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The average salary of merchandisers is $60000 per year with a standard deviation of $5000. a. What is the probability that a merchandiser earns more than $71000 per year? (Round z-score computation to 2 decimal places ond the final answer to 4 decimal places.) Probability b. What is the probability that a merchandiser earns less than $49 o00 per year? (Round z-score computation to 2 decimal places and the final answer to 4 decimal places.) Probability c. What is the probability that a merchandiser earns between $56000 and $64000 per year? (Round z-score computation to 2 decimal places and the final answer to 4 decimal places.) Probability d. What is the probability that a merchandiser will earn between $51000 and $69 000 per year? (Round z-score computation to 2 decimal places and the final answer to 4 decimal places.) Probability e. What is the average salary below which 20% of the merchandisers earn? (Round the final answer to the nearest whole number) Salary f. What is the average salary above which the top 12% of the merchandisers earn? (Round the final answer to the nearest whole number.)

1 Answer

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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.

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