A university has an exclusive agreement with the coffee chain Monobeans. Several students and faculty complained about their lack of baked goods. Looking for a better provider of carbs and caffiene, the university management rents out for one trial semester a space within a newly built department to Monobeans' main competitor Old World Blend.
Based on data measured over many years, coffee stores on campus hada mean number of customers of 468.3 per day each, fluctuating with a standard deviation sigma of 207.5. At the end of the trial semester, Old World Blend's newly opened store featured a mean number of customers of 523.7, recorded over 78 business days in the semester.
1. Would you sign a new exclusive agreement with Old World Blend? Consider alpha = 0.05 as a level of significance, and assume that random fluctuations in number of customers may be either above or below Monobeans' average.
2. OWB signed the trial lease contract on the grounds that across the country, their mean number of customers is 11% higher than Monobeans. Given the local market on campus, how much power (1-Beta) did they have to prove their claim by one trial semester?
3. Regardless of how it ended up for OWB (result of question 2): to achieve 80% power, how many semesters of temporary lease would be necessary?
To answer these questions, we need to conduct hypothesis testing and calculate statistical power.
1. Hypothesis Testing:
The null hypothesis (H0) is that the mean number of customers for Old World Blend (OWB) is the same as the mean number of customers for Monobeans (MB). The alternative hypothesis (HA) is that the mean number of customers for OWB is significantly different from the mean number of customers for MB.
We can perform a two-sample t-test to compare the means of the two groups and determine if the difference is statistically significant.
Let's assume alpha (α) = 0.05, which means we are willing to accept a 5% chance of making a Type I error (rejecting a true null hypothesis).
The formula to calculate the t-value is:
t = (mean_OW - mean_MB) / sqrt((sigma^2_OW / n_OW) + (sigma^2_MB / n_MB))
Where:
mean_OW = mean number of customers for OWB (523.7)
mean_MB = mean number of customers for Monobeans (468.3)
sigma = standard deviation (207.5)
n_OW = number of business days for OWB (78)
n_MB = number of customers for Monobeans (unknown)
To calculate t, we need to know the number of customers for Monobeans. If we don't have that information, we cannot perform the hypothesis test and determine whether to sign a new exclusive agreement with OWB.
2. Statistical Power (1-Beta):
To calculate statistical power, we need additional information about the actual difference in means and the sample size. Specifically, we need to know the effect size (difference in means divided by the standard deviation) and the sample size.
Given that the mean number of customers for OWB is claimed to be 11% higher than Monobeans, we can calculate the effect size:
effect size = (mean_OW - mean_MB) / mean_MB
Assuming the local market on campus is representative, we can determine the statistical power within one trial semester. However, it is important to note that statistical power calculations usually require information about the sample size and the significance level. These details are not provided in the question, and therefore a specific power calculation cannot be performed.
3. Number of semesters for 80% power:
To achieve 80% power (typically considered acceptable), we need to determine the required sample size. However, we also need to know the effect size, standard deviation, and significance level to calculate the sample size using power analysis.
Unfortunately, the question does not provide the effect size or the significance level. Therefore, we cannot determine the required number of semesters of temporary lease to achieve 80% power.
In summary, without additional information, we cannot fully answer questions 1, 2, and 3.