Answer: a) 0.9857 b) 0.0139 c) 0.5762 d) 0.9282 e) $55,792 f) $65,875
Step-by-step explanation: To solve these problems, we need to use the standard normal distribution. We'll convert the given values into z-scores and then use z-tables or a calculator to find the probabilities.
a. To find the probability that a merchandiser earns more than $71,000 per year:
First, we need to calculate the z-score using the formula:
z = (X - μ) / σ
where X is the value we're interested in, μ is the mean, and σ is the standard deviation.
Using the given values:
X = $71,000
μ = $60,000
σ = $5,000
z = (71000 - 60000) / 5000
z = 2.2 (rounded to two decimal places)
Now, we can use the z-table or a calculator to find the probability corresponding to a z-score of 2.2. Looking up this value, we find that the probability is approximately 0.9857.
Therefore, the probability that a merchandiser earns more than $71,000 per year is 0.9857 (rounded to four decimal places).
b. To find the probability that a merchandiser earns less than $49,000 per year:
Again, we calculate the z-score:
z = (X - μ) / σ
Using the given values:
X = $49,000
μ = $60,000
σ = $5,000
z = (49000 - 60000) / 5000
z = -2.2 (rounded to two decimal places)
Using the z-table or a calculator, we find that the probability corresponding to a z-score of -2.2 is approximately 0.0139.
Therefore, the probability that a merchandiser earns less than $49,000 per year is 0.0139 (rounded to four decimal places).
c. To find the probability that a merchandiser earns between $56,000 and $64,000 per year:
We need to find the z-scores for both values and then calculate the difference between their probabilities.
z1 = (56000 - 60000) / 5000 = -0.8 (rounded to two decimal places)
z2 = (64000 - 60000) / 5000 = 0.8 (rounded to two decimal places)
Using the z-table or a calculator, we find that the probability corresponding to a z-score of -0.8 is approximately 0.2119, and the probability corresponding to a z-score of 0.8 is approximately 0.7881.
To find the probability between these two values, we subtract the lower probability from the higher probability:
0.7881 - 0.2119 = 0.5762 (rounded to four decimal places)
Therefore, the probability that a merchandiser earns between $56,000 and $64,000 per year is 0.5762 (rounded to four decimal places).
d. To find the probability that a merchandiser earns between $51,000 and $69,000 per year:
We follow the same process as in part c:
z1 = (51000 - 60000) / 5000 = -1.8 (rounded to two decimal places)
z2 = (69000 - 60000) / 5000 = 1.8 (rounded to two decimal places)
Using the z-table or a calculator, we find that the probability corresponding to a z-score of -1.8 is approximately 0.0359, and the probability corresponding to a z-score of 1.8 is approximately 0.9641.
To find the probability between these two values, we subtract the lower probability from the higher probability:
0.9641 - 0.0359 = 0.9282 (rounded to four decimal places)
Therefore, the probability that a merchandiser earns between $51,000 and $69,000 per year is 0.9282 (rounded to four decimal places).
e. To find the average salary below which 20% of the merchandisers earn:
We need to find the z-score corresponding to the given percentile (20%).
Using the z-table or a calculator, we find that the z-score corresponding to the 20th percentile is approximately -0.8416.
Now we can calculate the salary using the formula:
X = μ + z * σ
Using the given values:
μ = $60,000
σ = $5,000
z = -0.8416
X = 60000 + (-0.8416) * 5000
X ≈ $55,792 (rounded to the nearest whole number)
Therefore, the average salary below which 20% of the merchandisers earn is approximately $55,792.
f. To find the average salary above which the top 12% of the merchandisers earn:
We need to find the z-score corresponding to the given percentile (88%, which is the complement of 12%).
Using the z-table or a calculator, we find that the z-score corresponding to the 88th percentile is approximately 1.1750.
Using the same formula as in part e:
X = μ + z * σ
Using the given values:
μ = $60,000
σ = $5,000
z = 1.1750
X = 60000 + 1.1750 * 5000
X ≈ $65,875 (rounded to the nearest whole number)
Therefore, the average salary above which the top 12% of the merchandisers earn is approximately $65,875.