Do “props” make a difference when researchers interact with their subjects? Emily and Madi asked 100 people if they thought buying coffee at Starbucks was a waste of money. 50 of the subjects were asked while Emily and Madi were holding cups from Starbucks, and the other 50 of the subjects were asked when the girls were empty handed. The choice of holding or not holding the cups was determined at random for each subject. When holding the cups, 19 of 50 subjects agreed that buying coffee at Starbucks was a waste of money. When they weren’t holding the cups, 23 of 50 subjects said it was a waste of money. Calculate and interpret a 90% confidence interval for the difference in the proportion of people like the ones in this experiment who would say that buying coffee from Starbucks is a waste of money when asked by interviewers holding or not holding a cup from Starbucks. Use the 4 step process to complete your answer (State, Plan, Do, Conclude.)
The correct answer and explanation is :
Let’s go through the 4-step process for constructing a 90% confidence interval for the difference in the proportions of people who think buying coffee at Starbucks is a waste of money when interviewed by Emily and Madi holding a cup versus when they are not holding a cup.
1. State (the problem)
We are interested in estimating the difference in proportions between two groups of subjects: one group is asked when the interviewers (Emily and Madi) are holding Starbucks cups, and the other group is asked when the interviewers are not holding cups. Specifically, we are trying to calculate the 90% confidence interval for the difference in the proportions of subjects who agree that Starbucks coffee is a waste of money in these two conditions.
- Group 1: Emily and Madi holding cups.
- Group 2: Emily and Madi not holding cups.
We denote the proportions as follows:
- ( p_1 = \text{proportion of subjects who agree when holding a cup} )
- ( p_2 = \text{proportion of subjects who agree when not holding a cup} )
2. Plan (choose the method)
Since we are comparing two independent proportions, we will use the formula for the confidence interval for the difference in proportions. The formula is:
[
\text{CI} = \left( (p_1 – p_2) \right) \pm Z_{\alpha/2} \cdot \sqrt{\frac{p_1(1-p_1)}{n_1} + \frac{p_2(1-p_2)}{n_2}}
]
Where:
- ( p_1 = \frac{19}{50} = 0.38 ) (the proportion for the cup group),
- ( p_2 = \frac{23}{50} = 0.46 ) (the proportion for the no-cup group),
- ( n_1 = n_2 = 50 ) (sample size for each group),
- ( Z_{\alpha/2} = Z_{0.05} = 1.645 ) for a 90% confidence interval.
3. Do (perform the calculations)
Let’s plug in the values:
- ( p_1 = 0.38 ),
- ( p_2 = 0.46 ),
- ( n_1 = n_2 = 50 ),
- ( Z_{\alpha/2} = 1.645 ).
First, calculate the difference in proportions:
[
p_1 – p_2 = 0.38 – 0.46 = -0.08
]
Next, calculate the standard error (SE) for the difference in proportions:
[
SE = \sqrt{\frac{0.38(1 – 0.38)}{50} + \frac{0.46(1 – 0.46)}{50}} = \sqrt{\frac{0.38 \times 0.62}{50} + \frac{0.46 \times 0.54}{50}}
]
[
SE = \sqrt{\frac{0.2356}{50} + \frac{0.2484}{50}} = \sqrt{0.004712 + 0.004968} = \sqrt{0.00968} = 0.0984
]
Now, calculate the margin of error (ME):
[
ME = 1.645 \times 0.0984 = 0.161
]
Thus, the confidence interval is:
[
CI = -0.08 \pm 0.161
]
This gives the interval:
[
CI = (-0.08 – 0.161, -0.08 + 0.161) = (-0.241, 0.081)
]
4. Conclude (interpret the results)
We can interpret the 90% confidence interval for the difference in proportions as follows:
We are 90% confident that the true difference in the proportions of subjects who believe buying Starbucks coffee is a waste of money (between those who are asked when holding a cup and those who are asked when not holding a cup) is between -0.241 and 0.081. This means that the presence of the Starbucks cup may either decrease or increase the likelihood of subjects agreeing that buying Starbucks coffee is a waste of money, but based on this interval, we cannot conclusively say that the effect is large enough to be statistically significant.
Since 0 is included in the confidence interval, there is not enough evidence to suggest a significant difference in the proportion of subjects’ opinions based on whether Emily and Madi are holding a cup or not. Therefore, the “props” (in this case, the cups) do not appear to have a significant impact on the participants’ responses.