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Chapter 1: Problem 45

Exercise 33 in Section \(1.3\) presented a sample of 26 escape times for oil workers in a simulated escape exercise. Calculate and interpret the sample standard deviation. [Hint: \(\sum x_{i}=9638\) and \(\left.\sum x_{i}^{2}=3,587,566\right]\).

Short Answer

Expert verified
The sample standard deviation is approximately 136.61.

Step by step solution

01

Understand the Formula for Sample Standard Deviation

The formula for the sample standard deviation \( s \) is given by:\[ s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}} \]where \( x_i \) is each individual data point, \( \bar{x} \) is the sample mean, and \( n \) is the number of data points.
02

Calculate the Sample Mean

The sample mean \( \bar{x} \) can be found using the formula:\[ \bar{x} = \frac{\sum x_i}{n} \]Given \( \sum x_i = 9638 \) and \( n = 26 \), calculate \( \bar{x} \).\[ \bar{x} = \frac{9638}{26} \approx 370.69 \]
03

Calculate the Sum of Squared Deviations from Mean

Use the formula for the sum of squared deviations, \( \sum (x_i - \bar{x})^2 \), which can be computed from given data:\[ \sum (x_i - \bar{x})^2 = \sum x_i^2 - \frac{(\sum x_i)^2}{n} \]Substituting the given values:\[ \sum (x_i - \bar{x})^2 = 3,587,566 - \frac{(9638)^2}{26} \approx 461,361.92 \]
04

Compute the Sample Standard Deviation

Substitute \( \sum (x_i - \bar{x})^2 = 461,361.92 \) into the formula for \( s \):\[ s = \sqrt{\frac{461,361.92}{26-1}} = \sqrt{\frac{461,361.92}{25}} \approx 136.61 \]
05

Interpret the Result

The sample standard deviation is approximately \( 136.61 \). This measures the average distance that the escape times deviate from the mean escape time, indicating variability in the time it takes for workers to escape in the simulation.

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Key Concepts

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

Statistical Analysis
Statistical analysis is a powerful tool used to interpret and analyze data. It involves applying different statistical methods to draw meaningful conclusions from data sets.
The objective is to understand the characteristics of the data, uncover patterns, and make informed decisions based on the analysis. One of the main components of statistical analysis is determining how individual data points relate to the overall set, which helps in understanding variability and central tendency.
In the context of this exercise, statistical analysis helps us determine the sample standard deviation, which reflects how much the escape times vary from the average.
Variability
Variability refers to how spread out the data points are within a data set. It is a critical concept as it tells us how much individual items deviate from the average or mean.
High variability indicates that the data points are spread widely apart, while low variability suggests they are close to the mean. This information is crucial for understanding the reliability and predictability of the data.
In the case of the oil workers' escape times, the sample standard deviation tells us about the variability, showing the extent to which the times differ from the mean escape time. A higher standard deviation implies greater variation and less consistency in the escape times.
Sample Mean
The sample mean, often represented as \( \bar{x} \), provides a central value for a data set. It is calculated by summing up all data points and dividing by the number of points.
This gives us an overall picture of the dataset's central tendency.
For the escape times of oil workers, the sample mean is calculated as \( \bar{x} = \frac{9638}{26} \approx 370.69 \). This tells us that, on average, the escape time is approximately \( 370.69 \) seconds.
  • The sample mean helps to establish a baseline for comparing individual data points.
  • Without it, identifying variability and deviations would be more challenging.
Sum of Squared Deviations
The sum of squared deviations is a measure used to calculate the variance within a data set. It involves taking each data point, subtracting the sample mean, then squaring the result.
These squared differences are then summed, offering a clear measure of total deviation in the dataset.
In our example, the formula \( \sum (x_i - \bar{x})^2 = \sum x_i^2 - \frac{(\sum x_i)^2}{n} \) is applied, resulting in a value of approximately \( 461,361.92 \). This is an intermediary step in finding the sample variance and standard deviation.
  • This concept explains how spread out the escape times are around the mean.
  • Understanding this helps in estimating how each individual's escape time compares to the general trend.

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Most popular questions from this chapter

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