SOLUTION: Normally distributed observations such as a person's weight, height, or shoe size occur quite frequently in nature. Business people who are aware of this use it to their advantage.

Question 1168332: Normally distributed observations such as a person's weight, height, or shoe size occur quite frequently in nature. Business people who are aware of this use it to their advantage. A purchasing agent for a large retailer buying 15,000 pairs of women's shoes used the normal curve to decide on the order quantities for the various sizes. If women's average shoe size is 7.5 with a standard deviation of 1.5, how many pairs should be ordered between sizes 6.5 and 9?
Answer by CPhill(1959) (Show Source): You can put this solution on YOUR website!
Let's solve this problem step-by-step using the normal distribution.
**1. Understand the Problem**
* Women's shoe sizes are normally distributed.
* Mean (μ) = 7.5
* Standard deviation (σ) = 1.5
* Total number of pairs ordered = 15,000
* We need to find the number of pairs to order between sizes 6.5 and 9.
**2. Convert Shoe Sizes to Z-scores**
We need to find the z-scores corresponding to shoe sizes 6.5 and 9.
* **Z-score for 6.5:**
* z = (X - μ) / σ
* z = (6.5 - 7.5) / 1.5
* z = -1 / 1.5
* z = -2/3 ≈ -0.67
* **Z-score for 9:**
* z = (X - μ) / σ
* z = (9 - 7.5) / 1.5
* z = 1.5 / 1.5
* z = 1
**3. Find the Probabilities**
* **Probability for z = -0.67:**
* Using a z-table or calculator, the cumulative probability for z = -0.67 is approximately 0.2514.
* **Probability for z = 1:**
* Using a z-table or calculator, the cumulative probability for z = 1 is approximately 0.8413.
**4. Find the Probability Between 6.5 and 9**
* The probability of a shoe size being between 6.5 and 9 is the difference between the two cumulative probabilities.
* P(6.5 < X < 9) = P(z < 1) - P(z < -0.67)
* P(6.5 < X < 9) = 0.8413 - 0.2514
* P(6.5 < X < 9) = 0.5899
**5. Calculate the Number of Pairs**
* Multiply the probability by the total number of pairs ordered.
* Number of pairs = 0.5899 * 15,000
* Number of pairs = 8848.5
**6. Round to the Nearest Whole Number**
* Since we can't order fractions of pairs, round to the nearest whole number.
* Number of pairs ≈ 8849
**Therefore, the purchasing agent should order approximately 8849 pairs of shoes between sizes 6.5 and 9.**

RELATED QUESTIONS
Suppose that the weight X of individual male patients registered at a certain diet clinic (answered by Boreal)

shoe sizes and foot length are related by the formula S = 3F - 24, where S represents the (answered by Mathtut)

Hits on a new business website occur quite frequently. They occur randomly and... (answered by Boreal)

The weight of a sophisticated running shoe is normally distributed with a mean of 12... (answered by Boreal)

Variables such as height and weight are generally assumed to be normally distributed (for (answered by stanbon)

Height and Weight Suppose that the weight of a person is directly proportional to the... (answered by mananth,stanbon)

4. Given: μ = 100 and σ = 30 for a normally distributed population of... (answered by ewatrrr)

A shoe manufacturer collected data regarding men's shoe sizes and found that the... (answered by ikleyn)

A particular type of vacuum-packed coffee packet contains an average of 16 ounces. It has (answered by CPhill)