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 of each of the following sizes should be included in a batch of 2,000 shirts? (Assume that if your neck size falls below 14 you will purchase a size 14 shirt. Assume that if your neck size falls above 17 you will not purchase a shirt. Note: The answer blanks may not add up to exactly 2,000 shirts. Round your answers to the nearest whole number.)
To determine how many of each shirt size should be included in the batch of 2,000 shirts, we need to consider the distribution of neck sizes and the rounding policy for shirt purchases.
First, let's identify the range of neck sizes that will result in a shirt purchase:
- Neck sizes below 14 inches will purchase a size 14 shirt.
- Neck sizes between 14 and 17 inches will purchase the next larger shirt size.
Next, we will calculate the proportions of people falling into these ranges by using the normal distribution.
1. Calculate the z-scores for the lower and upper limits of each range.
- For the lower limit of 14 inches:
z1 = (14 - 15.5) / 0.5
- For the upper limit of 17 inches:
z2 = (17 - 15.5) / 0.5
2. Use a standard normal distribution table or a calculator to find the corresponding probabilities for these z-scores.
- P(Z ≤ z1) gives the proportion of people with neck sizes below 14 inches.
- P(Z ≤ z2) gives the proportion of people with neck sizes below 17 inches.
3. Subtract the probabilities obtained from step 2 to find the proportion of people falling between the two limits.
- P(14 ≤ X ≤ 17) = P(Z ≤ z2) - P(Z ≤ z1)
4. Multiply the proportion from step 3 by 2000 to get the approximate number of people falling between the limits of 14 and 17 inches.
- Number of people = P(14 ≤ X ≤ 17) * 2000
5. Using the rounding policy mentioned in the question, determine the number of shirts for each size based on the calculated number of people.
- Number of size 14 shirts: Round down the number of people falling below 14 inches.
- Number of size 15 shirts: Round up the number of people between 14 and 15 inches.
- Number of size 16 shirts: Round up the number of people between 15 and 16 inches.
- Number of size 17 shirts: Round up the number of people between 16 and 17 inches.
Please provide the values of z1 and z2 to continue the calculation.