A bicycle shop plans to offer 2 specially priced children\'s models at a sidewal
ID: 3066166 • Letter: A
Question
A bicycle shop plans to offer 2 specially priced children's models at a sidewalk sale. The basic model will return a profit of $130 and the deluxe model $180. Past experience indicates that sales of the basic model will have a mean of 5.7 bikes with a standard deviation of 1.2, and sales of the deluxe model will have a mean of 3.7 bikes with a standard deviation of 0.6 bikes. The cost of setting up for the sidewalk sale is $300. Complete parts a) through d).
a) Define the random variables and use them to express the bicycle shop's net profit.
A. B=number of basic bicycles sold, Upper D equals number of deluxe bicycles sold, Profit equals 130 Upper B plus 180 Upper D minus 300
B. B=price of the basic bicycle, Upper D equals price of a deluxe bicycle, Profit equals 5.7 Upper B plus 3.7 Upper D
C. B=price of the basic bicycle, Upper D equals price of a deluxe bicycle, Profit equals 5.7 Upper B plus 3.7 Upper D minus 300
D. B= number of basic bicycles sold, Upper D equals number of deluxe bicycles sold, Profit equals 130 Upper B plus 180 Upper D
b) What is the mean of the net profit? The mean profit is $-------
c) What is the standard deviation of the net profit? The standard deviation is $----------- (Round to two decimal places as needed.)
d) Do you need to make any assumptions in calculating the mean?
A. Yes; the sales must be independent.
B. Yes; the sales must be dependent.
C. Yes; the variables must be discrete.
D. No; no assumptions are made in calculating the mean.
Do you need to make any assumptions in calculating the standard deviation?
A. Yes; the variables must be discrete.
B. Yes; the sales must be dependent.
C. Yes; the sales must be independent.
D. No; no assumptions are made in calculating the standard deviation.
Explanation / Answer
a) Define the random variables and use them to express the bicycle shop's net profit.
If Upper B= number of basic bicycles sold, Upper D equals number of deluxe bicycles sold, then the profit is
Profit = Net Profit of basic bicycles * number of basic bicycles sold + Net Profit of deluxe bicycles * number of deluxe bicycles sold - cost of setting up for the sidewalk sale
P = 130 * Upper B + 180 * Upper D - 300
So, the correct option is,
A. B=number of basic bicycles sold, Upper D equals number of deluxe bicycles sold, Profit equals 130 Upper B plus 180 Upper D minus 300
b) What is the mean of the net profit? The mean profit is $-------
Mean of net Profit = E[P] = E[130 * Upper B + 180 * Upper D - 300]
= 130 * E[Upper B] + 180 * E[Upper D] - 300
= 130 * 5.7 + 180 * 3.7 - 300
= 1107
The mean profit is $1107
c) What is the standard deviation of the net profit? The standard deviation is $-----------
Var[P] = Var[130 * Upper B + 180 * Upper D - 300]
= 1302 * Var[Upper B] + 1802 * Var[Upper D] + Var[300]
= 1302 * 1.22 + 1802 * 0.62 + 0
= 36000
Standard deviation of net profit = sqrt(36000) = 189.74
The standard deviation is $189.74
d) Do you need to make any assumptions in calculating the mean?
D. No; no assumptions are made in calculating the mean.
Do you need to make any assumptions in calculating the standard deviation?
We need assumption of indepndence of sales. If they are dependent, we have to take covariance between sales of two types of bicycles into account to calculate the variance/standard deviation of net profit.
C. Yes; the sales must be independent.
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