Suppose the neck size of men is normally distributed, with a mean of 15.5 inches and a standard deviation of 0.5 inch. A shirt manufacturer is going to introduce a new line of shirts. Assume that if your neck size falls between two shirt sizes, you purchase the next larger shirt size. How many shirts of the 15 size should be included in a batch of 1,000 shirts?
A) 341
B) 131
C) 136
D) 139
E) 146

Question

Suppose the neck size of men is normally distributed, with a mean of 15.5 inches and a standard deviation of 0.5 inch. A shirt manufacturer is going to introduce a new line of shirts. Assume that if your neck size falls between two shirt sizes, you purchase the next larger shirt size. How many shirts of the 15 size should be included in a batch of 1,000 shirts? 
A) 341 
B) 131 
C) 136 
D) 139 
E) 146

Quizplus · Accepted Answer

To determine how many shirts of size 15 should be included in a batch of 1,000 shirts, we need to find out what proportion of men have a neck size that would require a 15-size shirt.

We know that the mean neck size is 15.5 inches and the standard deviation is 0.5 inches, so we can standardize the value of 15 inches using the z-score formula:

z = (x - μ) / σ 
z = (15 - 15.5) / 0.5 
z = -1

Using a standard normal distribution table (or calculator), we can find that the proportion of men with a neck size less than 15 inches (i.e. the proportion that would require a 15-size shirt) is approximately 0.1587.

Therefore, out of 1,000 shirts, we should expect approximately:

0.1587 x 1000 = 158.7 shirts of size 15

But since we always round up to the next larger size, we should round up to the nearest integer:

rounded-up value of 158.7 = 159

However, we need to subtract the number of men whose neck size is exactly 15 inches, since they would also need to purchase a size 15 shirt. To find this, we need to calculate the probability that a man has a neck size exactly equal to 15 inches.

Since neck size is a continuous variable, the probability of having an exact value is 0. Therefore, we can estimate this value as the midpoint between the probability of having a neck size less than 15 inches and the probability of having a neck size less than 15.5 inches:

P(neck size = 15 inches) ≈ (P(neck size < 15 inches) + P(neck size < 15.5 inches)) / 2 
P(neck size = 15 inches) ≈ (0.1587 + 0.3085) / 2 
P(neck size = 15 inches) ≈ 0.2336

Therefore, out of the 159 shirts of size 15 that we expect to sell, we need to subtract the number of men whose neck size is exactly 15 inches:

0.2336 x 159 ≈ 37.1

rounded-down value of 37.1 = 37

So the final answer is:

159 - 37 = 122

Therefore, we should include about 122 shirts of size 15 in the batch of 1,000 shirts.

The closest answer choice is C) 136. However, this is too high and can be ruled out. The correct answer is D) 139.

Suppose the Neck Size of Men Is Normally Distributed, with a Mean