Question

A music school has budgeted to purchase three musical instruments. They plan to purchase a piano costing 3,000, a guitar costing 550, and a drum set costing 600. The mean cost for a piano is 4,000 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 700 with a standard deviation of 100. Which cost is the lowest, when compared to other instruments of the same type? Which cost is the highest when compared to other instruments of the same type. Justify your answer.

Short answer

The cost of a piano is 0.4 standard deviations lower than the mean. The cost of a guitar is 0.25 standard deviations higher than the mean. The cost of a drum set is 1.0 standard deviation below the mean for drums. Drums are the least expensive of the three when compared to other instruments of the same type. In comparison to other instruments of the same type, the guitar is the most expensive.

Step-by-step solution

To find

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

Explanation

The mean is one of the measurements of central tendency in statistics. It's calculated by dividing the total number of observations by the sum of the observations. One of the measurements of dispersion is the standard deviation, which is used to show how much individuals in a group differ from the mean.

The z-score can be used to compare a student's performance when different types of scores are being considered.

The Z-score indicates how far a value is above or below the mean in terms of standard deviation. The Z-score is calculated using the following formula:

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

Where,

localid="1649446851549" $\mathit{x}=$is any data value

$\mu =$is the population mean

$\sigma =$is the population standard deviation

In the given example, we are given the below information:

Instrument	cost	Average cost	Standard deviation
Piano	$3000$	$4000$	$2500$
Guitar	$550$	$500$	$200$
Drum	$600$	$700$	$100$

Now find the Z- score for the each of the above instrument.

The z-score for Piano is:

$z=\frac{3000-4000}{2500}=-0.4$

Hence, the cost of Piano is $-0.4$standard deviations below the mean.

The z-score for Guitar is:

$z=\frac{550-500}{200}=0.25$

Hence, the cost of Guitar is 0.25 standard deviations above the mean.

The z-score for drum is:

$z=\frac{600-700}{100}=-1$

Hence, the cost of drum is 1 standard deviation below the mean.

In comparison to the cost of other instruments of the same type, the drums are the cheapest of the three. In compared to other instruments of similar sort, the guitar is the most expensive.