An art professor was interested in seeing what size group is best to prime the pump so to speak to encourage strangers to gather together in front of various exhibits at her art show, so she set up an experiment. In a municipal building off campus, she asked a group of two students to stand next to each other and stare intently at a wall decoration while simultaneously she asked a group of five students to do the same at the other end of the floor. For the next 30 minutes she observed passersby to see if they too looked at the decorations for more than five seconds or made comments about the artwork to members of the either group.
Here is what she found: She observed 9 people make comments or look at the decorations for more than five seconds with respect to group of two (G2). Down the hallway, she observed 26 people look at the decorations or make comments to the group of five (G5). However, she found that 34 people just passed by G5 without noticing the decoration or making a comment. A smaller number (31) just walked past G2. Because G5 had more passers than participants, she concluded that placing two of her students in front of art displays would encourage strangers to do the same. Was this a good decision? Please support your answer with an appropriate test.
To determine whether placing two students in front of the art displays would be more effective in encouraging strangers to gather than placing five students, we can conduct a hypothesis test.
First, let's define the hypothesis:
Null Hypothesis (H0): There is no difference in the proportion of passersby who make comments or look at the decorations for more than five seconds between the group of two students (G2) and the group of five students (G5).
Alternative Hypothesis (Ha): The proportion of passersby who make comments or look at the decorations for more than five seconds is higher for G2 than for G5.
To test this hypothesis, we can perform a one-sided z-test for proportions. We will compare the observed proportions of passersby who made comments or looked at the decorations for more than five seconds.
Let's calculate the z-value using the following formula:
z = (p1 - p2) / sqrt((p̂ * (1 - p̂) * (1/n1 + 1/n2)))
Where:
p1 = proportion of passersby who made comments or looked at the decorations for more than five seconds in G2
p2 = proportion of passersby who made comments or looked at the decorations for more than five seconds in G5
p̂ = (x1 + x2) / (n1 + n2)
x1 = number of passersby who made comments or looked at the decorations for more than five seconds in G2
x2 = number of passersby who made comments or looked at the decorations for more than five seconds in G5
n1 = total number of passersby in G2
n2 = total number of passersby in G5
Let's calculate the z-value:
p1 = 9 / (9 + 31) = 0.225
p2 = 26 / (26 + 34) = 0.433
n1 = 9 + 31 = 40
n2 = 26 + 34 = 60
p̂ = (9 + 26) / (40 + 60) = 0.25
z = (0.225 - 0.433) / sqrt((0.25 * (1 - 0.25) * (1/40 + 1/60)))
= -0.208 / sqrt(0.1875 * 0.8125 * (0.025 + 0.0167))
≈ -0.208 / sqrt(0.00303125)
≈ -0.208 / 0.055
z ≈ -3.7818
Now, we can look up the critical value for the significance level we desire (e.g., α = 0.05) in the standard normal distribution table. For a one-sided test, the critical value for α = 0.05 would be approximately 1.645.
Since the calculated z-value (-3.7818) is less than the critical value (1.645), we reject the null hypothesis. This means that there is sufficient evidence to conclude that the proportion of passersby who make comments or look at the decorations for more than five seconds is higher for G2 compared to G5.
Therefore, the art professor's decision to place two students in front of the art displays to encourage strangers to gather together was supported by the test.
To determine if the art professor's decision was a good one, we can conduct a hypothesis test using the observed data.
Let's define our null hypothesis (H0) as: The proportion of passersby who look at the decorations or make comments in the group of two (G2) is equal to or greater than the proportion in the group of five (G5).
And our alternative hypothesis (Ha) as: The proportion in G2 is less than the proportion in G5.
We can conduct a one-tailed test to determine if there is evidence to support the alternative hypothesis.
To test this hypothesis, we will use the proportions test. First, let's calculate the proportions of passersby who showed interest in the decorations for each group:
Proportion in G2 = 9/(9+31) = 0.225
Proportion in G5 = 26/(26+34) = 0.433
Next, we need to calculate the standard errors for each proportion:
Standard Error in G2 = sqrt((0.225 * (1-0.225))/(9+31)) = 0.070
Standard Error in G5 = sqrt((0.433 * (1-0.433))/(26+34)) = 0.058
Now, let's calculate the test statistic (z-score):
z = (Proportion in G2 - Proportion in G5) / sqrt(Standard Error in G2^2 + Standard Error in G5^2)
= (0.225 - 0.433) / sqrt(0.070^2 + 0.058^2)
= -4.415
Using a significance level of 0.05, the critical value for a one-tailed test is approximately -1.645 (obtained from a standard normal distribution table). Since the calculated test statistic (-4.415) is less than the critical value, we can reject the null hypothesis.
Therefore, there is evidence to support the art professor's decision that placing two students in front of art displays would encourage more passersby to look at the decorations or make comments compared to placing five students.