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Chapter 7: Problem 27

Assume that the mean hourly cost to operate a commercial airplane follows the normal distribution with a mean \(\$ 2,100\) per hour and a standard deviation of \(\$ 250 .\) What is the operating cost for the lowest 3 percent of the airplanes?

Short Answer

Expert verified
The operating cost for the lowest 3% of airplanes is approximately $1630.

Step by step solution

01

Identify the Problem

We need to find the operating cost for the lowest 3% of airplanes from a normally distributed cost with a mean of $2100 and a standard deviation of $250.
02

Understand the Normal Distribution

The problem states that operating costs follow a normal distribution. Thus, we can use properties of the normal distribution to find corresponding costs for specific percentages or percentiles.
03

Utilize Z-score Formula

The Z-score formula is used to find how many standard deviations an element is from the mean. The formula is: \( Z = \frac{X - \mu}{\sigma} \), where \( X \) is the value we are looking for, \( \mu \) is the mean, and \( \sigma \) is the standard deviation.
04

Find Z-score for Lowest 3%

Using a Z-table or calculator, find the Z-score that corresponds to the bottom 3%, which is approximately -1.88 (common value for the 3rd percentile in Z-tables).
05

Solve for X (Operating Cost)

Plug the Z-score back into the Z-score formula and solve for \( X \): \[ -1.88 = \frac{X - 2100}{250} \].
06

Calculate the Operating Cost

Rearrange the equation from the previous step: \( X - 2100 = -1.88 \times 250 \). Then \( X = 2100 - 470 = 1630 \). This means the operating cost for the lowest 3% of the airplanes is approximately \( \$1630 \).

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Key Concepts

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

Z-score
The Z-score is a crucial concept in statistics, especially when dealing with normal distributions. It tells us how many standard deviations an element is from the mean of the data set. Essentially, it's a way of standardizing scores on different scales to a common scale.
  • Why Use Z-scores? They help in understanding and comparing data from different normal distributions by bringing everything to a standard scale. Thus, comparisons become easier.
  • Z-score Formula: To calculate the Z-score, the formula is \( Z = \frac{X - \mu}{\sigma} \). Here, \(X\) is the value we want to analyze, \(\mu\) is the mean, and \(\sigma\) is the standard deviation.
For instance, in our problem, we want to find the operating costs that fall into the lowest 3%. By using the Z-score, we discover that this requires finding which value corresponds to a Z-score of approximately -1.88.
Percentile
A percentile provides a clear picture of where a particular value falls within a distribution. It describes the percentage of the data set that exists below a certain point.
  • Understanding Percentiles: If a value is in the 3rd percentile, this means that it is higher than 3% of all other values in the distribution, or conversely, 97% of the values exceed it.
  • Role in Our Example: To find the operating costs of the lowest 3% of airplanes, we used a percentile to understand which costs are typical for this proportion of the airplanes.
Percentiles are intuitive for grasping how an individual value compares against an overall distribution, especially useful in educational and scientific applications.
Standard Deviation
Standard deviation is a measure of the amount of variation or dispersion in a set of values. The lower the standard deviation, the closer the data points tend to be to the mean.
  • Importance: In normal distributions, standard deviation determines the spread of the data. It helps identify how much the typical data point differs from the mean.
  • Link to the Problem: In this exercise, a standard deviation of $250 means that most operating costs fall within $250 of the mean of $2100. This spread helps us gauge where a certain cost lies in relation to the mean value.
In conclusion, standard deviation is vital for interpreting data, especially when making inferences about a population from a sample, as seen in our problem.

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Most popular questions from this chapter

The amounts of money requested on home loan applications at Down River Federal Savings follow the normal distribution, with a mean of \(\$ 70,000\) and a standard deviation of \(\$ 20,000 .\) A loan application is received this morning. What is the probability: a. The amount requested is \(\$ 80,000\) or more? b. The amount requested is between \(\$ 65,000\) and \(\$ 80,000 ?\) c. The amount requested is \(\$ 65,000\) or more?

A normal distribution has a mean of 50 and a standard deviation of 4 a. Compute the probability of a value between 44.0 and 55.0 . b. Compute the probability of a value greater than 55.0 . c. Compute the probability of a value between 52.0 and 55.0 .

A recent article in the Cincinnati Enquirer reported that the mean labor cost to repair a heat pump is \(\$ 90\) with a standard deviation of \(\$ 22 .\) Monte's Plumbing and Heating Service completed repairs on two heat pumps this morning. The labor cost for the first was \(\$ 75\) and it was \(\$ 100\) for the second. Compute \(z\) values for each and comment on your findings.

A normal distribution has a mean of 50 and a standard deviation of \(4 .\) Determine the value below which 95 percent of the observations will occur.

WNAE, an all-news AM station, finds that the distribution of the lengths of time listeners are tuned to the station follows the normal distribution. The mean of the distribution is 15.0 minutes and the standard deviation is 3.5 minutes. What is the probability that a particular listener will tune in: a. More than 20 minutes? b. For 20 minutes or less? c. Between 10 and 12 minutes?
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