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Mark W. Tengler, M.S.

PSYC 2317

INFERENTIAL STATISTICS

Extra Credit #2

Directions:

Answer the questions with the data that is given and only using your formula

worksheets and the statistical tables from the class (t Distribution and F

Distribution tables). Please show the complete diagram of the research design

and all of your work. Please set up tables to do your calculations similarly to

how we did it in class, and then plug the numbers into the correct formulas.

Please work individually. This exercise is worth 10 points extra credit and is due

on the final exam day.

Single Sample Design:

Numerous studies have shown that IQ scores have been increasing, generation by

generation, for years (Flynn, 1984, 1999). The increase is called the Flynn Effect, and the

data indicate that the increase appears to be about 7 points per decade. To

demonstrate this phenomenon, a researcher obtains an IQ test that was written in

1980. At the time the test was prepared, it was standardized to produce a population

mean of Âµ = 100. The researcher administers the test to a random sample of n = 16 of

todayâ??s high school students and obtains a sample mean IQ of M = 121 with SS = 6000.

Is this result sufficient to conclude that todayâ??s sample scored significantly higher than

would be expected from a population with Âµ = 100? Use Î± = .01.

Two Independent Groups Design:

A biopsychologist studies the role of the brain chemical serotonin in aggression. One

sample of rats serves as a control group and receives a placebo. A second sample of rats

receives a drug that lowers brain levels of serotonin. Then the researcher tests the

animals by recording the number of aggressive responses each of the rats display. For

the data below, is there a significant effect of the drug on aggression? What, if any, role

does serotonin has in aggression? Use an alpha level of Î± = .05. (Note: is this a one-tail or

two-tail test?)

Lowers Serotonin

5

8

15

16

6

9

1

Control

3

6

5

6

8

4

7

Mark W. Tengler, M.S.

PSYC 2317

Three or More Independent Groups Design:

A developmental psychologist is examining problem-solving ability for grade school

children. Random samples of 5-year old, 6-year old, and 7-year old children are

obtained with n = 3 in each sample. Each child is given a standardized problem-solving

task, and the psychologist records the number of errors. The data are as follows:

5-year olds

5

4

6

a.

6-year olds

6

4

2

7-year olds

0

1

2

Does the data indicate whether there are any significant differences among the

three age groups? Use Î± = .05. Show the complete design, all your work, and

report your results professionally.

One Way ANOVA:

Complete the matrix for the following scenario:

A developmental psychologist is examining the development of language skills from age

2 to age 4. Three different groups of children are obtained, one for each age, with n =

18 children in each group. Each child is given a language-skills assessment test. The

resulting data were analyzed with an ANOVA to test for mean differences between age

groups. The results of the ANOVA are presented in the following table. Fill in all missing

values, using your knowledge of the relationships within the table. (Hint: Start with the

df values first, which you can calculate from your formula sheet.) Show your work and

all calculations.

Source

SS

Df

s2 or MS

F ratio

Between groups

48

______

______

F = ______

______

______

______

252

______

Within groups

Total

2

PSYC 2317

Mark W. Tengler, M.S.

DESCRIPTIVE STATISTICS

Extra Credit #1

Directions:

Answer the questions with the data that is given and only using your formula

worksheet. Please show your work. I recommend setting up tables to do your

calculations similar to how we did it in class, and then plug the numbers into the

correct formulas. Please work individually. This exercise is worth 10 points extra

credit and is due the day of the midterm exam.

The following IQ data has been obtained for 12 incoming graduate students:

X:

120 130 132 139 160 115 120 142 148 120

141

135

1.

What is the range of this sample?

2.

Create a frequency table and a histogram with four to five equal intervals.

3.

Compute the mean and standard deviation for this sample.

4.

Convert each of the raw scores to Z scores using the sample data.

5.

The IQ test, where these scores were derived, follows a normal distribution with a mean of

100 and a standard deviation of 15. Convert each of the raw scores to Z scores using the

populational data. Compare the Z scores for the sample data and the populational data.

Are they the same or different? Why or why not?

The same students in order above also had provided GPA data after one year of graduate study:

Y:

2.50 3.10 2.75 2.25 3.40 3.25 2.75 3.00 3.15 3.60 3.20 3.00

6.

Compute the mean and standard deviation for the GPA distribution above (Y data).

7.

Convert each of the Y raw scores into Z scores. Compare the Z scores for the sample IQ

data above to the Z scores for the GPA data for each student. How does each student=s Z

score compare in the two distributions.

Mark W. Tengler, M.S.

PSYC 4730

8.

Using the X (IQ) & the Y (GPA) data, compute the Pearson=s correlation coefficient (create

a table for your calculations). Is there a relationship? What kind of relationship? How

strong? Explain the relationship between the two variables, if any.

9.

Using regression formula #1 on the formula sheet, calculate the regression equation for the

two variables, X (IQ) and Y (GPA). Interpret the significance of the slope constant and the

Y-intercept in the regression equation.

10.

Using the regression equation, compute the predicted Y= (GPA) value obtained for each X

(IQ) value (put data into a table).

11.

If you were an admission officer looking at this data relationship and you were interested

in students who would succeed at your graduate school, what IQ score cut-off policy would

you institute for admission decisions to insure adequate academic performance of your

students at your school. Explain how you chose where to make the admission cut based on

IQ scores. What minimum GPA would your IQ cut-off score predict?

