Question

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?

A. 8640
B. 8864
C. 8940
D. 8849
E. 8664

Answer 1

The correct answer is D. To find out how many pairs of shoes to order between sizes 6.5 and 9, we calculate the corresponding z-scores and find the percentage of the population that falls between these sizes using the normal distribution. Multiplying this percentage by 15,000 pairs, we get 8849 pairs, which is the correct answer.

Step-by-step explanation:

To calculate the number of pairs of shoes to order between sizes 6.5 and 9, we need to use the properties of the normal distribution. The z-scores for sizes 6.5 and 9, with an average shoe size of 7.5 and standard deviation of 1.5, can be computed using the formula:

z = (X - μ) / σ

For size 6.5: z = (6.5 - 7.5) / 1.5 = -0.67
For size 9: z = (9 - 7.5) / 1.5 = 1.00

Next, we look up these z-scores in a standard normal distribution table or use a calculator with normal distribution functionality to find the corresponding percentile values. The percentile for z = -0.67 is approximately 0.2514 (25.14%), and for z = 1.00, it is approximately 0.8413 (84.13%). The difference between these two gives the proportion of the population that falls between these sizes:

0.8413 - 0.2514 = 0.5899 (58.99%)

Now, calculate the number of shoes:

15,000 pairs * 0.5899 = 8848.5 pairs

When rounded to the nearest whole number, we get 8849 pairs of shoes to order between sizes 6.5 and 9.