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Chapter 1: Problem 69

Two students determine the percentage of lead in a sample as a laboratory exercise. The true percentage is \(22.52 \%\). The students' results for three determinations are as follows: (1) 22.52,22.48,22.54 (2) 22.64,22.58,22.62 (a) Calculate the average percentage for each set of data and state which set is the more accurate based on the average. (b) Precision can be judged by examining the average of the deviations from the average value for that data set. (Calculate the average value for each data set; then calculate the average value of the absolute deviations of each measurement from the average.) Which set is more precise?

Short Answer

Expert verified
The more accurate data set is (1), with an average percentage of 22.5133, which is closer to the true percentage of 22.52%. Both data sets have an average absolute deviation of 0.0222, so they are equally precise.

Step by step solution

01

Calculate the average percentage for each data set

We will find the average percentage of each data set by summing the values and dividing by the number of values (3 in both cases). For data set (1): \(\frac{22.52+22.48+22.54}{3}=22.5133\) For data set (2): \(\frac{22.64+22.58+22.62}{3}=22.6133\)
02

Determine which set is more accurate

To determine which set is more accurate, we need to compare the average percentages to the true percentage (22.52%). The closer the average value to the true value, the more accurate the data set. Data set (1) average percentage: 22.5133 Data set (2) average percentage: 22.6133 Data set (1) is closer to the true percentage of 22.52%, so it is more accurate.
03

Calculate the average absolute deviation for each data set

We will now calculate the average absolute deviation to determine which set is more precise. First, find the absolute deviation of each value from its data set's average value. Then, find the average of these absolute deviations. For data set (1): - |22.52 - 22.5133| = 0.0067 - |22.48 - 22.5133| = 0.0333 - |22.54 - 22.5133| = 0.0267 The average absolute deviation for data set (1): \(\frac{0.0067+0.0333+0.0267}{3} = 0.0222\) For data set (2): - |22.64 - 22.6133| = 0.0267 - |22.58 - 22.6133| = 0.0333 - |22.62 - 22.6133| = 0.0067 The average absolute deviation for data set (2): \(\frac{0.0267+0.0333+0.0067}{3} = 0.0222\)
04

Determine which set is more precise

To determine which set is more precise, we need to compare the average absolute deviations. The lower the average absolute deviation, the more precise the data set. Data set (1) average absolute deviation: 0.0222 Data set (2) average absolute deviation: 0.0222 Both data sets have the same average absolute deviation, so they are equally precise.

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