Problem Set - Hypothesis testing & t-tests

Q1. Define the critical region for a hypothesis test, and explain how the critical region is related to the alpha level.

Q2. A sample is selected from a population with µ = 80.  After a treatment is administered to the individuals, the sample mean is found to be M = 75 and the variance is s² = 100.

a. If the sample has n = 4 scores, then calculate the estimated standard error and determine whether the sample is sufficient to conclude that the treatment has a significant effect?  Use a two-tailed test with α = .05.

S_M = __________________

t_critical = __________________

t_calculated =  ___________________________

Decision = __________________

b. If the sample has n = 25 scores, then calculate the estimated standard error and determine whether the sample is sufficient to conclude that the treatment has a significant effect?  Use a two-tailed test with α = .05.

S_M = __________________

t_critical =  __________________

t_calculated = ___________________________

Decision = __________________

c. Describe how increasing the size of the sample affects the standard error and the likelihood of rejecting the null hypothesis.

Q3. A sample of n = 16 individuals is selected from a population with µ = 30.  After a treatment is administered to the individuals, the sample mean is found to be M = 33.

a. If the sample variance is s2 = 16, then calculate the estimated standard error and determine whether the sample is sufficient to conclude that the treatment has a significant effect?  Use a two-tailed test with α = .05.

S_M = __________________

t_critical = __________________

t_calculated =  ___________________________

Decision = __________________

b. If the sample variance is s2 = 64, then calculate the estimated standard error and determine whether the sample is sufficient to conclude that the treatment has a significant effect?  Use a two-tailed test with α = .05.

S_M = __________________

t_critical = __________________

t_calculated = ___________________________

Decision = __________________

c. Describe how increasing variance affects the standard error and the likelihood of rejecting the null hypothesis.

Q4. A researcher is testing the effect of a new cold and flu medication on mental alertness.  A sample of n = 9 college students is obtained and each student is given the normal dose of the medicine.  Thirty minutes later, each student's performance is measured on a video game that requires careful attention and quick decision making.  The scores for the nine students are as follows:  6, 8, 10, 6, 7, 13, 5, 5, 3.     

a. Assuming that scores for students in the regular population average μ = 10, are the data sufficient to conclude that the medication has a significant effect on mental performance? Test at the .05 level of significance.

M =  ___________________________

SS =  ___________________________

s²  =  ___________________________

S_M =   __________________________________________

t_critical = ________________________

t_calculated =  ____________________________________

Decision = _______________________

b. Compute r², the percentage of variance explained by the treatment effect.

r² = ________________________

c. Write a sentence demonstrating how the outcome of the hypothesis test and the measure of effect size would be presented in a research report.

Q5. A researcher conducts an independent-measures study examining how the brain chemical serotonin is related to aggression. One sample of rats serves as a control group and receives a placebo that does not affect normal levels of serotonin. 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.  The data are as follows.

Control                 Low Serotonin  

n = 10                      n = 15

M = 14                    M = 19

SS = 180.5              SS = 130.0

a. Does the drug have a significant effect on aggression?   Use an alpha level of .05, two tails.

H₀ = ________________________________________

d_f = _________________________________________

t_critical = ____________________________________

s_p² = ________________________________________

S_(M1-M2) = ____________________________________

t_calculated =  _____________________________________________________

Decision = ___________________________________

b. Compute Cohen's d to measure the size of the treatment effect.

Cohen's d = ___________________________________

Q6.  An educational psychologist studies the effect of frequent testing on retention of class material.  In one section of an introductory course, students are given quizzes each week.  A second section of the same course receives only two tests during the semester.  At the end of the semester, both sections receive the same final exam, and the scores are summarized below. 

Frequent Quizzes       Two Exams

n = 20                             n = 20

M = 73                            M = 68

a. If the first sample variance is s2 = 38 and the second sample has s2 = 42, do the data indicate that testing frequency has a significant effect on performance?  Use a two-tailed test at the .05 level of significance. (Note: Because the two sample are the same size, the pooled variance is simply the average of the two sample variances.)

t_critical = ____________________________________

s_p² = ____________________________________

S_(M1-M2) = ____________________________________

t_calculated =  _____________________________________________________

Decision = ___________________________________

b. If the first sample variance is s² = 84 and the second sample has s² = 96, do the data indicate that testing frequency has a significant effect?  Again, use a two-tailed test with α = .05.

t_critical =  ____________________________________

s_p² = ________________________________________

S_(M1-M2) = ____________________________________

t_calculated =  _____________________________________________________

Decision = ___________________________________

Q7. The following data are from an independent-measures experiment comparing two treatment conditions.

