Question

A researcher is interested in examining whether different practice strategies have an influence on the accuracy of basketball players' free throws. The researcher collected data from a sample of 18 participants who were classified in one of three groups: (1) 6 players who received no special practice, (2) 6 players who spent 30 minutes just imagining that they were shooting baskets, and (3) 6 players who spent 30 minutes actually practicing shooting baskets. After participating in the experiment for two weeks, the players attempted 10 free throws. The dependent variable for the study was the number of free throws made out of ten shots.  The researcher selected an alpha level of .01. The researcher found the following descriptive statistics:

1. Participants who received no special practice made an average of 3.50 free throws (SD = 1.38)

2. Participants who imagined shooting baskets made an average of 5.00 free throws (SD = 1.26)

3. Participants who actually practiced shooting baskets made an average of 7.00 free throws (SD = .89)

The researcher then ran an analysis of variance to determine whether at least two of these means were significantly different. The researcher found the following:

1. Sum of squares between groups = 37

2. Sum of squares within groups = 21.45

Based on this information, match each of the following with the correct answer:      

The degrees of freedom between groups?

1.96       18.50            3            6.36            1.43            15            2            19.93            12.94            17

The degrees of freedom within groups?

1.96       18.50            3            6.36            1.43            15            2            19.93            12.94            17

Mean squares between groups?

1.96       18.50            3            6.36            1.43            15            2            19.93            12.94            17

Mean squares within groups?   

1.96       18.50            3            6.36            1.43            15            2            19.93            12.94            17

The obtained F-value?   

1.96       18.50            3            6.36            1.43            15            2            19.93            12.94            17

The critical F-value?   

1.96       18.50            3            6.36            1.43            15            2            19.93            12.94            17

What should the researcher conclude?  

A: Fail to reject the null hypothesis; the groups did not differ significantly in the number of successful free throws that they completed.

B: Reject the null hypothesis; at least two groups differ in the average number of successful free throws that they completed.  

Answer 1

The problem given to us is a problem of One-Way Analysis of Variance since the output variable is a one-way classified variable.

Let our model be: y_ij=a_i+e_ij (i=1,2,3 j=1(1)6) (i=1 => the group that didn't receive any special practice.

  i=2 => the group that received imaginary practice

i=3 => the group that received actual practice)

where y_ij=  the free throw made by the jth participant in the ith group

a_i= the mean effect of the ith group on the free throw

e_ij= the random error corresponding to the free throw by the jth participant from the ith group. We assume e_ij~N(0, ) where is unknown.

Our hypothesis to be tested is : vs at least one pair for (i,j = 1,2,3).

The MSB (Mean square variation between the groups) and MSE (Mean square variation between the groups) is given as:

MSB = SSB/(k-1) = 37/2 = 18.5

MSE = SSE/(n-k) = 21.45/15= 1.43

Our test statistic for the testing problem for k=3 and n = (6*3) = 18 is :

F_observed = MSB/MSE ~ F_{0.01; 2,15} under Ho.

Our testing rule for this problem at 0.01 level of significance would be : Reject Ho, if   F_observed > F_{0.01; 2,15} (which is the tabulated value of this F-statistic at 0.01 level of significance)

So we see for our problem

F_observed = 12.94

F_{0.01; 2,15} = 11.34 (from F table)

So we see that   F_{observed >} F_{0.01; 2,15} . Hence we reject the null hypothesis and can conclude that there is statistically significant difference between at least two pairs of groups of different methods.

So, for this question

1. The degrees of freedom between the groups is (3-1) = 2
2. The degrees of freedom within the groups is (18-3) = 15
3. Mean square between the groups = 18.5
4. Mean square between the groups = 1.43
5. The obtained F value = 12.94
6. The critical F value = 11.34
7. The researcher should conclude that he should reject the null hypothesis: at least two groups differe in the average number of successful free throws they completed.