Question

Another researcher is interested in how caffeine will affect the speed with which people read but decides to include a third condition, a placebo group (a group that gets a pill that looks like the caffeine group does, but it does not contain caffeine). The researcher randomly assigns 12 people into one of three groups: 50mg Caffeine (n1=4), No Caffeine (n2=4), and Placebo (n3=4). An hour after the treatment, the 12 participants in the study are asked to read from a book for 1 minute; the researcher counts the number of words each participant finished reading. The following are the data for each group:

50mg Caffeine (group 1)

450 400 500 450

No Caffeine (group 2)

400 410 430 440

Placebo (group 3)

400 410 430 440

Answer the following questions using the Analysis of Variance instead of the t-test

a. What is the research hypothesis?

b. What is the null hypothesis?

c. What is dfbetween and dfwithin? What is the total df for this problem?

d. What is SSbetween and SSwithin? What is the total SS for this problem?

e. What is MSbetween and MSwithin?

f. Calculate F.

Use an a-level of .05 to answer the questions below

g. Draw a picture of the F distribution for dfbetween and dfwithin above, and locate F on the x-axis.

h. What is the critical value of F, given dfbetween and dfwithin? Indicate the critical value of F (and its value) in your drawing. Also indicate what the area is in the tail beyond the critical value of F.

i. Can you reject the null hypothesis?

j. Can you accept the research hypothesis?

Answer 1

a.

Research hypothesis Ha: At least one of the mean number of words each participant finished reading is different among the three caffeine groups

b.

Null hypothesis H0: Mean number of words each participant finished reading is same for all caffeine groups.

c.

dfbetween = Number of groups - 1 = 3 - 1 = 2

dfwithin = Number of observations - Number of groups = 12 - 3 = 9

total df = dfbetween + dfwithin = 2 + 9 = 11

d.

Let T_i be the total number of words for group i, n_i be number of observations of group i.

Let G be the total number of words of all observations and N be total number of observations.

ΣX² is sum of all squares of all number of words (observations)

From the data,

T1 = 1800, T2 = 1680 , T3 = 1680

G = 1800 + 1680 + 1680 = 5160

ΣX² = 815000 + 706600 + 706600 = 2228200

SSTotal = ΣX² - G²/N = 2228200 - 5160²/12 = 9400

SSbetween = ΣT²/n - G²/N = (1800² /4 + 1680² /4 + 1680² /4 ) - 5160²/12 = 2400

SSwithin = SSTotal - SSbetween = 9400 - 2400 = 7000

Total SS = 9400

e.

MSbetween =  SSbetween / dfbetween = 2400/2 = 1200

MSwithin = SSwithin / dfwithin = 7000/ 9 = 777.7778

f.

F = MSbetween / MSwithin = 1200 / 777.7778 = 1.542857

g.

F distribution for dfbetween = 2 and dfwithin = 9 is,

h.

Critical value of F for dfbetween = 2 and dfwithin = 9 and a-level of .05 is 4.26

Indicated as "C" in the above plot.

i.

Since the observed F is less than the critical value, we fail to reject the null hypothesis.

No, we fail to reject the null hypothesis.

j.

We cannot accept the research hypothesis.