 -4

6 62 63 -1

1. Find 2. Using a calculator : 3. and ( two – tailed )




Difference

1 58 60 -2

2 61 64 -3

3 53 52 1

4 60 65 -5

5 71 75 -4

6 62 63 -1


4.




Difference

1 58 60 -2

2 61 64 -3

3 53 52 1

4 60 65 -5

5 71 75 -4

6 62 63 -1


4. 5. ( two – tailed ) = between 0.02 and 0.05




Difference

1 58 60 -2

2 61 64 -3

3 53 52 1

4 60 65 -5

5 71 75 -4

6 62 63 -1


4. 5. ( two – tailed ) = between 0.02 and 0.05 6. Since our interval containing we reject

Documents

Patient Score before counseling(B) Score after counseling(A) Jan12176 Tom93 Diane10564 Becky115117 Fred13082 John9880 Craig14279