Bicycles arrive at a bike shop in boxes. Before they can be sold, they must be u

Question

Bicycles arrive at a bike shop in boxes. Before they can be sold, they must be unpacked, assembled and tuned (lubricated, adjusted, etc.). Based on past experience, the shop manager assumes the times for each setup phase are independent and follow a normal distribution. The means and standard deviations of the times (in minutes) are as shown:

Unpacking and Assembly Mean= 25.1 SD=2.4

Tuning Mean=12.3 SD=2.6

What is the mean total bicycle setup time?

Round to one decimal as needed.What is the standard deviation of the total bicycle setup time?

Round to two decimals as needed.

A customer decides to buy a bike like one of the display models but wants a different color. The shop has one, still in the box. The manager says they can have it ready in half an hour. What is the chance the bike will be set up and ready to go as promised?

Round to four decimals as needed.

The bicycle shop will be offering 2 specially priced children’s models at a sidewalk sale. The basic model will sell for $120 and the deluxe model for $150. Past experience indicates that sales of the basic model will have a mean of 5.4 bikes with a standard deviation of 1.2, and sales of the deluxe model will have a mean of 3.2 bikes with a standard deviation of 0.8 bikes. What’s the chance the shop will sell more deluxe than basic models?

Round to four decimals as needed.Assume the shop sells no deluxe models and the cost to set up the sidewalk sale is $500. What’s the chance that the sales from the basic models will cover these costs?

Round to four decimals as needed.

Explanation / Answer

1)

The mean total bicycle setup time = mean time for assembly & unpacking + mean time for tuning = 25.1min+ 12.3min = 37.4 minutes

2)

The standard deviation of the total bicycle setup time = standard deviation of time for assembly & unpacking + standard deviation of time for tuning = 2.4min+ 2.6min = 5 minutes

3)

Probability of total bicycle set up time less than 30 mins : P( X < 30)

X = 30, Mean = 37.4 , SD = 5

Z score = X - Mean / SD = 30 - 37.4 / 5 = -7.4 / 5 = -1.48

We need to find P ( Z < -1.48)

Referring to the standard normal Z Table :

P ( Z < -1.48 ) = 0.0694

Therefore, there is a probability of 0.0694 or 6.94% chance that the bike will be set up in less than half an hour.

4) The chance that the shop will sell more deluxe than basic :

X = 3.2 , mean = 5.4 , standard deviation = 1.2

Z = 3.2 - 5.4 / 1.2 = -2.2 / 1.2 = -1.83

P ( Z < -1.83 ) = 0.0336 = 3.36%

Ans: There is a 0.0336 probability that Deluxe will sell more than Basic.

5)

P(Sales from basic model = $500) :

Price of basic model = $120 , No of units of basic model needed to sell = 500 / 120 = 4.16

Therefore, the no of units needed to sell are 5 units to make up the sidewalk cost of $500.

We need to find P ( no of basic units sold > 5 ) :

Z = 5 - 5.4 / 1.2 = -0.4 / 1.2 = -0.33

P ( Z > -0.33 ) = 1 - P (Z < -0.33)

= 1 - 0.3707 = 0.6293

Therefore, there is a 0.6293 probabillity or 62.93% chance that the sales from the basic models will cover up the sidewalk cost.

Cheers!

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