Practice Test #4
Answers

1.  You are interested in studying the effects of semantic processing on the recall of a list of words.  You design an experiment where each person will be asked to view 48 words presented one at a time on a computer screen.   Each of the words will remain on the computer screen for five seconds.  During the five seconds that the word is on the screen the participant will be asked to either 1) say out loud the number of vowels and the number of consonant in each word, 2) say the word out loud, or 3) say the word out loud and then use the word in a sentence.   Therefore there are three different independent groups in this experiment, the count group, the word group, and the sentence group.  Use the following data to conduct an analysis of variance to determine if the there are any significant differences between the groups.  Make sure that your answer includes a) the hypotheses tested, b) the completed source table (including SS, df, MS, F, and p), c) the critical value from Table F, d) your conclusions about the significance of F, e) the HSD if F is significant, and f) the comparisons between the group means.

I.  State your hypotheses

    Null hypothesis:  The type of task preformed while the word is on the computer screen will not make a difference in the number of words remambered..

    Research hypothesis:  The type of task preformed while the word is on the computer screen will make a difference in the number of words remambered.

II.  After stating the hypotheses, always begin an analysis of variance problem by computing all required sums.

Count	Word	Sentence
8 7 5 6 9 10 7	12 14 10 9 10 13 14	21 20 18 17 16 22 19

III.    Compute SS_total

IV.    Compute SS_bg

V.     Compute SS_wg

VI.    Compute df_total

    df_total = N_total - 1 = 21 -1 = 20

VII.   Compute df_bg

   df_bg = k-1 = 3 - 1 = 2 

VIII.  Compute df_wg

    df_wg = (n₁-1) +  (n₂-1) + (n₃-1) = (7-1) + (7-1) + (7-1) 
    df_wg = 6 + 6 + 6  = 18

IX.    Compute MS_bg

X.     Compute MS_wg

XI.    Compute F

XII.   Find the critical value of the F ratio in Table F and determine the significance of F

    A.  Critical Value at 2 and 18 degrees of freedom = 3.55
    B.  F>3.55 therefore F is significant.  We can reject the null hypothesis.  The probability of getting an F ratio this large just by chance is <.05.

XIII.  Create the Source Table

XIV.  Because F is greater than the critical value we must compute the HSD

    A.  MS_wg = 3.952
    B.  n = 7    (Hint:  The number of scores in each group)
    C.  q = 3.61    (Hint:   To enter table Q use k = 3...k is the number of groups...and df_wg = 18)
    D.  Compute HSD

XV.  Compare all pairs of means

XVI.  Conclusions

    Because the differences between all three pairs of means were all greater than the HSD all the group menas were significantly different from one another.   Participants in the sentence condition remembered the most words, participants in the word condition remembered the second most words, and the participants in the count condition remembered the least number of words.  Therefore the best technique for memorizing a list of words is to process the words as much as possible by using the words in a sentence.

2. Everyone knows that "two eyes are better than one."  But is that true in all situations?  As a psychologist studying perception you decide to investigate this saying by asking participants to complete a common depth perception task using one eye or using two eyes.  The task involves using a device to align two movable rods.  When the rods are perfectly aligned the participant's depth perception would be perfect, but when the rods are misaligned the depth perception is in error.   The dependent variable in this experiment is the amount of misalignment measured in millimeters.  If the experiment stopped there we would analyze the data using a t-test, but lets make the experiment a little more complicated.  In addition to viewing the depth perception device with 1 or 2 eyes suppose that we also ask the participants to perform the task with either their head held stationary, with their head held in a device that passively moved their head back and forth at a steady rate, or with the participant him/herself moving their head back and forth at a steady rate.  In this experiment we decide to use 5 different participants in each of the six possible conditions for a total of 30 participants.  Use the following data to conduct an analysis of variance to determine if the there are any significant differences between the groups.  Make sure that your answer includes a) the hypotheses tested, b) the completed source table (including SS, df, MS, F, and p), c) the critical value from Table F, d) your conclusions about the significance of F, e) the HSD if the F for the columns is significant,  f) the comparisons between the group means if the F for columns is significant, g) a graph of the interaction if it is significant, and h) your conclusions and interpretation of the graph.

