Question

question 14 please!

14) (20) Does the presence of others affect people's willingness to assume responsibility for the welfare of others? For example, does the presence of other people affect our reaction to potentially dangerous smoke seeping from a wall vent? To answer this question, a researcher measured how long in seconds) it took subjects to ask for help as smoke gradually filled a waiting room. The room was occupied by a subject plus "crowds" of either zero, two, or four experimental confederates (confederates are research assistants who pretend they are subjects) who acted as if they were regular subjects but ignored the smoke. Here are some summary data: zero confederates two confederates four confederates T = 15 T= 39 T= 54 n=3 n=3 n=3 SS = 14 S5 = 14 S5 = 8 a. (2) What is the independent variable in this experiment, and how many levels does it have? b. (2) What is the dependent variable in this experiment and how is it measured? c. (8) Complete the following summary table. SStotal has already been computed! Source SS df MS Between Within 11 Total 294 d. (1) State the critical value for the F-ratio Set a =.05. e. (1) What is your decision about Ho: Reject or retain? f. (6) Tukey's HSD = 6.14. Use this value to draw specific conclusions about this experiment. Please show any work you do.

Answer 1

Objective: To test whether presence of others affect people's willingness to assume responsibility for the welfare of others. From this statement,

a. The independent variable would be - No. of Confederates which is considered to be the treatment with three categories - 0, 2 and 4 confederates

b. The dependent variable would be - The time taken by subjects to ask for help.

c.

Source	SS	df	MS	F
Between	SSB	k-1	MSB=SSB / (k-1)	MSB / MSE
Within	SSW	N-k	MSE=SSE / (N-k)
Total	294	N-1

where,

Within Sum of Squares = SS₁ + SS₂ + SS₃

= 14 + 14 + 8

SSW = 36

Between Sum of Squares

$=\sum_{i=1}^{k}\frac{T_{i}^{2}}{n_{i}}-\frac{G^{2}}{N}$

where, G = 15 + 39 + 54 = 108, N = 3 + 3 + 3 = 9

$=\frac{15^{2}+39^{2}+54^{2}}{3}-\frac{108^{2}}{9}$

= 1554 - 1296

= 258

We may verify the result obtained be determining whether the sum of the explained (SSB) and unexplained variation (SSW) equals total variation (SST).

Here SSB + SSW = 258 + 36 = 294 which is equal to the Total Sum of Squares.

Next,

No. of treatments compared = k = 3

No. of observations = N = 9

Completing the ANOVA table,

Source	SS	df	MS	F
Between	258	3-1=2	MSB=258/2=129	129/6=21.5
Within	36	9-3=6	MSE=36/6=6
Total	294	9-1=8

The test statistic F ~ F_(2,6), with critical / rejection region given by, F > F_critical.

d. The critical value of F can be obtained from F tables by looking for the intersection of row and column corresponding to 2 and 6 df respectively at 5% level of significance.

