/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 104 A music school has budgeted to p... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 104

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

Expert verified
The drum set cost is the lowest, and the guitar cost is the highest compared to their average prices.

Step by step solution

01

Calculate Z-score for Piano

To determine how the price of the piano compares with the average price, we calculate the Z-score using the formula: \( Z = \frac{X - \mu}{\sigma} \), where \( X \) is the piano's cost, \( \mu \) is the mean price for a piano, and \( \sigma \) is the standard deviation.Here, \( X = 3000 \), \( \mu = 4000 \), \( \sigma = 2500 \).\[ Z_{piano} = \frac{3000 - 4000}{2500} = \frac{-1000}{2500} = -0.4 \]
02

Calculate Z-score for Guitar

Next, calculate the Z-score for the guitar using the same formula: \( Z = \frac{X - \mu}{\sigma} \).For the guitar, \( X = 550 \), \( \mu = 500 \), \( \sigma = 200 \).\[ Z_{guitar} = \frac{550 - 500}{200} = \frac{50}{200} = 0.25 \]
03

Calculate Z-score for Drums

Finally, calculate the Z-score for the drum set:For the drum set, \( X = 600 \), \( \mu = 700 \), \( \sigma = 100 \).\[ Z_{drum} = \frac{600 - 700}{100} = \frac{-100}{100} = -1 \]
04

Analyze Z-scores

The Z-scores indicate how many standard deviations the price of each instrument is from the average. A lower Z-score indicates a relatively lower price compared to the average.- For the piano, \( Z = -0.4 \).- For the guitar, \( Z = 0.25 \).- For the drums, \( Z = -1 \).The drum set has the lowest Z-score, meaning it is the most underspending cost compared to other drum sets. The guitar has the highest Z-score, meaning it is priced above the average guitar cost.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Musical Instrument Costs
When budgeting for musical instruments, it鈥檚 important to compare the planned purchase costs against the typical costs using Z-score analysis.
The music school is considering three instruments: a piano, a guitar, and a drum set.
  • The piano costs $3,000.
  • The guitar is priced at $550.
  • The drum set costs $600.
By examining these costs in relation to typical prices, you can determine how competitively priced each instrument is.
Z-scores are used to evaluate how much an instrument's cost deviates from the average, allowing the school to assess if a purchase is a good deal or not.
Standard Deviation
Standard deviation is a statistical concept that helps us understand variability in data.
In the context of musical instruments, it provides insight into how much the prices typically vary from the average price within the market for each type of instrument.

The standard deviations provided are:
  • Piano: $2,500
  • Guitar: $200
  • Drums: $100
A higher standard deviation suggests more variability in prices. The piano has the highest standard deviation at $2,500, indicating that prices for pianos vary widely, while the drum set has the lowest, meaning prices are more consistent.
Mean Comparison
Comparing the purchase price of an instrument to its mean or average market price gives a better understanding of the value being offered.
The mean prices for the instruments are:
  • Piano: $4,000
  • Guitar: $500
  • Drums: $700
The mean acts as a benchmark to determine if the instrument is being purchased above or below the typical market rate.
For the piano priced at $3,000, which is less than its mean, this implies that purchases are economically favorable unless standard deviation indicates otherwise. Conversely, a guitar costing more than its mean suggests a higher-than-average purchase price, alerting potential overpayment.
Piano, Guitar, Drum Set Costs
To evaluate which instrument is priced most competitively, we analyze Z-scores for each purchase:
  • Piano: Z-score of -0.4
  • Guitar: Z-score of 0.25
  • Drums: Z-score of -1
The Z-score formula \[ Z = \frac{X - \mu}{\sigma} \] indicates how many standard deviations the purchase price is from the mean.
A negative Z-score indicates a price below the mean, highlighting potential savings:
  • The drum set (Z = -1) represents the best deal, being the furthest below the mean price of similar instruments.
  • The piano also offers savings with its negative Z-score, although not as pronounced as the drums.
  • The guitar's positive Z-score highlights it as over its typical market price, suggesting a cautious approach to its purchase.

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Most popular questions from this chapter

Following are the published weights (in pounds) of all of the team members of the San Francisco 49ers from a previous year. 177; 205; 210; 210; 232; 205; 185; 185; 178; 210; 206; 212; 184; 174; 185; 242; 188; 212; 215; 247; 241; 223; 220; 260; 245; 259; 278; 270; 280; 295; 275; 285; 290; 272; 273; 280; 285; 286; 200; 215; 185; 230; 250; 241; 190; 260; 250; 302; 265; 290; 276; 228; 265 a. Organize the data from smallest to largest value. b. Find the median. c. Find the first quartile. d. Find the third quartile. e. Construct a box plot of the data. f. The middle 50\(\%\) of the weights are from ____ to ____ . g. If our population were all professional football players, would the above data be a sample of weights or the population of weights? Why? h. If our population included every team member who ever played for the San Francisco 49ers, would the above data be a sample of weights or the population of weights? Why? i. Assume the population was the San Francisco 49ers. Find: $$\begin{array}{l}{\text { i. the population mean, } \mu} \\ {\text { ii. the population standard deviation, } \sigma \text { . }} \\ {\text { iii. the weight that is two standard deviations below the mean. }} \\ {\text { iv. When Steve Young, quarterback, played football, he weighed 205 pounds. How many standard }} \\ {\text { deviations above or below the mean was he? }}\end{array}$$ j. That same year, the mean weight for the Dallas Cowboys was 240.08 pounds with a standard deviation of 44.38 pounds. Emmit Smith weighed in at 209 pounds. With respect to his team, who was lighter, Smith or Young? How did you determine your answer?

Use the following information to answer the next three exercises: The following data show the lengths of boats moored in a marina. The data are ordered from smallest to largest: 16; 17; 19; 20; 20; 21; 23; 24; 25; 25; 25; 26; 26; 27; 27; 27; 28; 29; 30; 32; 33; 33; 34; 35; 37; 39; 40 Identify the mode.

Use the following information to answer the next two exercises. X = the number of days per week that 100 clients use a particular exercise facility. $$\begin{array}{|l|l|}\hline x & {\text { Frequency }} \\ \hline 0 & {3} \\\ \hline 1 & {12} \\ \hline 2 & {33} \\ \hline 3 & {28} \\ \hline 4 & {11} \\\ \hline 5 & {9} \\ \hline 6 & {4} \\ \hline\end{array}$$ The \(80^{\text { th }}\) percentile is ____ a. 5 b. 80 c. 3 d. 4

Which is the greatest, the mean, the mode, or the median of the data set? 11; 11; 12; 12; 12; 12; 13; 15; 17; 22; 22; 22

On an exam, would it be more desirable to earn a grade with a high or low percentile? Explain.
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