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.**
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