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Chapter 13: Problem 42

An inspection of 12 specimens of material from inside a reactor vessel revealed the following percentages of impurities: $$ \begin{aligned} &2.3,1.9,2.1,2.8,2.3,3.6,1.4,1.8,2.1,3.2 \\ &2,0,1.9 \end{aligned} $$ Find (a) the sample average and both versions of the sample standard deviation, (b) the sample median and range.

Short Answer

Expert verified
The sample average is 2.37%. The sample standard deviation is 0.68, population standard deviation is 0.65. Median is 2.1 and range is 2.2.

Step by step solution

01

Calculate the Sample Average

To find the sample average, sum all the impurity percentages and then divide by the total number of specimens. The data points are: 2.3, 1.9, 2.1, 2.8, 2.3, 3.6, 1.4, 1.8, 2.1, 3.2, 2.0, and 1.9. First, calculate the sum: 2.3 + 1.9 + 2.1 + 2.8 + 2.3 + 3.6 + 1.4 + 1.8 + 2.1 + 3.2 + 2.0 + 1.9 = 28.4. Then, divide the sum by 12 (the number of specimens): \(\bar{x} = \frac{28.4}{12} = 2.37\). Hence, the sample average is 2.37%.
02

Calculate the Sample Standard Deviation (SD)

For the sample standard deviation, use the formula \( s = \sqrt{\frac{1}{n-1} \sum (x_i - \bar{x})^2} \). Here, \(n = 12\) and the sample mean \(\bar{x} = 2.37\). First, find the squared deviations from the mean for each data point, then sum them. After that, divide by \(n-1\), which is 11, and then take the square root. Calculating this gives the sample standard deviation \( s = 0.68 \).
03

Calculate the Population Standard Deviation

In a population context, the standard deviation is calculated using \( \sigma = \sqrt{\frac{1}{n} \sum (x_i - \bar{x})^2} \). Use the sum of squared deviations from the mean obtained in the previous step. This time, divide by \(n = 12\) instead of \(n-1\). The population standard deviation is \( \sigma = 0.65 \).
04

Find the Median

To find the median, organize the data points in ascending order: 1.4, 1.8, 1.9, 1.9, 2.0, 2.1, 2.1, 2.3, 2.3, 2.8, 3.2, 3.6. If there is an even number of observations (here, 12), the median is the average of the two middle numbers. These two numbers are 2.1 and 2.1, so the median is \(\frac{2.1+2.1}{2} = 2.1\).
05

Calculate the Range

The range is the difference between the maximum and minimum values in the dataset. From the ordered data: the maximum value is 3.6 and the minimum is 1.4. Therefore, the range is \(3.6 - 1.4 = 2.2\).

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Key Concepts

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

Sample Average
The sample average, often referred to as the mean, is calculated by adding up all the values in a sample and then dividing by the number of observations. This gives us a measure that represents a "central" value for the dataset. To find the sample average in our example, the impurity percentages were summed up to get 28.4. This total was then divided by 12, the number of specimens. So, the equation for the sample average is \[ \bar{x} = \frac{28.4}{12} = 2.37 \]This tells us that, on average, the percentage of impurities in the specimens is 2.37%. This value gives a good sense of the general impurity level in the dataset.
Sample Standard Deviation
The sample standard deviation is useful to understand how much variation or spread there is from the sample average. Start by calculating the difference between each value and the sample average (2.37%), square these differences, and sum them all up. This sum helps in finding out how varied the data points are by seeing how far each point is from the average squared, eliminating any negative differences.
  • Next, divide by \(n - 1 = 11\), where \(n\) is the number of observations, to ensure an unbiased estimate of the population variance.
  • The final step is to take the square root of the result to bring the units back to the original scale, giving a standard deviation of \(0.68\).
This result helps us see that there is a moderate variation in impurity percentages among the specimens.
Median
The median is a measure of central tendency that divides the dataset into two equal halves. It鈥檚 especially useful when datasets contain outliers that might skew the mean. To find the median, order the data from smallest to largest. With an even number of observations, like our dataset with 12 entries, the median is the average of the two middle numbers.
For the ordered impurities:
  • The two middle values are both 2.1.
Thus, the median is simply 2.1. This indicates that half of the specimens have impurity percentages below this value, fulfilling its role as a robust measure of central tendency.
Range
The range provides a basic sense of spread in a dataset by indicating the difference between the maximum and minimum values. Unlike other measurements which focus on central tendency, the range provides insight into the extent of variation.
To determine the range, simply subtract the smallest value in your dataset from the largest. For our impurity percentages:
  • The highest impurity is 3.6%.
  • The lowest is 1.4%.
  • Thus, the range is \(3.6 - 1.4 = 2.2\).
This range tells us that there is a 2.2% variation between the least and most impure specimens, giving us a measure of variability in the data.

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Most popular questions from this chapter

Find the median and the mode for the Rayleigh distribution $$ f_{X}(x)=\left\\{\begin{array}{cl} \frac{x}{a} \exp \left(-\frac{x^{2}}{2 a}\right) & (x>0) \\ 0 & \text { (otherwise) } \end{array}\right. $$ (see Question 31 in Exercises \(13.4 .5\) ). Also show that the mean is given by $$ \mu_{X}=\int_{0}^{\infty} \exp \left(-\frac{x^{2}}{2 a}\right) \mathrm{d} x $$ which can be shown to be \(\sqrt{\left(\frac{1}{2} \pi a\right) \text {. Compare these }} \end{array}\) quantities when \(a=6\), and find the interquartile range.

Find the distribution of the sum of the numbers when a pair of dice is tossed.

If there are 720 personal computers in an office building and they each break down independently with probability \(0.002\) per working day, use the Poisson approximation to the binomial distribution to find the probability that more than four of these computers will break down in any one working day,

You arrive at a railway station knowing only that trains leave for your destination at intervals of one hour. Find the mean and standard deviation of your waiting time.

A large number \(N\) of people are subjected to a blood investigation to test for the presence of an illegal drug. This investigation is carried out by mixing the blood of \(k\) persons at a time and testing the mixture. If the result of the analysis is negative then this is sufficient for all \(k\) persons. If the result is positive then the blood of each person must be analysed separately, making \(k+1\) analyses in all. Assume that the probability \(p\) of a positive result is the same for each person and that the results of the analyses are independent. Find the expected number of analyses, and minimize with respect to \(k\). In particular, find the optimum value of \(k\) when \(p=0.01\), and the expected saving compared with a separate analysis for all \(N\) people.
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