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Chapter 3: Problem 49

Do the following: a. Compute the sample variance. b. Determine the sample standard deviation. Dave's Automatic Door, referred to in Exercise 37 . installs automatic garage door openers. Based on a sample, following are the times, in minutes, required to install 10 door openers: \(28,32,24,46,44,40,54,38,\) \(32,\) and 42

Short Answer

Expert verified
Sample variance is approximately 82.67, sample standard deviation is approximately 9.09.

Step by step solution

01

Calculate the Mean

To calculate the sample variance and standard deviation, start by finding the mean of the data set. Sum all the data points and divide by the number of data points. The data points are: 28, 32, 24, 46, 44, 40, 54, 38, 32, and 42. \[\text{Mean} = \frac{28 + 32 + 24 + 46 + 44 + 40 + 54 + 38 + 32 + 42}{10} = 38\]
02

Calculate Deviations from the Mean

For each data point, calculate the deviation from the mean computed in Step 1. The deviations are:- \(28 - 38 = -10\)- \(32 - 38 = -6\)- \(24 - 38 = -14\)- \(46 - 38 = 8\)- \(44 - 38 = 6\)- \(40 - 38 = 2\)- \(54 - 38 = 16\)- \(38 - 38 = 0\)- \(32 - 38 = -6\)- \(42 - 38 = 4\)
03

Square the Deviations

Square each of the deviations calculated in Step 2.The squared deviations are:- \((-10)^2 = 100\)- \((-6)^2 = 36\)- \((-14)^2 = 196\)- \(8^2 = 64\)- \(6^2 = 36\)- \(2^2 = 4\)- \(16^2 = 256\)- \(0^2 = 0\)- \((-6)^2 = 36\)- \(4^2 = 16\)
04

Compute the Sample Variance

Add up all the squared deviations from Step 3 and divide by the number of data points minus 1 (due to the sample correction, \(n-1\)).\[\text{Sample Variance} = \frac{100 + 36 + 196 + 64 + 36 + 4 + 256 + 0 + 36 + 16}{9} = \frac{744}{9} \approx 82.67\]
05

Determine the Sample Standard Deviation

The sample standard deviation is the square root of the sample variance computed in Step 4.\[\text{Sample Standard Deviation} = \sqrt{82.67} \approx 9.09\]

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Key Concepts

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

Sample Standard Deviation
The sample standard deviation is a measure that helps us understand the amount of variation or dispersion in a set of data. This measure is particularly useful for summarizing how far away data points are from the mean. To calculate the sample standard deviation:
  • Begin by finding the sample variance.
  • Once you have the sample variance, take the square root to obtain the sample standard deviation.
For example, if we have already calculated a sample variance of approximately 82.67, the sample standard deviation would be the square root of this number, which is roughly 9.09. This value suggests that, on average, the individual data points differ from the mean by 9.09 minutes. The sample standard deviation is commonly used to understand the spread or dispersion of data points in relation to the mean.
Mean Calculation
Calculating the mean of a data set is one of the first steps in statistical analysis. The mean provides a central value around which data points are dispersed. To calculate the mean:
  • Add up all the values in the data set.
  • Divide the sum by the number of data points.
For example, with data points such as 28, 32, 24, 46, 44, 40, 54, 38, 32, and 42, the process is as follows: Sum of values: 28 + 32 + 24 + 46 + 44 + 40 + 54 + 38 + 32 + 42 = 380. Number of data points: 10. Mean: 380/10 = 38. This mean value of 38 minutes serves as a reference point for assessing how close or far each individual data point is from the average installation time.
Data Deviation
Data deviation helps in identifying how much each value in a data set differs from the mean. This step is crucial in various statistical calculations, such as variance and standard deviation. Follow these steps:
  • Calculate the mean of your dataset.
  • Subtract the mean from each data point to find the deviations.
For example, if our mean is 38, deviations would be calculated as follows: - 28 - 38 = -10 - 32 - 38 = -6 - 24 - 38 = -14 - 46 - 38 = 8 - 44 - 38 = 6 - 40 - 38 = 2 - 54 - 38 = 16 - 38 - 38 = 0 - 32 - 38 = -6 - 42 - 38 = 4. These deviations indicate how each individual data point compares to the mean, serving as a basis for further computations like squared deviations.
Squared Deviations
Squared deviations are calculated to remove negative deviations and emphasize larger deviations. This method is essential in calculating variance and involves squaring each individual deviation from the mean:
  • Take each deviation calculated previously.
  • Square this deviation to get the squared deviation.
For instance, given deviations: - (-10)^2 = 100 - (-6)^2 = 36 - (-14)^2 = 196 - 8^2 = 64 - 6^2 = 36 - 2^2 = 4 - 16^2 = 256 - 0^2 = 0 - (-6)^2 = 36 - 4^2 = 16. By squaring, we eliminate negative numbers and set a foundation for variance calculation. Summing all squared values gives us the numerator for the variance calculation, which is later adjusted by dividing by (n-1) for samples.

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Most popular questions from this chapter

In 1996 a total of 14,968,000 taxpayers in the United States filed their individual tax returns electronically. By the year 2006 the number increased to 73,300,000 . What is the geometric mean annual increase for the period?

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