91Ó°ÊÓ

The propagation of fatigue cracks in various aircraft parts has been the subject of extensive study in recent years. The accompanying data consists of propagation lives (flight hours \(/ 10^{4}\) ) to reach a given crack size in fastener holes intended for use in military aircraft ("Statistical Crack Propagation in Fastener Holes under Spectrum Loading," J. Aircraft, 1983: 1028-1032): \(\begin{array}{rrrrrrrr}.736 & .863 & .865 & .913 & .915 & .937 & .983 & 1.007 \\\ 1.011 & 1.064 & 1.109 & 1.132 & 1.140 & 1.153 & 1.253 & 1.394\end{array}\) a. Compute and compare the values of the sample mean and median. b. By how much could the largest sample observation be decreased without affecting the value of the median?

Step by step solution

01

Compute the Sample Mean

First, add all the given propagation lives together: \[0.736 + 0.863 + 0.865 + 0.913 + 0.915 + 0.937 + 0.983 + 1.007 + 1.011 + 1.064 + 1.109 + 1.132 + 1.140 + 1.153 + 1.253 + 1.394 = 16.475\]Next, divide the sum by the number of observations, which is 16:\[\text{Mean} = \frac{16.475}{16} = 1.0297\]

02

Compute the Sample Median

To find the median, arrange the data in ascending order: 0.736, 0.863, 0.865, 0.913, 0.915, 0.937, 0.983, 1.007, 1.011, 1.064, 1.109, 1.132, 1.140, 1.153, 1.253, 1.394. Since the number of observations (16) is even, calculate the median by taking the average of the 8th and 9th observations:\[\text{Median} = \frac{1.007 + 1.011}{2} = 1.009\]

03

Determine Maximum Decrease Without Affecting Median

To adjust the largest observation without affecting the median, ensure the median position isn't altered. The median remains at the average of the 8th and 9th values (between 1.007 and 1.011), regardless of how the observed largest value (1.394) is lowered. Thus, the largest observation can be decreased to any number above 1.153 (the next largest value). Therefore, it could be decreased by:\[1.394 - 1.153 = 0.241\]without affecting the median.

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Key Concepts

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

Descriptive Statistics

Descriptive statistics provide a way to summarize and interpret a collection of data points to understand its main features. In statistics, there are several measures to describe data:

• Central Tendency: This includes measures like mean and median, which provide a central value for the data.
• Variability: This tells us about the spread or dispersion of data, including measures like range and standard deviation.
• Shape of Distribution: This involves assessing the symmetry or skewness of data distribution.

Understanding these traits helps in making decisions based on data insights. This exercise focuses on calculating the sample mean and median, which are pivotal components in descriptive statistics.

Data Analysis

Data analysis involves a systematic process of inspecting, cleaning, transforming, and modeling data to discover useful information and support decision-making. In this exercise, we're analyzing propagation lives of aircraft parts, focusing on their centrality - the mean and median.

• Gathering Data: We start by collecting all necessary data points, which are often numerical values in real-world scenarios.
• Organizing Data: Arranging the data in ascending or descending order helps in plotting and calculating more accurately.
• Computing Statistics: Calculating descriptive statistics, like mean and median, aids in summarizing the main characteristics of the data set.

A detailed data analysis like this helps in predicting trends and making informed decisions for future aircraft maintenance.

Median Calculation

Calculating the median involves finding the "middle" value of a data set when it is ordered from smallest to largest. The median is a measure of central tendency that provides insights distinct from the mean.

• Ordering Data: To find the median, first arrange the data values in ascending order.
• Even Number of Observations: As in our exercise, if the number of data points is even, the median is the average of the two middle numbers.
• Importance: The median is useful for understanding skewed distributions or those with outliers, as it is not affected by extreme values.

Knowing how to calculate the median is crucial whenever you need a robust measure of central tendency.

Mean Calculation

The mean, often called the average, is the sum of all data points divided by their number. It provides a measure of central tendency, indicating where most data points lie.

• Adding Data Points: Sum all the values from the data set.
• Dividing by Count: Divide this sum by the total number of data points to find the mean.
• Sensitivity to Outliers: Unlike the median, the mean can be influenced by extremely high or low values, which can skew the result.

Calculating the mean gives a quick insight into the general level of the data but should be considered with other measures to get a full understanding of data distribution.