To determine which store's daily sales are closest to the mean, we follow these steps:
1. Calculate the overall mean of the daily sales for all the stores.
Given the average daily sales:
- Store 1: 84
- Store 2: 75
- Store 3: 93
- Store 4: 104
- Store 5: 68
The overall mean (μ) is calculated as:
[tex]\[
\mu = \frac{84 + 75 + 93 + 104 + 68}{5} = \frac{424}{5} = 84.8
\][/tex]
2. Compute the z-score for the average daily sales of each store.
The z-score measures how far each store's average daily sales is from the overall mean in terms of standard deviations. The formula for the z-score is:
[tex]\[
z = \frac{X - \mu}{\sigma}
\][/tex]
where [tex]\(X\)[/tex] is the average daily sales for the store, [tex]\( \mu \)[/tex] is the overall mean, and [tex]\( \sigma \)[/tex] is the standard deviation.
3. Calculate the z-scores for each store:
- Store 1:
[tex]\[
z_1 = \frac{84 - 84.8}{2.2} \approx -0.36
\][/tex]
- Store 2:
[tex]\[
z_2 = \frac{75 - 84.8}{1.7} \approx -5.76
\][/tex]
- Store 3:
[tex]\[
z_3 = \frac{93 - 84.8}{1.5} \approx 5.47
\][/tex]
- Store 4:
[tex]\[
z_4 = \frac{104 - 84.8}{2.0} \approx 9.6
\][/tex]
- Store 5:
[tex]\[
z_5 = \frac{68 - 84.8}{1.8} \approx -9.78
\][/tex]
4. Determine the store with the z-score closest to 0.
The closer the z-score is to 0, the closer the store's sales are to the overall mean.
Comparing the absolute values of the z-scores, we find:
- |[tex]\(z_1\)[/tex]| = |−0.36| = 0.36
- |[tex]\(z_2\)[/tex]| = |−5.76| = 5.76
- |[tex]\(z_3\)[/tex]| = 5.47
- |[tex]\(z_4\)[/tex]| = 9.6
- |[tex]\(z_5\)[/tex]| = 9.78
The smallest absolute z-score is 0.36 for Store 1.
Conclusion:
Store 1's daily sales are closest to the mean.
