Question 5.7: Clothing Transactions Purchase Value Study Assume that the p......
Clothing Transactions Purchase Value Study
Assume that the purchase value of transactions, x, at a national clothing store such as Edgars, is normally distributed with a mean of R244 and a standard deviation of R68.
(a) What is the minimum purchase value of transactions for the highest-spending 15% of clothing store customers?
(b) What purchase value of transactions separates the lowest-spending 20% of clothing store customers from the remaining customers?
(a) This question requires that a specific transaction value, x, be identified such that above this value the top 15% of high-spending customers are found.
Step 1: Sketch the normal curve, showing the transaction value, x, that identifies the area corresponding to the 15% of highest spenders. This is shown in Figure 5.14.
Step 2: From the z-table, find the z-value that corresponds to an area of 0.15 in the top tail of the standard normal distribution.
To use the z-table, the appropriate area to read off is 0.5 – 0.15 = 0.35 (i.e. the middle area). The closest z-value is 1.04.
Step 3: Find the x-value associated with the identified z-value in Step 2.
Substitute z = 1.04, μ = 244 and σ = 68 into the z transformation Formula 5.6, and solve for x.
z=\frac{x-\mu }{\sigma } (Solve for x: x = μ + zσ)
1.04 = \frac{x-244 }{68 }
x = 244 + (1.04 × 68) = 244 + 70.72 = R314.72
Thus the highest-spending 15% of clothing store customers spend at least R314.72 per transaction.
(b) This question requires that a specific transaction value, x, be identified such that the area below this value is 20%, which represents the lowest spending 20% of clothing store customers.
Step 1: Sketch the normal curve, showing the transaction value, x, that identifies the area corresponding to the 20% of lowest spenders. This is shown in Figure 5.15.
Step 2: From the z-table, find the z-value that corresponds to an area of 0.20 in the bottom tail of the standard normal distribution.
To use the z-table, the area that must be found is 0.5 – 0.20 = 0.30 (i.e. the middle area).
The closest z-value read off the z-table is 0.84. However, since the required z-value is below its mean, the z-value will be negative. Hence z = −0.84.
Step 3: Find the x-value associated with the identified z-value in Step 2.
Substitute z = −0.84, μ = 244 and σ = 68 into the z transformation
Formula 5.6 and solve for x.
–0.84 = \frac{x-244 }{68 }
x = 244 + (−0.84 × 68) = 244 – 57.12 = R186.88
Thus the lowest-spending 20% of clothing store customers spend at most R186.88 per transaction.
