How Do Sales Representatives' Earnings Vary in a Canadian Clothing Chain?

[adeel]

A clothing store has offices in Toronto, Montreal, Vancouver and Calgary, and each office has 25 sales representatives. The weekly sales for the sales representatives are normally distributed with a mean of $12,000 and a standard deviation of $ 2,000. Within what range about the mean is the probability 0.9973 that:

1. A given representatives weekly sales will fall
2. The average weekly sales per representative of the Montreal office will fall
3. The average weekly sales per representative of the company will fall.

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I have the answers, but I have no idea from where to even begin. If someone can help me understand and explain this id be very appreciative.

 

[juvenal]

I don't quite understand the wording of (1) or the rest of the question about the range of the mean enough to give a complete answer.

If a representative has sales of x for week one, then the probability of him having less than x sales for the next week is the integral of the normal distribution from -infinity to x.

If you give the answer to (1) maybe one could work backward and understand the wording better.

 

[tacman]

Sure, I'd be happy to help you understand this problem and its solutions. Let's break it down step by step.

1. A given representative's weekly sales will fall:
The probability that a given representative's weekly sales will fall within a certain range is 0.9973. This means that there is a 99.73% chance that the representative's weekly sales will fall within this range. To determine this range, we need to use the mean and standard deviation provided.

The mean is $12,000 and the standard deviation is $2,000. Since we are looking for a range, we need to find the upper and lower limits. To do this, we can use the following formula:

Upper limit = mean + (3 x standard deviation)
Lower limit = mean - (3 x standard deviation)

Plugging in the numbers, we get:
Upper limit = $12,000 + (3 x $2,000) = $18,000
Lower limit = $12,000 - (3 x $2,000) = $6,000

Therefore, the range around the mean of $12,000 where the probability is 0.9973 is $6,000 to $18,000.

2. The average weekly sales per representative of the Montreal office will fall:
To find the range for the average weekly sales per representative of the Montreal office, we need to use the same formula as above, but with a different mean. The mean for the Montreal office is not given, but we can calculate it by dividing the total weekly sales for the office by the number of representatives (25).

Total weekly sales for the Montreal office = $12,000 x 25 = $300,000
Mean for the Montreal office = $300,000 / 25 = $12,000

Using the same formula as before, we get:
Upper limit = $12,000 + (3 x $2,000) = $18,000
Lower limit = $12,000 - (3 x $2,000) = $6,000

Therefore, the range around the mean of $12,000 for the average weekly sales per representative of the Montreal office is $6,000 to $18,000.

3. The average weekly sales per representative of the company will fall:
To find the range for the average weekly sales per representative of the company, we need to use the same formula as above