Treatment 1       Treatment 2

4                          19

5                          11

12                        18

10                        10

10                        12

7                          14

a. Do these data indicate a significant difference between the treatments at the .05 level of significance?

M₁ =  _____________________________

SS₁ = _____________________________

M₂ = _____________________________

SS₂ = _____________________________

t_critical =     _________________________

s_p² = _____________________________

S_(M1-M2) = _________________________

t_calculated =  _________________________

Decision = ________________________

b. Compute r² to measure the size of the treatment effect.

r² = ______________________________________________

c. Write a sentence demonstrating how the outcome of the hypothesis test and the measure of effect size would appear in a research report.

Q8. Briefly explain the advantages and disadvantages of using a repeated-measures design as opposed to an independent-measures design.

Q9.  Calculate the difference scores and find MD for the following data from a repeated-measures study.

Subject   Treatment 1   Treatment 2

A              12                          14

B                7                           15

C              10                             8

D                8                           12

∑D = _________________________________

M_D = _________________________________

Q10. One of the major advantages of a repeated-measures design is that it removes individual differences from the variance and, therefore, reduces the standard error.  The following two sets of data demonstrate this fact.  The first set of data represents the original results from a repeated-measures study.  To create the second set of data we started with the original scores but increased the individual differences by adding 10 points to each score for subject B, adding 20 points to each score for subject C, and adding 30 points to each score for subject D.  Note that this change produces a huge increase in the differences from one subject to another and a huge increase in the variability of the scores within each treatment condition.

              Set 1                                      Set 2

Subject    I            II             Subject    I              II 

A             12         14           A             12             14

B             7           17           B              17             27

C             11         13           C              31             33

D            10          12           D             40              42

M = 10   M = 14                     M = 25    M = 29

SS = 14  SS = 14                    SS = 494  SS = 414

a. Find the difference scores for each set of data and compute the mean and variance for each sample of difference scores.

Set 1 M_D = _________________________

Set 1 SS = _________________________

Set 1 s² = __________________________

Set 2 M_D = _________________________

Set 2 SS = _________________________

Set 2 s² = _________________________

b. You should find that both sets of data produce the same mean difference and the same variance for the difference scores.  Explain what happened to the huge individual differences that were added to the second set of data.

Q11. A researcher would like to examine how the chemical tryptophan, contained in foods such as turkey, can affect mental alertness.  A sample of n = 9 college students is obtained and each student's performance on a familiar video game is measured before and after eating a traditional Thanksgiving dinner including roast turkey.  The average score dropped by M = 14 points after the meal with SS = 1152 for the difference scores.

a. Is there is significant difference in performance before eating versus after eating?  Use a two-tailed test with α = .05.

S_MD = ___________________________

t_critical =     ________________________

t_calculated =    ________________________

Decision = _______________________

b. Compute r² to measure the size of the effect.

r² = _____________________________

c. Write a sentence demonstrating how the outcome of the test and the measure of effect size would appear in a research report.

Q12. A researcher would like to determine if relaxation training will affect the number of headaches for chronic headache sufferers. For a week prior to training, each participant records the number of headaches suffered. Participants then receive relaxation training and for the week following training the number of headaches is again measured.  The data are as follows:

Before                  After

6                              4

5                              1

3                              3

3                              1

6                              2

2                              1

4                              3

4                              2

a. Compute the mean and variance for the sample of difference scores.

M_D = _____________________________

SS = ______________________________

s² = _______________________________

b. Do the results indicate a significant difference?  Use a two-tailed test with α = .05

S_MD = _____________________________

t_critical =     __________________________

t_calculated =    __________________________

Decision = _________________________

c. Compute Cohen's d to measure the size of the effect.

Cohen's d = ________________________

Q13. A teacher gives a third grade class of n = 16 a reading skills test at the beginning of the school year.  To evaluate the changes that occur during the year, students are tested again at the end of the year.  Their test scores revealed an average improvement of MD = 4.7 points

a. If the variance for the difference scores is s2 = 144, are the results sufficient to conclude that there is significant improvement?  Use a one-tailed test with α = .05.

s² = ________________________________

S_MD = ______________________________

t_critical =     ___________________________

t_calculated =  ___________________________

Decision = __________________________

b. If the variance for the difference scores is reduced to s² = 64, are the results sufficient to conclude that there is significant improvement?  Use a one-tailed test with α = .05.

S_MD = ______________________________

t_critical =     ___________________________

t_calculated =    ___________________________

Decision = __________________________