The CVS Pharmacy located on US 17 in Murrells Inlet has been one of the busiest pharmaceutical retail stores in South Carolina for many years. To try and capture more business in the area, CVS top management opened another store about 6 miles west on SC 707. After a few months, CVS management decided to compare the business volume at the two stores. One way to measure business volume is to count the number of cars in the store parking lots on random days and times. The results of the survey from the last 3 months of the year are reported below. To explain, the first observation was on October 2 at 20:52 military time (8:52 p.m.). At that time, there were four cars in the US 17 lot and nine cars in the SC 707 lot. At the 0.05 significance level, is it reasonable to conclude that, based on vehicle counts, the US 17 store has more business volume than the SC 707 store? Vehicles Count Date Time US 17 SC 707 Oct 2 20:52 4 9 Oct 11 19:30 5 7 Oct 15 22:08 9 12 Oct 19 11:42 4 5 Oct 25 15:32 10 8 Oct 26 11:02 9 15 Nov 3 11:22 13 7 Nov 5 19:09 20 3 Nov 8 15:10 15 14 Nov 9 13:18 15 11 Nov 15 22:38 13 11 Nov 17 18:46 16 12 Nov 21 15:44 17 8 Nov 22 15:34 15 3 Nov 27 21:42 20 6 Nov 29 9:57 17 13 Nov 30 17:58 5 9 Dec 3 19:54 7 13 Dec 15 18:20 11 6 Dec 16 18:25 14 15 Dec 17 11:08 8 8 Dec 22 21:20 10 3 Dec 24 15:21 4 6 Dec 25 20:21 7 9 Dec 30 14:25 19 4 Click here for the Excel Data File State the decision rule: H0: μ US 17 – μ SC 707 = μd ≤ 0 H1: μd > 0. (Round your answer to 3 decimal places.) Compute the value of the test statistic. (Round your answer to 3 decimal places.)
The Correct Answer and Explanation is:
To test whether the US 17 store has more business volume than the SC 707 store, we will perform a paired t-test. The hypothesis is:
- Null Hypothesis (H₀): The difference in the number of cars between the two stores is zero or negative. In other words, the business volume at US 17 is equal to or less than the SC 707 store. H0:μd≤0H_0: \mu_d \leq 0H0:μd≤0
- Alternative Hypothesis (H₁): The US 17 store has a greater business volume, i.e., a positive difference in the number of cars. H1:μd>0H_1: \mu_d > 0H1:μd>0
Where:
- μd\mu_dμd is the population mean of the differences between the counts at US 17 and SC 707.
Steps to compute the test statistic:
- Find the differences between the vehicle counts at the two locations for each observation: di=(US 17 count)−(SC 707 count)d_i = (\text{US 17 count}) – (\text{SC 707 count})di=(US 17 count)−(SC 707 count) These differences are calculated for each observation.
- Compute the sample mean and standard deviation of these differences:
- Let dˉ\bar{d}dˉ represent the sample mean of the differences.
- Let sds_dsd represent the sample standard deviation of the differences.
- Calculate the t-statistic using the formula: t=dˉsd/nt = \frac{\bar{d}}{s_d / \sqrt{n}}t=sd/ndˉ Where:
- dˉ\bar{d}dˉ is the mean difference.
- sds_dsd is the standard deviation of the differences.
- nnn is the number of paired observations (here, there are 20 observations).
- Degrees of freedom: Since we are using a t-test for paired samples, the degrees of freedom is n−1n – 1n−1, which is 20−1=1920 – 1 = 1920−1=19.
- Critical value: From a t-distribution table or using software, find the critical t-value for a one-tailed test at a significance level of 0.05 with 19 degrees of freedom.
- Decision rule:
- If the calculated t-statistic is greater than the critical value, reject the null hypothesis H0H_0H0, indicating that the US 17 store has significantly more business volume than the SC 707 store.
- If the calculated t-statistic is less than or equal to the critical value, fail to reject H0H_0H0, meaning there isn’t sufficient evidence to conclude that US 17 has more business volume.
Sample Calculation of Differences
Let’s take a few of the differences as examples:
| Date | US 17 | SC 707 | Difference (d) |
|---|---|---|---|
| Oct 2 | 4 | 9 | -5 |
| Oct 11 | 5 | 7 | -2 |
| Oct 15 | 9 | 12 | -3 |
| Oct 19 | 4 | 5 | -1 |
| Oct 25 | 10 | 8 | 2 |
Continue calculating the differences for all observations.
Once you’ve calculated all the differences, compute the mean dˉ\bar{d}dˉ and the standard deviation sds_dsd. Then, use the formula for the t-statistic.
Interpretation:
If the test statistic is larger than the critical value, we reject the null hypothesis and conclude that the US 17 store has more business volume based on vehicle counts. If the test statistic is smaller, we do not have enough evidence to support this conclusion.
