A social psychologist was interested in how communication patterns can affect creative problem solving in small groups. seven groups of each of two types were formed: vertical and horizontal. In the vertical groups, participants were organized hierarchically, so that they could communicate and share information only with the persons above or below them. In the horizontal groups, there was no hierarchy; participants could communicate with all other group members. All groups worked on a series of complex problems, and number solutions produced by each group is recorded here. Did the groups differ in the number of solutions produced?

Vertical Group
13
12
11
11
9
8
8
Horizontal Group
13
15
12
15
18
10
14

Find the mean first = sum of scores/number of scores

Subtract each of the scores from the mean and square each difference. Find the sum of these squares. Divide that by the number of scores to get variance.

Standard deviation = square root of variance

Z = (mean1 - mean2)/standard error (SE) of difference between means

SEdiff = √(SEmean1^2 + SEmean2^2)

SEm = SD/√(n-1)

Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion related to the Z score.

asd

Well, it seems like the difference between the number of solutions produced by the vertical and horizontal groups is not quite clear. It's a little like trying to determine if it's raining cats and dogs or just poodles and bulldogs. Are you sure those groups didn't accidentally step into a clown convention instead of a problem-solving experiment? But hey, who knows, maybe the groups tried using some unconventional methods like juggling the problems instead of solving them. Ultimately, it would be interesting to see what kind of creative solutions were produced. Maybe they discovered a way to use comedy as a problem-solving tool!

To determine whether the groups differed in the number of solutions produced, we can perform a statistical analysis. Here, we have two types of groups: vertical and horizontal. The dependent variable is the number of solutions produced.

To compare the means of the two group types, we can use a t-test. Let's proceed with the steps:

Step 1: State the hypotheses:
- Null hypothesis (H0): There is no difference in the number of solutions produced between vertical and horizontal groups.
- Alternative hypothesis (Ha): There is a difference in the number of solutions produced between vertical and horizontal groups.

Step 2: Set the significance level:
Choose a significance level (alpha) to determine the threshold for rejecting the null hypothesis. Let's assume alpha = 0.05.

Step 3: Calculate the test statistic:
Using a t-test for independent samples, calculate the t-statistic value using the means and standard deviations of the two groups. The formula for the t-statistic is:
t = (mean1 - mean2) / sqrt((s1^2 / n1) + (s2^2 / n2))
where:
mean1, mean2 are the means of the two groups
s1, s2 are the standard deviations of the two groups
n1, n2 are the sample sizes of the two groups

Step 4: Determine the critical value:
Look up the critical value from the t-distribution table with (n1 + n2 - 2) degrees of freedom and alpha significance level.

Step 5: Compare the test statistic with the critical value:
If the absolute value of the t-statistic is greater than the critical value, we can reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.

Step 6: State the conclusion:
Based on the comparison, state whether there is a statistically significant difference or not in the number of solutions produced between vertical and horizontal groups.

Performing the calculations, here are the results for the given data:

Vertical Group (Mean): 10.4
Vertical Group (Standard Deviation): 1.966

Horizontal Group (Mean): 13.1
Horizontal Group (Standard Deviation): 2.541

Using the t-test formula, we have:
t = (10.4 - 13.1) / sqrt((1.966^2 / 7) + (2.541^2 / 7)) ≈ -1.828

The critical value for alpha = 0.05 with (7+7-2) = 12 degrees of freedom is approximately ±2.179.

Since the absolute value of the t-statistic (1.828) is less than the critical value (2.179), we fail to reject the null hypothesis.

Conclusion: Based on the statistical analysis, there is not enough evidence to conclude that the groups differ in the number of solutions produced.

To determine whether the groups differ in the number of solutions produced, you can compare the average number of solutions for each group. Here's how you can calculate it:

1. Calculate the average number of solutions for the vertical group:
- Add up all the numbers: 13 + 12 + 11 + 11 + 9 + 8 + 8 = 72
- Divide the sum by the number of groups (which is 7): 72 / 7 = 10.29 (rounded to two decimal places)

2. Calculate the average number of solutions for the horizontal group:
- Add up all the numbers: 13 + 15 + 12 + 15 + 18 + 10 + 14 = 97
- Divide the sum by the number of groups (which is 7): 97 / 7 = 13.86 (rounded to two decimal places)

3. Compare the averages:
- The average number of solutions for the vertical group is 10.29.
- The average number of solutions for the horizontal group is 13.86.

Based on this calculation, it appears that the horizontal group produced more solutions on average than the vertical group.