Question

A music school has budgeted to purchase 3 musical instruments. They plan to purchase a piano costing $4,000, a guitar costing $600, and a drum set costing $750. The mean cost for a piano is $4,500 with a standard deviation of $2,500. The mean cost for a guitar is $500 with a standard deviation of $200. The mean cost for drums is $850 with a standard deviation of $100. (Enter your answers to two decimal places.)
1.) How many standard deviations above or below the average piano cost is the piano?

2.) How many standard deviations above or below the average guitar cost is the guitar?

3.) How many standard deviations above or below the average drum set cost is the drum set?

4.) Which cost is the lowest, when compared to other instruments of the same type?

5.) Which cost is the highest when compared to other instruments of the same type?

Answer 1

Answer:

1) $z=\frac{4000-4500}{2500}=-0.2$

That means 0.2 deviations below the mean

2) $z=\frac{600-500}{200}=0.5$

That means 0.5 deviations above the mean

3) $z=\frac{750-850}{100}=-1$

That means 1 deviations below the mean

4) For this case the lowest instrument compared to the other instruments of the same type is the drum set since have the lowest z score z =-1

5) For this case the highest instrument compared to the other instruments of the same type is guitar since have the highest z score z =0.5

Step-by-step explanation:

They plan to purchase a piano costing $4,000, a guitar costing $600, and a drum set costing $750.

Previous concepts

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".  

Part 1

Let X the random variable that represent the cost of for a piano, and for this case we know this  

Where $\mu=4500$ and $\sigma=2500$

And the best way to solve this problem is using the normal standard distribution and the z score given by:

$z=\frac{x-\mu}{\sigma}$

If we replace the formula for X= 4000 we got:

$z=\frac{4000-4500}{2500}=-0.2$

That means 0.2 deviations below the mean

Part 2

Let X the random variable that represent the cost of for a guitar, and for this case we know this  

Where $\mu=500$ and $\sigma=200$

And the best way to solve this problem is using the normal standard distribution and the z score given by:

$z=\frac{x-\mu}{\sigma}$

If we replace the formula for X= 600 we got:

$z=\frac{600-500}{200}=0.5$

That means 0.5 deviations above the mean

Part 3

Let X the random variable that represent the cost of for drum set, and for this case we know this  

Where $\mu=850$ and $\sigma=100$

And the best way to solve this problem is using the normal standard distribution and the z score given by:

$z=\frac{x-\mu}{\sigma}$

If we replace the formula for X= 750 we got:

$z=\frac{750-850}{100}=-1$

That means 1 deviations below the mean

Part 4

For this case the lowest instrument compared to the other instruments of the same type is the drum set since have the lowest z score z =-1

Part 5

For this case the highest instrument compared to the other instruments of the same type is guitar since have the highest z score z =0.5