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 perform hypothesis testing and use statistical calculations. Here's how you can approach each question:
1. Hypothesis Testing:
To determine whether the mean number of customers at Old World Blend (OWB) is significantly different from Monobeans, we can perform a one-sample t-test. The null hypothesis (H0) is that the mean number of customers at OWB is the same as Monobeans, and the alternative hypothesis (H1) is that it is significantly different.
α (alpha) is set at 0.05 as the significance level, which means we are willing to accept a 5% chance of making a Type I error (rejecting H0 when it is true).
To calculate this test statistic, we will need the sample mean, sample standard deviation, number of observations, and the assumed population mean and standard deviation.
The formula for the t-test statistic is:
t = (sample mean - population mean) / (sample standard deviation / sqrt(n))
Here are the given values:
Monobeans:
- Sample mean = 468.3
- Sample standard deviation = 207.5
OWB (trial semester):
- Sample mean = 523.7
- Number of observations (business days) = 78
Since we do not have the population mean and standard deviation for OWB, we will assume that they are equal to Monobeans (based on the given data).
Now, we can calculate the t-test statistic and compare it to the critical t-value at α = 0.05 (two-tailed test) with degrees of freedom (df) = n - 1.
If the calculated t-value is outside the critical t-value range, we reject the null hypothesis and conclude that there is a significant difference in means.
2. Power Calculation:
To calculate the power (1-Beta) of the statistical test, we need the following values:
- Significance level (α)
- Effect size (the difference between the mean number of customers at OWB and Monobeans)
- Sample size for OWB (number of observations)
The formula to calculate power depends on the specific test used. Considering the given scenario, we would use a one-sample t-test for comparing means.
To estimate the effect size, OWB claims that their mean number of customers is 11% higher than Monobeans' mean. Convert this percentage increase into a decimal (11% = 0.11) and multiply it by Monobeans' mean.
Now, using tools like a power calculator or statistical software, we can input all these values to compute the power of the test.
3. Calculating the Number of Semesters:
To achieve 80% power, we need to determine the required sample size (number of semesters) for the temporary lease.
Using the same formula as question 2, we specify the desired power level (80%), set the significance level (α), and determine the effect size (the difference between means) based on OWB's claim.
By rearranging the power formula and solving for the sample size (number of semesters), we can find the answer.
Note: It's important to ensure that assumptions for each statistical test are met, such as normality and independence of data. Also, real-life factors, like the market's response to a new provider, may need to be considered beyond these statistical calculations.