PSYC 2317

Mark W. Tengler, M.S.

Assignment #13 Repeated Measures t-Test

(For Extra Credit)

13.1

What is the primary advantage of a repeated-measures design over an

independent-measures design?

13.2

Research has shown that losing even one nightâ??s sleep can have a significant effect

on performance of complex tasks such as problem solving (Linde & Bergstroem,

1992). To demonstrate this phenomenon, a sample of n = 30 college students was

given a problem-solving task at noon on one day and again at noon on the

following day. The students were not permitted any sleep between the two tests.

For each student, the difference between the first and second score was recorded.

For this sample the students averaged MD = 6.3 points better on the first test, with

SS for the difference scores (i.e. SSD) equal to 3480.

a.

Do the data demonstrate a significant change in problem-solving ability?

Use a two-tailed test with ” = .01.

13.3

A variety of research results suggest that visual images interfere with visual

perception. In one study, Segal and Fusella (1970) had participants watch a

screen, looking for brief presentations of a small blue arrow. On some trials, the

participants were also asked to form a mental image (for example, imagine a

volcano). The results show that participants made more errors while forming

images than while not forming images. Data similar to the Segal and Fusella

results are as follows. Do the data indicate a significant difference between the

two conditions? Use a two-tailed test with ” = .05.

Errors

Errors

Participant

with Image

without Image

A

13

4

B

9

2

C

12

10

D

7

8

E

10

6

F

8

6

G

9

4

Repeated Measures or Matched Subjects Designs

Two Related Samples t-test

I.

Assumptions for t-test

A.

Populations

1.

population from which the sample is selected is normal

2.

two populations must have equal variances (i.e. are the same population)

B.

One random sample (each tested twice) with each subject experiencing two conditions

1.

Individuals in one treatment are directly related (one-to-one) to individuals in

the other treatment

a.

Repeated measures (same person does both treatments)

(1)

danger of order effect (i.e. carryover, practice, fatigue)

(2)

must counterbalance to erase order effect

b.

Matched subjects (matched twins substitute for same personâ??s scores)

C.

Data values

1.

Sample values known (mean, standard deviation)

2.

Difference (D) values between two treatments utilized (mean, standard dev)

3.

Populational values (mean, standard deviation) not known

II.

Diagramming your research (shows the whole logic and process of hypothesis testing)

a.

Draw a picture of your research design (see diagramming your research handout).

b.

There are always two explanations (i.e. hypotheses) of your research results, the

wording of which depends on whether the research question is directional (one-tailed)

or non-directional (two-tailed). State them as logical opposites.

c.

For statistical testing, ignore the alternative hypothesis and focus on the null hypothesis,

since the null hypothesis claims that the research results happened by chance through

sampling error.

d.

Assuming that the null is true (i.e. that the research results occurred by chance through

sampling error) allows one to do a probability calculation (i.e. all statistical tests are

nothing more than calculating the probability of getting your research results by chance

through sampling error).

e.

Observe that there are two outcomes which may occur from the results of the

probability calculation (high or low probability of getting your research results by

chance, depending on the alpha (Î±) level).

f.

Each outcome will lead to a decision about the null hypothesis, whether the null is

probably true (i.e. we then accept the null to be true) or probably not true (i.e. we then

reject the null as false).

III.

Hypotheses

A.

Two-tailed (non-directional research question)

1.

Alternative hypothesis (H1): The independent variable causes a difference in

performance between the two treatments.

2.

Null hypothesis (H0): The independent variable does not cause a difference in

performance between the two treatments other than by sampling error.

B.

One-tailed (directional research question)

1.

Alternative hypothesis (H1): The independent variable causes one treatment to

perform better or less than the other.

2.

Null hypothesis (H0): The independent variable causes the one treatment to

1

perform in an opposite effect than expected or no change in performance.

IV.

Determine critical regions (i.e. critical t value between high & low probability) using table A-27

A.

Significance level (should be given or decided prior to experiment)

1.

Î± or p = .05, .01, or .001

B.

One- or two-tailed test

1.

One-tailed: use the first row across the top

2.

Two-tailed: use the second row across the top

C.

Degrees of freedom

1.

df = n – 1

D.

With degrees of freedom and one- or two-tailed p values, find the critical t value

1.

If two-tailed, then critical t value is Â± t value

2.

If one-tailed, then determine if critical t value is or – t

V.

Calculate t-test statistic

A.

t-test formula for two related samples (note: D is the difference of the two raw scores

per person or per paired person)

t = DM

standard error

B.

Calculations

1.

Compute variance

s2 =

2.

3.

(â??ð·)2

ð??

ð??â??1

or

DM = â??D

n

ð??ð??ð·

ð??ð??

ð?

= â??

2

ð??

Compute t-test statistic

t=

VI.

â??ð·2 â??

&

Compute standard error (Note: standard error is simply an estimate of the

average sampling error which may occur by chance, since a sample can never

give a totally accurate picture of a population

ð? ð·

B.

where D = x1 – x2

ð·ð??

ð? ð·

Compare calculated t-statistic to critical t-value & make decision

1.

Reject null and accept alternative

or

2.

Accept null

Reporting the results of a related samples (repeated measures) t test

â??The group performed better after experiencing treatment (M = 25, SD = 4.22) than before

experiencing treatment (M = 19, SD = 4.71). This difference was significant, t(18) = 3.00, p < .05,
two-tailed.â?
2
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