TABLE E F critical values Degrees of freedom in the numerator р 2 3 4 5 6 7 8 9 .100 .050 .025 .010 .001 39.86 161.45 647.79 4052.2 405284 49.50 199.50 799.50 4999.5 500000 53.59 215.71 864.16 5403.4 540379 55.83 224.58 899.58 5624.6 562500 57.24 230.16 921.85 5763.6 576405 58.20 233.99 937.11 5859.0 585937 58.91 236.77 948.22 5928.4 592873 59.44 238.88 956.66 5981.1 598144 59.86 240.54 963.28 6022.5 602284 2 .100 .050 .025 .010 .001 8.53 18.51 38.51 98.50 998.50 9.00 19.00 39.00 99.00 999.00 9.16 19.16 39.17 99.17 999.17 9.24 19.25 39.25 99.25 999.25 9.29 19.30 39.30 99.30 999.30 9.33 19.33 39.33 99.33 999.33 9.35 19.35 39.36 99.36 999.36 9.37 19.37 39.37 99.37 999.37 9.38 19.38 39.39 99.39 999.39 3 .100 .050 .025 .010 .001 5.54 10.13 17.44 34.12 167.03 5.46 9.55 16.04 30.82 148.50 5.39 9.28 15.44 29.46 141.11 5.34 9.12 15.10 28.71 137.10 5.31 9.01 14.88 28.24 134.58 5.28 8.94 14.73 27.91 132.85 5.27 8.89 14.62 27.67 131.58 5.25 8.85 14.54 27.49 130.62 5.24 8.81 14.47 27.35 129.86 Degrees of freedom in the denominator 4 .100 .050 .025 .010 .001 4.54 7.71 12.22 21.20 74.14 4.32 6.94 10.65 18.00 61.25 4.19 6.59 9.98 16.69 56.18 4.11 6.39 9.60 15.98 53.44 4.05 6.26 9.36 15.52 51.71 4.01 6.16 9.20 15.21 50.53 3.98 6.09 9.07 14.98 49.66 3.95 6.04 8.98 14.80 49.00 3.94 6.00 8.90 14.66 48,47 5 .100 .050 .025 .010 .001 4.06 6.61 10.01 16.26 47.18 3.78 5.79 8.43 13.27 37.12 3.62 5.41 7.76 12.06 33.20 3.52 5.19 7.39 11.39 31.09 3.45 5.05 7.15 10.97 29.75 3.40 4.95 6.98 10.67 28.83 3.37 4.88 6.85 10.46 28.16 3.34 4.82 6.76 10.29 27.65 3.32 4.77 6.68 10.16 27.24 6 .100 .050 .025 .010 .001 3.78 5.99 8.81 13.75 35.51 3.46 5.14 7.26 10.92 27.00 3.29 4.76 6.60 9.78 23.70 3.18 4.53 6.23 9.15 21.92 3.11 4.39 5.99 8.75 20.80 3.05 4.28 5.82 8.47 20.03 3.01 4.21 5.70 8.26 19.46 2.98 4.15 5.60 8.10 19.03 2.96 4.10 5.52 7.98 18.69

F_critical = 5.14

e. Since, the test statistic F = 21.5 > 5.14 lies in the rejection region of the test, we may reject H₀ at 5% level. We may conclude that the data does provides sufficent evidence to support the claim that presence of others affect people's willingness to assume responsibility for the welfare of others.

f. Tukey's HSD is a post hoc test conducted on obtaining a signifcant difference in the means, to determine which pair differs significantly. It is computed using the formula:

$HSD = \frac{M_{i}-M_{j}}{\sqrt{\frac{MSW}{n}}}$

And the value thus obtained is compared with the critical value of HSD for (k,N-k) df and conclude the difference to be significant if HSD > Critical value. Obtaiing the critical value for (3,16) df from critical value table for Tukey's HSD:

df.k 2 3 4 5 6 7 8 9 10 5 3.64 4.60 5.22 5.67 6.03 6.33 6.58 6.80 6.99 6 3.46 4.34 4.90 5.30 5.63 5.90 6.12. 6.32 6.49 7 3.34 4.16 4.68 5.06 5.36 5.61 5.82 6.00 6.16 8 3.26 4.04 4.53 4.89 5.17 5.40 5.60 5.77 5.92 9 3.20 3.95 4.41 4.76 5.02 5.24 5.43 5.59 5.74 10 3.15 3.88 4.33 4.65 4.91 5.12 5.30 5.46 5.60 11 3.11 3.82 4.26 4.57 4.82 5.03 5.20 5.35 5.49 12 3.08 3.77 4.20 4.51 4.75 4.95 5.12 5.27 5.39 13 3.06 3.73 4.15 4.45 4.69 4.88 5.05 5.19 5.32 14 3.03 3.70 4.11 4.41 4.64 4.83 4.99 5.13 5.25 л 3.01 3.67 4.08 4.37 4.59 4.78 4.94 5.08 5.20 16 3.00 3.65 4.05 4.33 4.56 4.74 4.90 5.03 5.15 17 2.98 3.63 4.02 4.30 4.52 4.70 4.86 4.99 5.11

Since Tukey's HSD = 6.14 > 3.65, we may conclude that the means compared were significantly different.