Mens shirt sizes are determined by neck sizes. suppose the mens nexck sizes are approximatley normally distributed with mean 15.7 inches and standard deviation 0.7 inch.retailer sells men shirts in S,M,L,XL where the shirt sizes are defined in the table below.
S: 14 ≤ neck size <15
M: 15 ≤ neck size <16
L: 16 ≤ neck size <17
XL:17 ≤ neck size <18
because the retailer only stocks the sizes listed above what proportion of customers will find the retailer does not carry any shirts in their size?
To find the proportion of customers who will find that the retailer does not carry any shirts in their size, we will calculate the probability that a customer's neck size falls outside the given ranges.
Let's first calculate the z-scores for the lower and upper bounds of each size range:
For size S:
Lower bound: z = (14 - 15.7) / 0.7
Upper bound: z = (15 - 15.7) / 0.7
For size M:
Lower bound: z = (15 - 15.7) / 0.7
Upper bound: z = (16 - 15.7) / 0.7
For size L:
Lower bound: z = (16 - 15.7) / 0.7
Upper bound: z = (17 - 15.7) / 0.7
For size XL:
Lower bound: z = (17 - 15.7) / 0.7
Upper bound: z = (18 - 15.7) / 0.7
Now, we can use the standard normal distribution table or a calculator to find the probabilities associated with these z-scores.
For size S:
P(neck size < 14) = P(z < (14 - 15.7) / 0.7)
P(neck size > 15) = 1 - P(z < (15 - 15.7) / 0.7)
For size M:
P(neck size < 15) = P(z < (15 - 15.7) / 0.7)
P(neck size > 16) = 1 - P(z < (16 - 15.7) / 0.7)
For size L:
P(neck size < 16) = P(z < (16 - 15.7) / 0.7)
P(neck size > 17) = 1 - P(z < (17 - 15.7) / 0.7)
For size XL:
P(neck size < 17) = P(z < (17 - 15.7) / 0.7)
P(neck size > 18) = 1 - P(z < (18 - 15.7) / 0.7)
Finally, we can calculate the proportion of customers who will find that the retailer does not carry any shirts in their size by summing up the probabilities for all the size ranges:
Proportion = P(neck size < 14) + P(neck size > 15) + P(neck size < 15) + P(neck size > 16) + P(neck size < 16) + P(neck size > 17) + P(neck size < 17) + P(neck size > 18)
Please note that the neck sizes outside the given range will not have any corresponding size options available, so they are excluded from consideration here.
To find the proportion of customers who will not find the retailer's size, we need to calculate the probability that a customer's neck size falls outside the ranges of the available shirt sizes.
Let's consider each size range individually:
For size S: The range is 14 ≤ neck size < 15. In terms of standard deviations from the mean, this corresponds to (14 - 15.7) / 0.7 ≤ Z < (15 - 15.7) / 0.7, which simplifies to -2.43 ≤ Z < -1.
To find the probability of Z falling within this range, we can use a standard normal table or a calculator. From the standard normal table or calculator, we find that the probability of Z falling in this range is approximately 0.0228.
For size M: The range is 15 ≤ neck size < 16. The corresponding standard deviation range is (-1, 0), so the probability of Z falling within this range is 0.3413 (which is the difference between the cumulative probabilities of Z = 0 and Z = -1).
For size L: The range is 16 ≤ neck size < 17. The corresponding standard deviation range is (0, 1), so the probability of Z falling within this range is again 0.3413.
For size XL: The range is 17 ≤ neck size < 18. The corresponding standard deviation range is (1, 2.43), so the probability of Z falling within this range is approximately 0.0228.
Now, let's calculate the probability of a customer's neck size falling outside the available ranges:
P(neck size ≠ S, M, L, XL) = P(neck size < 14) + P(neck size ≥ 18)
To calculate these probabilities, we need to find the corresponding Z scores for neck sizes 14 and 18:
Z1 = (14 - 15.7) / 0.7
Z2 = (18 - 15.7) / 0.7
Using these Z scores, we can find the probabilities using the standard normal table or calculator:
P(neck size < 14) = P(Z < Z1)
P(neck size ≥ 18) = P(Z ≥ Z2)
Calculating the probabilities using the standard normal table or calculator, we find:
P(neck size ≠ S, M, L, XL) ≈ P(Z < -2.43) + P(Z ≥ 2.43) ≈ 0.0072 + 0.0072 = 0.0144
Therefore, the proportion of customers who will find that the retailer does not carry any shirts in their size is approximately 0.0144, or about 1.44%.
P(X < 14.0) = P((x - 15.7) / 0.7) < (14.0 - 15.7) / 0.7) = P(Z < -2.43) = 0.0076
P(X < 18.0) = P((x - 15.7) / 0.7) < (18.0 - 15.7) / 0.7) = P(Z < 3.29) = 0.9995
or use calc by typing in normalcdf(-99,-2.43) and normalcdf(-99,3.29)you get the same probabilities of .0076 and .9995
P(14.0 < x < 18.0) = 0.9995 - 0.0076 = .9919 have their shirts in stock, 1-.9919 = .0081 do not have shirts in stock