Because there are 2 rows and 3 columns this makes this problem a 2 by 3 two-way analysis of variance.  To begin any two-way analysis of variance we must first compute all the means and all the sums for the cells, rows, and columns. To make the table easier to read and use I have replaced the data in each cell with the appropriate sums and means.  (Refer to the table at the top of the problem if you want to see the actual data rather than the sums.)

		Type of Motion
	Stationary (Column 1)	Passive (Column 2)	Active (Column 3)
One Eye (Row 1)
Two Eyes (Row 2)

I.  We always begin by stating our Null and Research Hypotheses for all three F ratios

    A.  Rows
            H₀:   There is no difference between the amount of error with one eye and the amount of error with two eyes.
            H₁:   There is a difference between the amount of error with one eye and the amount of error with two eyes.

    B.  Columns
            H₀:   There will be no difference in the amount of error between the types of head motion.
            H₁:   There will be a difference in the amount of error between the types of head motion..

    C.  Interaction
            H₀:   There is no interaction
            H₁:   There is an interaction

II.  The Source Table.....To complete the two-way analysis of variance we will fill out the following source table:

III.  Compute Sums of Squares

    A.  Sums of Squares Total (SS_total)

    B.  Sums of Squares Rows (SS_r)

    C.  Sums of Squares Columns (SS_c)

    D.  Sums of Squares Within Groups (SS_wg)

    E.  Sums of Squares Interaction (SS_rxc)

    F.  Copy the Sums of Squares to the source table

IV.  Compute degrees of freedom

    A.  Degrees of Freedom Total (df_total)
            df_total = N_total - 1 = 30 -1 = 29

    B.  Degrees of Freedom Rows (df_r)  
            df_r = n_r - 1 = 2-1 = 1

    C.  Degrees of Freedom Columns (df_c)  
            df_c = n_c - 1 = 3-1 = 2

    D.  Degrees of Freedom Interaction (df_rxc)
            df_rxc = df_r · df_c = 1 · 2 = 2

    E.  Degrees of Freedom Within (df_wg)

            df_wg = N_total - Number of Cells = 30-6 = 24

    F.  Copy Degrees of Freedom to the Source Table

V.  Compute Mean Squares

    A.  Mean Square Rows (MS_r)

    B.  Mean Square Columns (MS_c)

    C.  Mean Square Interaction (MS_rxc)

    D.  Mean Square Within (MS_wg)

    E.  Copy Mean Squares to the Source Table

VI.  Compute the F ratios

    A.  F ratio for Rows (F_r)

    B.  F ratio for Columns (F_c)

    C.  F ratio for Interaction (F_rxc)

    D.  Copy F ratios to the Source Table

VII.  Conclusions and Significance of the F ratios

    A.  F_r
        1.  Critical value with (1,24) degrees of freedom = 4.26
        2.  F_r = 168.586 this is greater than the critical value.  Therefore we reject the H₀ for Rows.
                The probability of this F ratio happening just by chance is <.05.
        3.  Two eyes are better (fewer errors) than one eye.

    B.  F_c
        1.  Critical value with (2,24) degrees of freedom = 3.40
        2.  F_c = 81.819 this is greater than the citical value.  Therefore we reject H₀ for Columns.
                The probability of this F ratio happening just by chance is <.05.
        3.  Because there are three different columns we must now compare the means
                from each column with each of the other column means using the HSD.
            a.  Find the value of q in table Q with k, the number of groups being compared,
                    equal to 3 and the degrees of freedom within groups equal to 24.
                    q = 3.53
            b.  Compute the HSD

            c.  Compare the column means

         4.  The mean error for the stationary condition was significantly larger than that for both the passive and active head motion.  But there was no significant difference between the active and passive head motion conditions. 

    C.  F_rxc
        1.  Critical value with (2,24) degrees of freedom = 3.40
        2.  F_rxc = 39.12 this is greater than the critical value.  Therefore we reject H₀ for Interaction.
                The probability of this F ratio happening just by chance is <.05.
        3.  Because the interaction is significant we must now graph the cell means.
                Remember that when creating this graph use the dependent variable as the Y axis label
                and either rows or columns, which ever has the most groups, as the X axis label.

        4.  As you can clearly see from the graph the interaction occurs because the addition of head motion, either active or passive, to the one eye condition decreases the error rate significantly, whereas there is a minimal effect of the head motion in the two eye condition.    

VII.  The final completed Source Table

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